Mercurial > hg > mpdl-xml-content
view texts/XML/archimedes/la/newto_philo_039_la_1713.xml @ 10:d7b79f6537bb
Version vom 2009-02-14
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:08:12 +0200 |
| parents | 22d6a63640c6 |
| children |
line wrap: on
line source
<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> <author>Newton, Isaac</author> <title>Philosophia naturalis principia mathematica</title> <date>1713</date> <place>Cambridge</place> <translator></translator> <lang>la</lang> <cvs_file>newto_philo_039_la_1713.xml</cvs_file> <cvs_version></cvs_version> <locator>039.xml</locator> </info> <text> <front> </front> <body> <chap> <pb xlink:href="039/01/001.jpg"></pb> <p type="main"> <s><emph type="center"></emph>PHILOSOPHIÆ <lb></lb>NATURALIS <lb></lb>PRINCIPIA <lb></lb>MATHEMATICA.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>AUCTORE <lb></lb>ISAACO NEWTONO, <lb></lb>EQUITE A RATO.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>EDITIO SECUNDA AUCTIOR ET EMENDATIOR.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>CANTABRIGIÆ, MDCCXIII.<emph.end type="center"></emph.end></s></p><pb xlink:href="039/01/002.jpg"></pb><pb xlink:href="039/01/003.jpg"></pb> <p type="main"> <s><emph type="center"></emph>ILLUSTRISSIMÆ <lb></lb>SOCIETATI REGALI, <lb></lb>A <lb></lb>SERENISSIMO REGE <lb></lb>CAROLO II <lb></lb>AD PHILOSOPHIAM PROMOVENDAM <lb></lb>FUNDATÆ, <lb></lb>ET <lb></lb>AUSPICIIS <lb></lb>AUGUSTISSIMÆ REGINÆ <lb></lb>ANNÆ <lb></lb>FLORENTI,<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>TRACTATUM HUNC D.D.D.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>JS. NEWTONUS.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/004.jpg"></pb></chap><chap><pb xlink:href="039/01/005.jpg"></pb> <p type="main"> <s><emph type="center"></emph>IN <lb></lb>VIRI PRÆSTANTISSIMI <lb></lb>ISAACI NEWTONI <lb></lb>OPUS HOCCE <lb></lb>MATHEMATICO PHYSICUM<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Sæculi Gentiſque noſtræ Decus egregium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>EN tibi norma Poli, & divæ libramina Molis, <lb></lb>Computus en Jovis; & quas, dum primordia rerum. </s> <s><lb></lb>Conderet, omnipotens ſibi Leges ipſe Creator <lb></lb>Dixerit, atque operum quæ fundamenta locarit. </s> <s><lb></lb>Intima panduntur victi penetralia Cæli, <lb></lb>Nec latet, extremos quæ Vis circumrotet Orbes. </s> <s><lb></lb>Sol ſolio reſidens ad ſe jubet omnia prono <lb></lb>Tendere deſcenſu, nec recto tramite currus <lb></lb>Sidereos patitur vaſtum per inane moveri; <lb></lb>Sed rapit immotis, ſe centro, ſingula gyris. </s> <s><lb></lb>Hinc patet, horrificis qua ſit via flexa Cometis: <lb></lb>Diſcimus hinc tandem, qua cauſa argentea Phœbe <lb></lb>Paſſibus haud æquis eat, & cur ſubdita nulli <lb></lb>Hactenus Aſtronomo numerorum fræna recuſet: <lb></lb>Cur remeent Nodi, curque Auges progrediantur. </s> <s><lb></lb>Diſcimus, & quantis refluum vaga Cynthia Pontum <lb></lb>Viribus impellat; feſſis dum fluctibus ulvam <lb></lb>Deſerit, ac nautis ſuſpectas nudat arenas; <lb></lb>Alterniſve ruens ſpumantia littora pulſat. <pb xlink:href="039/01/006.jpg"></pb>Quæ toties animos veterum torſere Sophorum, <lb></lb>Quæque Scholas hodie rauco certamine vexant, <lb></lb>Obvia conſpicimus; nubem pellente Matheſi: <lb></lb>Quæ ſuperas penetrare domos, atque ardua Cæli, <lb></lb>NEWTONI auſpiciis, jam dat contingere Templa. </s> <s><lb></lb>Surgite Mortales, terrenas mittite curas; <lb></lb>Atque hinc cæligenæ vites cognoſcite Mentis, <lb></lb>A pecudum vita longe longeque remotæ. </s> <s><lb></lb>Qui ſcriptis primus Tabulis compeſcere Cædes, <lb></lb>Furta & Adulteria, & perjuræ crimina Fraudis; <lb></lb>Quive vagis populis circumdare mœnibus Urbes <lb></lb>Auctor erat; Cereriſve beavit munere gentes; <lb></lb>Vel qui curarum lenimen preſſit ab Uva; <lb></lb>Vel qui Niliaca monſtravit arundine pictos <lb></lb>Conſociare ſonos, oculiſque exponere Voces; <lb></lb>Humanam ſortem minus extulit; utpote pauca <lb></lb>In commune ferens miſeræ ſolatia vitæ. </s> <s><lb></lb>Jam vero Superis convivæ admittimur, alti <lb></lb>Jura poli tractare licet, jamque abdita diæ <lb></lb>Clauſtra patent Naturæ, & rerum immobilis ordo; <lb></lb>Et quæ præteritis latuere incognita ſæclis. </s> <s><lb></lb>Talia monſtrantem juſtis celebrate Camænis, <lb></lb>Vos qui cæleſti gaudetis nectare veſci, <lb></lb>NEWTONUM clauſi reſerantem ſcrinia Veri, <lb></lb>NEWTONUM Muſis carum, cui pectore puro <lb></lb>Phœbus adeſt, totoQ.E.I.ceſſit Numine mentem: <lb></lb>Nec fas eſt propius Mortali attingere Divos. <lb></lb><emph type="italics"></emph>EDM. HALLET.<emph.end type="italics"></emph.end></s></p></chap><chap><pb xlink:href="039/01/007.jpg"></pb> <p type="main"> <s><emph type="center"></emph>AUCTORIS <lb></lb>PRÆFATIO <lb></lb>AD <lb></lb>LECTOREM.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>CUM Veteres<emph.end type="italics"></emph.end>Mechanicam (<emph type="italics"></emph>uti Auctor eſt<emph.end type="italics"></emph.end>Pappus) <emph type="italics"></emph>in rerum <lb></lb>Naturalium inveſtigatione maximi fecerint; & Recentiores, <lb></lb>miſſis formis ſubſtantialibus & qualitatibus occultis, Phænomena <lb></lb>Naturæ ad leges Mathematicas revocare aggreſſi fint: Viſum eſt <lb></lb>in hoc Tractatu<emph.end type="italics"></emph.end>Matheſin <emph type="italics"></emph>excolere, quatenus ea ad<emph.end type="italics"></emph.end>Philoſophiam <lb></lb><emph type="italics"></emph>ſpectat.<emph.end type="italics"></emph.end>Mechanicam <emph type="italics"></emph>vero duplicem Veteres conſtituerunt<emph.end type="italics"></emph.end>: Ra<lb></lb>tionalem <emph type="italics"></emph>quæ per Demonſtrationes accurate procedit, &<emph.end type="italics"></emph.end>Practi<lb></lb>cam. <emph type="italics"></emph>Ad Practicam ſpectant Artes omnes Manuales, a quibus <lb></lb>utique<emph.end type="italics"></emph.end>Mechanica <emph type="italics"></emph>nomen mutuata eſt. </s> <s>Cum autem Artifices pa<lb></lb>rum accurate operari ſoleant, fit ut<emph.end type="italics"></emph.end>Mechanica <emph type="italics"></emph>omnis a<emph.end type="italics"></emph.end>Geome<lb></lb>tria <emph type="italics"></emph>ita diſtinguatur, ut quicquid accuratum ſit ad<emph.end type="italics"></emph.end>Geometriam <lb></lb><emph type="italics"></emph>referatur, quicquid minus accuratum ad<emph.end type="italics"></emph.end>Mechanicam. <emph type="italics"></emph>Attamen <lb></lb>errores non ſunt Artis ſed Artificum. </s> <s>Qui minus accurate ope<lb></lb>ratur, imperfectior eſt Mechanicus, & ſi quis accuratiſſime ope<lb></lb>rari poſſet, hic foret Mechanicus omnium perfectiſſimus. </s> <s>Nam & <lb></lb>Linearum rectarum & Circulorum deſcriptiones in quibus<emph.end type="italics"></emph.end>Geo<lb></lb>metria <emph type="italics"></emph>fundatur, ad<emph.end type="italics"></emph.end>Mechanicam <emph type="italics"></emph>pertinent. </s> <s>Has lineas deſcri<lb></lb>bere<emph.end type="italics"></emph.end>Geometria <emph type="italics"></emph>non docet ſed poſtulat. </s> <s>Poſtulat enim ut Tyro <lb></lb>eaſdem accurate deſcribere prius didicerit quam linen attingat<emph.end type="italics"></emph.end><lb></lb>Geometriæ; <emph type="italics"></emph>dein, quomodo per has operationes Problemata ſol<lb></lb>uantur, docet. </s> <s>Rectas & Circulos deſcribere Problemata ſunt,<emph.end type="italics"></emph.end><pb xlink:href="039/01/008.jpg"></pb><emph type="italics"></emph>ſed non Geometrica. </s> <s>Ex<emph.end type="italics"></emph.end>Mechanica <emph type="italics"></emph>poſtulatur horum ſolutio, in<emph.end type="italics"></emph.end><lb></lb>Geometria <emph type="italics"></emph>docetur ſolutorum uſus. </s> <s>Ac gloriatur<emph.end type="italics"></emph.end>Geometria <lb></lb><emph type="italics"></emph>quod tam paucis principiis aliunde petitis tam multa præſtet. </s> <s>Fun<lb></lb>datur igitur<emph.end type="italics"></emph.end>Geometria <emph type="italics"></emph>in praxi Mechanica, & nihil aliud eſt <lb></lb>quam<emph.end type="italics"></emph.end>Mechanicæ univerſalis <emph type="italics"></emph>pars illa quæ artem menſurandi ac<lb></lb>curate proponit ac demonſtrat. </s> <s>Cum autem artes Manuales in <lb></lb>corporibus movendis præcipue verſentur, fit ut<emph.end type="italics"></emph.end>Geometria <emph type="italics"></emph>ad mag<lb></lb>nitudinem,<emph.end type="italics"></emph.end>Mechanica <emph type="italics"></emph>ad motum vulgo referatur. </s> <s>Quo ſenſu<emph.end type="italics"></emph.end>Me<lb></lb>chanica rationalis <emph type="italics"></emph>erit Scientia Motuum qui ex viribus quibuſ<lb></lb>cunque reſultant, & Virium quæ ad motus quoſcunque requirun<lb></lb>tur, accurate propoſita ac demonſtrata. </s> <s>Pars hæc<emph.end type="italics"></emph.end>Mechanicæ <emph type="italics"></emph>a <lb></lb>Veteribus in<emph.end type="italics"></emph.end>Potentiis quinque <emph type="italics"></emph>ad artes manuales ſpectantibus <lb></lb>exculta fuit, qui Gravitatem (cum potentia manualis non ſit) vix <lb></lb>aliter quam in ponderibus per potentias illas movendis conſiderarunt. </s> <s><lb></lb>Nos autem non Artibus ſed Philoſophiæ conſulentes, deque poten<lb></lb>tiis non manualibus ſed naturalibus ſcribentes, ea maxime tracta<lb></lb>mus quæ ad Gravitatem, Levitatem, vim Elaſticam, reſiſtentiam <lb></lb>Fluidorum & ejuſmodi vires ſeu attractivas ſeu impulſivas ſpe<lb></lb>ctant: Et ea propter, hæc noſtra tanquam Philoſophiæ principia <lb></lb>Mathematica proponimus. </s> <s>Omnis enim Philoſophiæ difficultas in <lb></lb>eo verſari videtur, ut a Phænomenis motuum inveſtigemus vires <lb></lb>Naturæ, deinde ab his viribus demonſtremus phænomena reliqua. </s> <s><lb></lb>Et huc ſpectant Propoſitiones generales quas Libro primo & ſecundo <lb></lb>pertractavimus. </s> <s>In Libro autem tertio Exemplum hujus rei propo<lb></lb>ſuimus per explicationem Syſtematis mundani. </s> <s>Ibi enim, ex phæ<lb></lb>nomenis cæleſtibus, per Propoſitiones in Libris prioribus Mathe<lb></lb>matice demonſtratas, derivantur vires Gravitatis quibus corpora <lb></lb>ad Solem & Planetas ſingulos tendunt. </s> <s>Deinde ex his viribus <lb></lb>per Propoſitiones etiam Mathematicas, deducuntur motus Planeta<lb></lb>rum, Cometarum, Lunæ & Maris. </s> <s>Utinam cætera Naturæ phæ<lb></lb>nomena ex principiis Mechanicis eodem argumentandi genere deri<lb></lb>vare liceret. </s> <s>Nam multa me movent ut nonnihil ſuſpicer ea om<emph.end type="italics"></emph.end><pb xlink:href="039/01/009.jpg"></pb><emph type="italics"></emph>nia ex viribus quibuſdam pendere poſſe, quibus corporum particulæ <lb></lb>per cauſas nondum cognitas vel in ſe mutuo impelluntur & ſe<lb></lb>cundum figuras regulares cohærent, vel ab invicem fugantur & <lb></lb>recedunt: quibus viribus ignotis, Philoſophi hactenus Naturam fru<lb></lb>ſtra tentarunt. </s> <s>Spero autem quod vel huic Philoſophandi modo, <lb></lb>vel veriori alicui, Principia hic poſita lucem aliquam præbebunt.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>In his edendis, Vir acutiſſimus & in omni literarum genere <lb></lb>eruditiſſimus<emph.end type="italics"></emph.end>Edmundus Halleius <emph type="italics"></emph>operam navavit, nec ſolum <lb></lb>Typothetarum Sphalmata correxit & Schemata incidi curavit, ſed <lb></lb>etiam Auctor fuit ut horum editionem aggrederer. </s> <s>Quippe cum <lb></lb>demonſtratam a me Figuram Orbium cæleſtium impetraverat, ro<lb></lb>gare non deſtitit ut eandem cum<emph.end type="italics"></emph.end>Societate Regali <emph type="italics"></emph>communicarem, <lb></lb>Quæ deinde hortatibus & benignis ſuis auſpiciis effecit ut de ea<lb></lb>dem in lucem emittenda cogitare inciperem. </s> <s>At poſtquam Mo<lb></lb>tuum Lunarium inæqualitates aggreſſus eſſem, deinde etiam ælia <lb></lb>tentare cæpiſſem quæ ad leges & menſuras Gravitatis & aliarum <lb></lb>virium, & Figuras a corporibus ſecundum datas quaſcunque leges <lb></lb>attractis deſcribendas, ad motus corporum plurium inter ſe, ad <lb></lb>motus corporum in Mediis reſiſtentibus, ad vires, denſitates & <lb></lb>motus Mediorum, ad Orbes Cometarum & ſimilia ſpectant, edi<lb></lb>tionem in aliud tempus differendam eſſe putavi, ut cætera rima<lb></lb>rer & una in publicum darem. </s> <s>Quæ ad motus Lunares ſpectant, <lb></lb>(imperfecta cum ſint,) in Corollariis Propoſitionis<emph.end type="italics"></emph.end>LXVI. <emph type="italics"></emph>ſimul <lb></lb>complexus ſum, ne ſingula methodo prolixiore quam pro rei digNI<lb></lb>tate proponere, & ſigillatim demonſtrare tenerer, & ſeriem reli<lb></lb>quarum Propoſitionum interrumpere. </s> <s>Nonnulla ſero inventa lo<lb></lb>cis minus idoneis inſerere malui, quam numerum Propoſitionum <lb></lb>& citationes mutare. </s> <s>Ut omnia candide legantur, & defectus, <lb></lb>in materia tam difficili non tam reprehendantur, quam novis Le<lb></lb>ctorum conatibus inveſtigentur, & benigne ſuppleantur, enixe rogo.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Dabam <emph type="italics"></emph>Cantabrigiæ,<emph.end type="italics"></emph.end>e Collegio <lb></lb><emph type="italics"></emph>S. Trinitatis,<emph.end type="italics"></emph.end>Maii 8. 1686. </s></p> <p type="main"> <s><emph type="italics"></emph>IS. NEWTON.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/010.jpg"></pb> <p type="main"> <s><emph type="italics"></emph>IN hac Secunda Principiorum Editione, multa ſparſim emen<lb></lb>dantur & nonnulla adjiciuntur. </s> <s>In Libri primi Sectione<emph.end type="italics"></emph.end>II, <lb></lb><emph type="italics"></emph>Inventio virium quibus corpora in Orbibus datis revolvi poſſint, <lb></lb>facilior redditur & amplior. </s> <s>In Libri ſecundi Sectione<emph.end type="italics"></emph.end>VII, <lb></lb><emph type="italics"></emph>Theoria reſiſtentiæ Fluidorum accuratius inveſtigatur & novis <lb></lb>Experimentis confirmatur. </s> <s>In Libro tertio Theoria Lunæ & Præ<lb></lb>ceſſio Æquinoctiorum ex Principiis ſuis plenius deducuntur, & <lb></lb>Theoria Cometarum pluribus & accuratius computatis Orbium <lb></lb>exemplis confirmatur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Dabam <emph type="italics"></emph>Londini,<emph.end type="italics"></emph.end><lb></lb>Mar. </s> <s>28. 1713. </s></p> <p type="main"> <s><emph type="italics"></emph>IS. NEWTON.<emph.end type="italics"></emph.end></s></p></chap><chap><pb xlink:href="039/01/011.jpg"></pb> <p type="main"> <s><emph type="center"></emph>EDITORIS <lb></lb>PRÆFATIO.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>NEWTONIANÆ Philoſophiæ novam tibi, Lector benevole, <lb></lb>diuQ.E.D.ſideratam Editionem, plurimum nunc emenda<lb></lb>tam atque auctiorem exhibemus. </s> <s>Quæ potiſſimum conti<lb></lb>neantur in hoc Opere celeberrimo, intelligere potes ex Indicibus <lb></lb>adjectis: quæ vel addantur vel immutentur, ipſa te fere docebit <lb></lb>Auctoris Præfatio. </s> <s>Reliquum eſt, ut adjiciantur nonnulla de Me<lb></lb>thodo hujus Philoſophiæ. </s></p> <p type="main"> <s>Qui Phyſicam tractandam ſuſceperunt, ad tres fere claſſes re<lb></lb>vocari poſſunt. </s> <s>Extiterunt enim, qui ſingulis rerum ſpeciebus Quali<lb></lb>tates ſpecificas & occultas tribuerint; ex quibus deinde corporum <lb></lb>ſingulorum operationes, ignota quadam ratione, pendere volue<lb></lb>runt. </s> <s>In hoc poſita eſt ſumma doctrinæ Scholaſticæ, ab <emph type="italics"></emph>Ariſtotele<emph.end type="italics"></emph.end><lb></lb>& Peripateticis derivatæ: Affirmant utique ſingulos Effectus ex <lb></lb>corporum ſingularibus Naturis oriri; at unde ſint illæ Naturæ <lb></lb>non docent; nihil itaQ.E.D.cent. </s> <s>Cumque toti ſint in rerum no<lb></lb>minibus, non in ipſis rebus; Sermonem Q.E.D.m Philoſophicum <lb></lb>cenſendi ſunt adinveniſſe, Philoſophiam tradidiſſe non ſunt cen<lb></lb>ſendi. </s></p> <p type="main"> <s>Alii ergo melioris diligentiæ laudem conſequi ſperarunt, rejecta <lb></lb>Vocabulorum inutili farragine. </s> <s>Statuerunt itaque Materiam uNI<lb></lb>verſam homogeneam eſſe, omnem vero Formarum varietatem, quæ <lb></lb>in corporibus cernitur, ex particularum componentium ſimpliciſſi<lb></lb>mis quibuſdam & intellectu facillimis affectionibus oriri. </s> <s>Et recte <lb></lb>quidem progreſſio inſtituitur a ſimplicioribus ad magis compoſita, <lb></lb>ſi particularum primariis illis affectionibus non alios tribuunt mo<lb></lb>dos, quam quos ipſa tribuit Natura. </s> <s>Verum ubi licentiam ſibi <lb></lb>aſſumunt, ponendi quaſcunque libet ignotas partium figuras & <lb></lb>magnitudines, incertoſque ſitus & motus; quin & fingendi Fluida <lb></lb>quædam occulta, quæ corporum poros liberrime permeent, omNI<lb></lb>potente prædita ſubtilitate, motibuſque occultis agitata; jam ad <lb></lb>ſomnia delabuntur, neglecta rerum conſtitutione vera: quæ fane <lb></lb>fruſtra petenda eſt ex fallacibus conjecturis, cum vix etiam per <lb></lb>certiſſimas Obſervationes inveſtigari poſſit. </s> <s>Qui ſpeculationum <pb xlink:href="039/01/012.jpg"></pb>ſuarum fundamentum deſumunt ab Hypotheſibus, etiamſi deinde <lb></lb>ſecundum leges Mechanicas accuratiſſime procedant; Fabulam qui<lb></lb>dem elegantem forte & venuſtam, Fabulam tamen concinnare di<lb></lb>cendi ſunt. </s></p> <p type="main"> <s>Relinquitur adeo tertium genus, qui Philoſophiam ſcilicet Ex<lb></lb>perimentalem profitentur. </s> <s>Hi quidem ex ſimpliciſſimis quibus <lb></lb>poſſunt principiis rerum omnium cauſas derivandas eſſe volunt: <lb></lb>nihil autem Principii loco aſſumunt, quod nondum ex Phænome<lb></lb>nis comprobatum fuerit. </s> <s>Hypotheſes non comminiſcuntur, neque <lb></lb>in Phyſicam recipiunt, niſi ut Quæſtiones de quarum veritate diſ<lb></lb>putetur. </s> <s>Duplici itaque Methodo incedunt, Analytica & Syn<lb></lb>thetica. </s> <s>Naturæ vires legeſque virium ſimpliciores ex ſelectis <lb></lb>quibuſdam Phænomenis per Analyſin deducunt, ex quibus deinde <lb></lb>per Syntheſin reliquorum conſtitutionem tradunt. </s> <s>Hæc illa eſt <lb></lb>Philoſophandi ratio longe optima, quam præ ceteris merito am<lb></lb>plectendam cenſuit Celeberrimus Auctor noſter. </s> <s>Hanc ſolam uti<lb></lb>Q.E.D.gnam judicavit, in qua excolenda atque adornanda operam <lb></lb>ſuam collocaret. </s> <s>Hujus igitur illuſtriſſimum dedit Exemplum, <lb></lb>Mundani nempe Syſtematis explicationem e Theoria Gravitatis <lb></lb>feliciſſime deductam. </s> <s>Gravitatis virtutem univerſis corporibus in<lb></lb>eſſe, ſuſpicati ſunt vel finxerunt alii: primus Ille & ſolus ex Ap<lb></lb>parentiis demonſtrare potuit, & ſpeculationibus egregiis firmiſſi<lb></lb>mum ponere fundamentum. </s></p> <p type="main"> <s>Scio equidem nonnullos magni etiam nominis Viros, præjudiciis <lb></lb>quibuſdam plus æquo occupatos, huic novo Principio ægre aſſen<lb></lb>tiri potuiſſe, & certis incerta identidem prætuliſſe. </s> <s>Horum famam vel<lb></lb>licare non eſt animus: Tibi potius, Benevole Lector, illa paucis ex<lb></lb>ponere lubet, ex quibus Tute ipſe judicium non iniquum feras. </s></p> <p type="main"> <s>Igitur ut Argumenti ſumatur exordium a ſimpliciſſimis & pro<lb></lb>ximis; deſpiciamus pauliſper qualis ſit in Terreſtribus natura Gra<lb></lb>vitatis, ut deinde tutius progrediamur ubi ad corpora Cæleſtia, lon<lb></lb>giſſime a ſedibus noſtris remota, perventum fuerit. </s> <s>Convenit jam <lb></lb>inter omnes Philoſophos, corpora univerſa circumterreſtria gra<lb></lb>vitare in Terram. </s> <s>Nulla dari corpora vere levia, jamdudum <lb></lb>confirmavit Experientia multiplex. </s> <s>Quæ dicitur Levitas relativa, <lb></lb>non eſt vera Levitas, ſed apparens ſolummodo; & oritur a præ<lb></lb>pollente Gravitate corporum contiguorum. </s></p> <p type="main"> <s>Porro, ut corpora univerſa gravitant in Terram, ita Terra viciſ<lb></lb>ſim in corpora æqualiter gravitat; Gravitatis enim actionem eſſe <lb></lb>mutuam & utrinque æqualem, ſic oſtenditur. </s> <s>Diſtinguatur Terræ <pb xlink:href="039/01/013.jpg"></pb>totius moles in binas quaſcunque partes, vel æquales vel utcunque <lb></lb>inæquales: jam ſi pondera partium non eſſent in ſe mutuo æqua<lb></lb>lia; cederet pondus minus majori, & partes conjunctæ pergerent <lb></lb>recta moveri ad infinitum, verſus plagam in quam tendit pondus <lb></lb>majus: omnino contra Experientiam. </s> <s>ItaQ.E.D.cendum erit, pon<lb></lb>dera partium in æquilibrio eſſe conſtituta: hoc eſt, Gravitatis <lb></lb>actionem eſſe mutuam & utrinque æqualem. </s></p> <p type="main"> <s>Pondera corporum, æqualiter a centro Terræ diſtantium, ſunt ut <lb></lb>quantitates materiæ in corporibus. </s> <s>Hoc utique colligitur ex <lb></lb>æquali acceleratione corporum omnium, e quiete per ponderum <lb></lb>vires cadentium: nam vires quibus inæqualia corpora æqualiter <lb></lb>accelerantur, debent eſſe proportionales quantitatibus materiæ <lb></lb>movendæ. </s> <s>Jam vero corpora univerſa cadentia æqualiter acce<lb></lb>lerari, ex eo patet, quod in Vacuo <emph type="italics"></emph>Boyliano<emph.end type="italics"></emph.end>temporibus æqualibus <lb></lb>æqualia ſpatia cadendo deſcribunt, ſublata ſcilicet Aeris reſiſtentia: <lb></lb>accuratius autem comprobatur per Experimenta Pendulorum. </s></p> <p type="main"> <s>Vires attractivæ corporum, in æqualibus diſtantiis, ſunt ut <lb></lb>quantitates materiæ in corporibus. </s> <s>Nam cum corpora in Ter<lb></lb>ram & Terra viciſſim in corpora momentis æqualibus gravitent; <lb></lb>Terræ pondus in unumquodque corpus, ſeu vis qua corpus Ter<lb></lb>ram attrahit, æquabitur ponderi corporis ejuſdem in Terram. </s> <s><lb></lb>Hoc autem pondus erat ut quantitas materiæ in corpore: itaque <lb></lb>vis qua corpus unumquodque Terram attrahit, ſive corporis vis <lb></lb>abſoluta, erit ut eadem quantitas materiæ. </s></p> <p type="main"> <s>Oritur ergo & componitur vis attractiva corporum integrorum <lb></lb>ex viribus attractivis partium: ſiquidem aucta vel diminuta mole <lb></lb>materiæ, oſtenſum eſt, proportionaliter augeri vel diminui ejus vir<lb></lb>tutem. </s> <s>Actio itaque Telluris ex conjunctis partium Actionibus <lb></lb>conflari cenſenda erit; atque adeo corpora omnia Terreſtria ſe <lb></lb>mutuo trahere oportet viribus abſolutis, quæ ſint in ratione ma<lb></lb>teriæ trahentis. </s> <s>Hæc eſt natura Gravitatis apud Terram: videa<lb></lb>mus jam qualis ſit in Cælis. </s></p> <p type="main"> <s>Corpus omne perſeverare in ſtatu ſuo vel quieſcendi vel movendi <lb></lb>uniformiter in directum, niſi quatenus a viribus impreſſis cogitur <lb></lb>ſtatum illum mutare; Naturæ lex eſt ab omnibus recepta Philoſo<lb></lb>phis. </s> <s>Inde vero ſequitur, corpora quæ in Curvis moventur, atque <lb></lb>adeo de lineis rectis Orbitas ſuas tangentibus jugiter abeunt, Vi <lb></lb>aliqua perpetuo agente retineri in itinere curvilineo. </s> <s>Planetis <lb></lb>igitur in Orbibus curvis revolventibus neceſſario aderit Vis aliqua, <lb></lb>per cujus actiones repetitas indeſinenter a Tangentibus deflectantur. </s></p><pb xlink:href="039/01/014.jpg"></pb> <p type="main"> <s>Jam illud concedi æquum eſt, quod Mathematicis rationibus <lb></lb>colligitur & certiſſime demonſtratur; Corpora nempe omnia, quæ <lb></lb>moventur in linea aliqua curva in plano deſcripta, quæque radio <lb></lb>ducto ad punctum vel quieſcens vel utcunque motum deſcribunt <lb></lb>areas circa punctum illud temporibus proportionales, urgeri a <lb></lb>Viribus quæ ad idem punctum tendunt. </s> <s>Cum igitur in confeſſo <lb></lb>ſit apud Aſtronomos, Planetas primarios circum Solem, ſecunda<lb></lb>rios vero circum ſuos primarios, areas deſcribere temporibus pro<lb></lb>portionales; conſequens eſt ut Vis illa, qua perpetuo detorquen<lb></lb>tur a Tangentibus rectilineis, & in Orbitis curvilineis revolvi co<lb></lb>guntur, verſus corpora dirigatur quæ ſita ſunt in Orbitarum cen<lb></lb>tris. </s> <s>Hæc itaque Vis non inepte vocari poteſt, reſpectu quidem <lb></lb>corporis revolventis, Centripeta; reſpectu autem corporis cen<lb></lb>tralis, Attractiva; a quacunQ.E.D.mum cauſa oriri fingatur. </s></p> <p type="main"> <s>Quin & hæc quoque concedenda ſunt, & Mathematice demon<lb></lb>ſtrantur: Si corpora plura motu æquabili revolvantur in Circulis <lb></lb>concentricis, & quadrata temporum periodieorum ſint ut cubi di<lb></lb>ſtantiarum a centro communi; Vires centripetas revolventium <lb></lb>fore reciproce ut quadrata diſtantiarum. </s> <s>Vel, ſi corpora revol<lb></lb>vantur in Orbitis quæ ſunt Circulis finitimæ, & quieſcant Orbita<lb></lb>rum Apſides; Vires centripetas revolventium fore reciproce ut <lb></lb>quadrata diſtantiarum. </s> <s>Obtinere caſum alterutrum in Planetis <lb></lb>univerſis conſentiunt Aſtronomi. </s> <s>Itaque Vires centripetæ Plane<lb></lb>tarum omnium ſunt reciproce ut quadrata diſtantiarum ab Or<lb></lb>bium centris. </s> <s>Si quis objiciat Planetarum, & Lunæ præſertim, <lb></lb>Apſides non penitus quieſcere; ſed motu quodam lento ferri in <lb></lb>conſequentia: reſponderi poteſt, etiamſi concedamus hunc mo<lb></lb>tum tardiſſimum exinde profectum eſſe quod Vis centripetæ pro<lb></lb>portio aberret aliquantum a duplicata, aberrationem illam per <lb></lb>computum Mathematicum inveniri poſſe & plane inſenſibilem <lb></lb>eſſe. </s> <s>Ipſa enim ratio Vis centripetæ Lunaris, quæ omnium ma<lb></lb>xime turbari debet, paululum quidem duplicatam ſuperabit; ad <lb></lb>hanc vero ſexaginta fere vicibus propius accedet quam ad tripli<lb></lb>catam. </s> <s>Sed verior erit reſponſio, ſi dicamus hanc Apſidum progreſ<lb></lb>ſionem, non ex aberratione a duplicata proportione, ſed ex alia <lb></lb>prorſus diverſa cauſa oriri, quemadmodum egregie commonſtratur <lb></lb>in hac Philoſophia. </s> <s>Reſtat ergo ut Vires centripetæ, quibus Pla<lb></lb>netæ primarii tendunt verſus Solem & ſecundarii verſus primarios <lb></lb>ſuos, ſint accurate ut quadrata diſtantiarum reciproce. </s></p><pb xlink:href="039/01/015.jpg"></pb> <p type="main"> <s>Ex iis quæ hactenus dicta ſunt, conſtat Planetas in Orbitis ſuis <lb></lb>retineri per Vim aliquam in ipſos perpetuo agentem: conſtat <lb></lb>Vim illam dirigi ſemper verſus Orbitarum centra: conſtat hujus <lb></lb>efficaciam augeri in acceſſu ad centrum, diminui in receſſu ab eo<lb></lb>dem: & augeri quidem in eadem proportione qua diminuitur qua<lb></lb>dratum diſtantiæ, diminui in eadem proportione qua diſtantiæ <lb></lb>quadratum augetur. </s> <s>Videamus jam, comparatione inſtituta inter <lb></lb>Planetarum Vires centripetas & Vim Gravitatis, annon ejuſdem <lb></lb>forte ſint generis. </s> <s>Ejuſdem vero generis erunt, ſi deprehendan<lb></lb>tur hinc & inde leges eædem eædemque affectiones. </s> <s>Primo ita<lb></lb>que Lunæ, quæ nobis proxima eſt, Vim centripetam expendamus. </s></p> <p type="main"> <s>Spatia rectilinea, quæ a corporibus e quiete demiſſis dato tem<lb></lb>pore ſub ipſo motus initio deſeribuntur, ubi a viribus quibuſcun<lb></lb>que urgentur, proportionalia ſunt ipſis viribus: Hoc utique con<lb></lb>ſequitur ex ratiociniis Mathematicis. </s> <s>Erit igitur Vis centripeta <lb></lb>Lunæ in Orbita ſua revolventis, ad Vim Gravitatis in ſuperficie <lb></lb>Terræ, ut ſpatium quod tempore quam minimo deſcriberet Luna <lb></lb>deſcendendo per Vim centripetam verſus Terram, ſi circulari om<lb></lb>ni motu privari fingeretur, ad ſpatium quod eodem tempore quam <lb></lb>minimo deſcribit grave corpus in vicinia Terræ, per Vim gravita<lb></lb>tis ſuæ cadendo. </s> <s>Horum ſpatiorum prius æquale eſt arcus a Luna <lb></lb>per idem tempus deſcripti ſinui verſo, quippe qui Lunæ tranſla<lb></lb>tionem de Tangente, factam a Vi centripeta, metitur; atque adeo <lb></lb>computari poteſt ex datis tum Lunæ tempore periodico tum di<lb></lb>ſtantia ejus a centro Terræ. </s> <s>Spatium poſterius invenitur per Ex<lb></lb>perimenta Pendulorum, quemadmodum docuit <emph type="italics"></emph>Hugenius.<emph.end type="italics"></emph.end>Inito <lb></lb>itaque calculo, ſpatium prius ad ſpatium pofterius, ſeu vis cen<lb></lb>tripeta Lunæ in Orbita ſua revolventis ad vim Gravitatis in ſu<lb></lb>perficie Terræ, erit ut quadratum ſemidiametri Terræ ad Orbitæ <lb></lb>ſemidiametri quadratum. </s> <s>Eandem habet rationem, per ea quæ <lb></lb>ſuperius oſtenduntur, vis centripeta Lunæ in Orbita ſua revol<lb></lb>ventis ad vim Lunæ centripetam prope Terræ ſuperficiem. </s> <s>Vis <lb></lb>itaque centripeta prope Terræ ſuperficiem æqualis eſt vi Gravita<lb></lb>tis. </s> <s>Non ergo diverſæ ſunt vires, ſed una atque eadem: ſi enim <lb></lb>diverſæ eſſent, corpora viribus conjunctis duplo celerius in Ter<lb></lb>ram caderent quam ex vi ſola Gravitatis. </s> <s>Conſtat igitur Vim <lb></lb>illam centripetam, qua Luna perpetuo de Tangente vel trahitur <lb></lb>vel impellitur & in Orbita retinetur, ipſam eſſe vim Gravitatis <lb></lb>terreſtris ad Lunam uſque pertingentem. </s> <s>Et rationi quidem con<lb></lb>ſentaneum eſt ut ad ingentes diſtantias illa ſeſe Virtus extendat, <pb xlink:href="039/01/016.jpg"></pb>cum nullam ejus ſenſibilem imminutionem, vel in altiſſimis montium <lb></lb>cacuminibus, obſervare licet. </s> <s>Gravitat itaque Luna in Terram: <lb></lb>quin & actione mutua, Terra viciſſim in Lunam æqualiter gravitat: <lb></lb>id quod abunde quidem confirmatur in hac Philoſophia, ubi agi<lb></lb>tur de Maris æſtu & Æquinoctiorum præceſſione, ab actione tum <lb></lb>Lunæ tum Solis in Terram oriundis. </s> <s>Hinc & illud tandem edo<lb></lb>cemur, qua nimirum lege vis Gravitatis decreſcat in majoribus a <lb></lb>Tellure diſtantiis. </s> <s>Nam cum Gravitas non diverſa ſit a Vi cen<lb></lb>tripeta Lunari, hæc vero ſit reciproce proportionalis quadrato <lb></lb>diſtantiæ; diminuetur & Gravitas in eadem ratione. </s></p> <p type="main"> <s>Progrediamur jam ad Planetas reliquos. </s> <s>Quoniam revolu<lb></lb>tiones primariorum circa Solem & ſecundariorum circa Jovem & <lb></lb>Saturnum ſunt Phænomena generis ejuſdem ac revolutio Lunæ <lb></lb>circa Terram, quoniam porro demonſtratum eſt vires centripetas <lb></lb>primariorum dirigi verſus centrum Solis, ſecundariorum verſus <lb></lb>centra Jovis & Saturni, quemadmodum Lunæ vis centripeta verſus <lb></lb>Terræ centrum dirigitur; adhæc, quoniam omnes illæ vires ſunt <lb></lb>reciproce ut quadrata diſtantiarum a centris, quemadmodum vis <lb></lb>Lunæ eſt ut quadratum diſtantiæ a Terra: concludendum erit <lb></lb>eandem eſſe naturam univerſis. </s> <s>Itaque ut Luna gravitat in Ter<lb></lb>ram, & Terra viciſſim in Lunam; ſic etiam gravitabunt omnes <lb></lb>ſecundarii in primarios ſuos, & primarii viciſſim in ſecundarios; <lb></lb>ſic & omnes primarii in Solem, & Sol viciſſim in primarios. </s></p> <p type="main"> <s>Igitur Sol in Planetas univerſos gravitat & univerſi in Solem. </s> <s><lb></lb>Nam ſecundarii dum primarios ſuos comitantur, revolvuntur in<lb></lb>terea circum Solem una cum primariis. </s> <s>Eodem itaque argumento, <lb></lb>utriuſque generis Planetæ gravitant in Solem, & Sol in ipſos. </s> <s><lb></lb>Secundarios vero Planetas in Solem gravitare abunde inſuper <lb></lb>conſtat ex inæqualitatibus Lunaribus; quarum accuratiſſimam <lb></lb>Theoriam, admiranda ſagacitate patefactam, in tertio hujus Operis <lb></lb>libro expoſitam habemus. </s></p> <p type="main"> <s>Solis virtutem attractivam quoquoverſum propagari ad ingen<lb></lb>tes uſQ.E.D.ſtantias, & ſeſe diffundere ad ſingulas circumjecti ſpa<lb></lb>tii partes, apertiſſime colligi poteſt ex motu Cometarum; qui ab <lb></lb>immenſis intervallis profecti feruntur in viciniam Solis, & non<lb></lb>nunquam adeo ad ipſum proxime accedunt ut Globum ejus, in <lb></lb>Periheliis ſuis verſantes, tantum non contingere videantur. </s> <s>Ho<lb></lb>rum Theoriam ab Aſtronomis antehac fruſtra quæſitam, noſtro <lb></lb>tandem ſæculo feliciter inventam & per Obſervationes certiſ<lb></lb>ſime demonſtratam, Præſtantiſſimo noſtro Auctori debemus. </s> <s>Patet <pb xlink:href="039/01/017.jpg"></pb>igitur Cometas in Sectionibus Conicis umbilicos in centro Solis <lb></lb>habentibus moveri, & radiis ad Solem ductis areas temporibus <lb></lb>proportionales deſcribere. </s> <s>Ex hiſce vero Phænomenis manife<lb></lb>ſtum eſt & Mathematice comprobatur, vires illas, quibus Cometæ <lb></lb>retinentur in orbitis ſuis, reſpicere Solem & eſſe reciproce ut qua<lb></lb>drata diſtantiarum ab ipſius centro. </s> <s>Gravitant itaque Cometæ <lb></lb>in Solem: atque adeo Solis vis attractiva non tantum ad corpora <lb></lb>Planetarum in datis diſtantiis & in eodem fere plano collocata, <lb></lb>ſed etiam ad Cometas in diverſiſſimis Cælorum regionibus & in <lb></lb>diverſiſſimis diſtantiis poſitos pertingit. </s> <s>Hæc igitur eſt natura <lb></lb>corporum gravitantium, ut vires ſuas edant ad omnes diſtantias in <lb></lb>omnia corpora gravitantia. </s> <s>Inde vero ſequitur, Planetas & Co<lb></lb>metas univerſos ſe mutuo trahere, & in ſe mutuo graves eſſe: <lb></lb>quod etiam confirmatur ex perturbatione Jovis & Saturni, Aſtro<lb></lb>nomis non incognita, & ab actionibus horum Planetarum in ſe in<lb></lb>vicem oriunda; quin & ex motu illo lentiſſimo Apſidum, qui ſu<lb></lb>pra memoratus eſt, quique a cauſa conſimili proficiſcitur. </s></p> <p type="main"> <s>Eo demum pervenimus ut dicendum ſit, & Terram & Solem & <lb></lb>corpora omnia cæleſtia, quæ Solem comitantur, ſe mutuo attrahere. </s> <s><lb></lb>Singulorum ergo particulæ quæque minimæ vires ſuas attractivas <lb></lb>habebunt, pro quantitate materiæ pollentes; quemadmodum ſu<lb></lb>pra de Terreſtribus oſtenſum eſt. </s> <s>In diverſis autem diſtantiis, <lb></lb>erunt & harum vires in duplicata ratione diſtantiarum reciproce: <lb></lb>nam ex particulis hac lege trahentibus componi debere Globos <lb></lb>eadem lege trahentes, Mathematice demonſtratur. </s></p> <p type="main"> <s>Concluſiones præcedentes huic innituntur Axiomati, quod a <lb></lb>nullis non recipitur Philoſophis; Effectuum ſcilicet ejuſdem ge<lb></lb>neris, quorum nempe quæ cognoſcuntur proprietates eædem ſunt, <lb></lb>eaſdem eſſe cauſas & eaſdem eſſe proprietates quæ nondum cog<lb></lb>noſcuntur. </s> <s>Quis enim dubitat, ſi Gravitas ſit cauſa deſcenſus <lb></lb>Lapidis in <emph type="italics"></emph>Europa,<emph.end type="italics"></emph.end>quin eadem ſit cauſa deſcenſus in <emph type="italics"></emph>America?<emph.end type="italics"></emph.end><lb></lb>Si Gravitas mutua fuerit inter Lapidem & Terram in <emph type="italics"></emph>Europa<emph.end type="italics"></emph.end>; <lb></lb>quis negabit mutuam eſſe in <emph type="italics"></emph>America?<emph.end type="italics"></emph.end>Si vis attractiva Lapidis <lb></lb>& Terræ componatur, in <emph type="italics"></emph>Europa,<emph.end type="italics"></emph.end>ex viribus attractivis partium; <lb></lb>quis negabit ſimilem eſſe compoſitionem in <emph type="italics"></emph>America?<emph.end type="italics"></emph.end>Si attractio <lb></lb>Terræ ad omnia corporum genera & ad omnes diſtantias propa<lb></lb>getur in <emph type="italics"></emph>Europa<emph.end type="italics"></emph.end>; quidni pariter propagari dicamus in <emph type="italics"></emph>America?<emph.end type="italics"></emph.end><lb></lb>In hac Regula fundatur omnis Philoſophia: quippe qua ſublata <lb></lb>nihil affirmare poſſimus de Univerſis. </s> <s>Conſtitutio rerum ſingula<lb></lb>rum innoteſcit per Obſervationes & Experimenta: inde vero non <pb xlink:href="039/01/018.jpg"></pb>niſi per hanc Regulam de rerum univerſarum natura judica<lb></lb>mus. </s></p> <p type="main"> <s>Jam cum Gravia ſint omnia corpora, quæ apud Terram vel in <lb></lb>Cælis reperiuntur, de quibus Experimenta vel Obſervationes in<lb></lb>ſtituere licet; omnino dicendum erit, Gravitatem corporibus uNI<lb></lb>verſis competere. </s> <s>Et quemadmodum nulla concipi debent cor<lb></lb>pora, quæ non ſint Extenſa, Mobilia, & Impenetrabilia; ita nulla <lb></lb>concipi debere, quæ non ſint Gravia. </s> <s>Corporum Extenſio, Mobi<lb></lb>litas, & Impenetrabilitas non niſi per Experimenta innoteſcunt: <lb></lb>eodem plane modo Gravitas innoteſcit. </s> <s>Corpora omnia de qui<lb></lb>bus Obſervationes habemus, Extenſa ſunt & Mobilia & Impene<lb></lb>trabilia: & inde concludimus corpora univerſa, etiam illa de qui<lb></lb>bus Obſervationes non habemus, Extenſa eſſe & Mobilia & Im<lb></lb>penetrabilia. </s> <s>Ita corpora omnia ſunt Gravia, de quibus Obſer<lb></lb>vationes habemus: & inde concludimus corpora univerſa, etiam <lb></lb>illa de quibus Obſervationes non habemus, Gravia eſſe. </s> <s>Si quis <lb></lb>dicat corpora Stellarum inerrantium non eſſe Gravia, quandoqui<lb></lb>dem eorum Gravitas nondum eſt obſervata; eodem argumento <lb></lb>dicere licebit neque Extenſa eſſe, nec Mobilia, nec Impenetrabilia, <lb></lb>cum hæ Fixarum affectiones nondum ſint obſervatæ. </s> <s>Quid opus <lb></lb>eſt verbis? </s> <s>Inter primarias qualitates corporum univerſorum vel <lb></lb>Gravitas habebit locum; vel Extenſio, Mobilitas, & Impenetra<lb></lb>bilitas non habebunt. </s> <s>Et natura rerum vel recte explicabitur <lb></lb>per corporum Gravitatem, vel non recte explicabitur per corpo<lb></lb>rum Extenſionem, Mobilitatem, & Impenetrabilitatem. </s></p> <p type="main"> <s>Audio nonnullos hanc improbare concluſionem, & de occultis <lb></lb>qualitatibus neſcio quid muſſitare. </s> <s>Gravitatem ſcilicet Occultum <lb></lb>eſſe quid, perpetuo argutari ſolent; occultas vero cauſas pro<lb></lb>cul eſſe ablegandas a Philoſophia. </s> <s>His autem facile reſpon<lb></lb>detur; occultas eſſe cauſas, non illas quidem quarum exiſtentia <lb></lb>per Obſervationes clariſſime demonſtratur, ſed has ſolum quarum <lb></lb>occulta eſt & ficta exiſtentia nondum vero comprobata. </s> <s>Gravitas <lb></lb>ergo non erit occulta cauſa motuum cæleſtium; ſiquidem ex Phæ<lb></lb>nomenis oſtenſum eſt, hanc virtutem revera exiſtere. </s> <s>Hi potius <lb></lb>ad occultas confugiunt cauſas; qui neſcio quos Vortices, materiæ <lb></lb>cujuſdam prorſus fictitiæ & ſenſibus omnino ignotæ, motibus <lb></lb>iiſdem regendis præficiunt. </s></p> <p type="main"> <s>Ideone autem Gravitas occulta cauſa dicetur, eoque nomine <lb></lb>rejicietur e Philoſophia, quod cauſa ipſius Gravitatis occulta eſt <lb></lb>& nondum inventa? </s> <s>Qui ſic ſtatuunt, videant nequid ſtatu<pb xlink:href="039/01/019.jpg"></pb>ant abſurdi, unde totius tandem Philoſophiæ fundamenta convel<lb></lb>lantur. </s> <s>Etenim cauſæ continuo nexu procedere ſolent a compo<lb></lb>ſitis ad ſimpliciora: ubi ad cauſam ſimpliciſſimam perveneris, jam <lb></lb>non licebit ulterius progredi. </s> <s>Cauſæ igitur ſimpliciſſimæ nulla <lb></lb>dari poteſt mechanica explicatio: ſi daretur enim, cauſa non<lb></lb>dum eſſet ſimpliciſſima. </s> <s>Has tu proinde cauſas ſimpliciſſimas <lb></lb>appellabis occultas, & exulare jubebis? </s> <s>ſimul vero exulabunt <lb></lb>& ab his proxime pendentes & quæ ab illis porro pendent, <lb></lb>uſQ.E.D.m a cauſis omnibus vacua fuerit & probe purgata Phi<lb></lb>loſophia. </s></p> <p type="main"> <s>Sunt qui Gravitatem præter naturam eſſe dicunt, & Miraculum <lb></lb>perpetuum vocant. </s> <s>Itaque rejiciendam eſſe volunt, cum in Phy<lb></lb>ſica præternaturales cauſæ locum non habeant. </s> <s>Huic ineptæ <lb></lb>prorſus objectioni diluendæ, quæ & ipſa Philoſophiam ſubruit <lb></lb>univerſam, vix operæ pretium eſt immorari. </s> <s>Vel enim Gravita<lb></lb>tem corporibus omnibus inditam eſſe negabunt, quod tamen dici <lb></lb>non poteſt: vel eo nomine præter naturam eſſe affirmabunt, quod <lb></lb>ex aliis corporum affectionibus atque adeo ex cauſis Mechanicis <lb></lb>originem non habeat. </s> <s>Dantur certe primariæ corporum affecti<lb></lb>ones; quæ, quoniam ſunt primariæ, non pendent ab aliis. </s> <s>Vide<lb></lb>rint igitur annon & hæ omnes ſint pariter præter naturam, eo<lb></lb>que pariter rejiciendæ: viderint vero qualis ſit deinde futura <lb></lb>Philoſophia. </s></p> <p type="main"> <s>Nonnulli ſunt quibus hæc tota Phyſica cæleſtis vel ideo minus <lb></lb>placet, quod cum <emph type="italics"></emph>Carteſii<emph.end type="italics"></emph.end>dogmatibus pugnare & vix conciliari <lb></lb>poſſe videatur. </s> <s>His ſua licebit opinione frui; ex æquo autem <lb></lb>agant oportet: non ergo denegabunt aliis eandem libertatem <lb></lb>quam ſibi concedi poſtulant. </s> <s>NEWTONIANAM itaque Philoſophi<lb></lb>am, quæ nobis verior habetur, retinere & amplecti licebit, & cauſas <lb></lb>ſequi per Phænomena comprobatas, potius quam fictas & nondum <lb></lb>comprobatas. </s> <s>Ad veram Philoſophiam pertinet, rerum naturas <lb></lb>ex cauſis vere exiſtentibus derivare: eas vero leges quærere, qui<lb></lb>bus voluit Summus opifex hunc Mundi pulcherrimum ordinem <lb></lb>ſtabilire; non eas quibus potuit, ſi ita viſum fuiſſet. </s> <s>Rationi enim <lb></lb>conſonum eſt, ut a pluribus cauſis, ab invicem nonnihil diverſis, <lb></lb>idem poſſit Effectus proficiſci: hæc autem vera erit cauſa, ex qua <lb></lb>vere atque actu proficiſcitur; reliquæ locum non habent in Philo<lb></lb>ſophia vera. </s> <s>In Horologiis automatis idem Indicis horarii mo<lb></lb>tus vel ab appenſo Pondere vel ab intus concluſo Elatere oriri po<lb></lb>teſt. </s> <s>Quod ſi oblatum Horologium revera ſit inſtructum Pondere; <pb xlink:href="039/01/020.jpg"></pb>ridebitur qui finget Elaterem, & ex Hypotheſi ſic præpropere con<lb></lb>ficta motum Indicis explicare ſuſcipiet: oportuit enim internam <lb></lb>Machinæ fabricam penitius perſcrutari, ut ita motus propoſiti prin<lb></lb>cipium verum exploratum habere poſſet. </s> <s>Idem vel non abſimile <lb></lb>feretur judicium de Philoſophis illis, qui materia quadam ſubti<lb></lb>liſſima Cælos eſſe repletos, hanc autem in Vortices indeſinenter <lb></lb>agi voluerunt. </s> <s>Nam ſi Phænomenis vel accuratiſſime ſatisfacere <lb></lb>poſſent ex Hypotheſibus ſuis; veram tamen Philoſophiam tradi<lb></lb>diſſe, & veras cauſas motuum cæleſtium inveniſſe nondum di<lb></lb>cendi ſunt; niſi vel has revera exiſtere, vel ſaltem alias non ex<lb></lb>iſtere demonſtraverint. </s> <s>Igitur ſi oſtenſum fuerit, univerſorum <lb></lb>corporum Attractionem habere verum locum in rerum natura; <lb></lb>quinetiam oſtenſum fuerit, qua ratione motus omnes cæleſtes ab<lb></lb>inde ſolutionem recipiant; vana fuerit & merito deridenda objectio, <lb></lb>ſi quis dixerit eoſdem motus per Vortices explicari debere, etiamſi <lb></lb>id fieri poſſe vel maxime conceſſerimus. </s> <s>Non autem concedimus: <lb></lb>Nequeunt enim ullo pacto Phænomena per Vortices explicari; <lb></lb>quod ab Auctore noſtro abunde quidem & clariſſimis rationibus <lb></lb>evincitur; ut ſomniis plus æquo indulgeant oporteat, qui inep<lb></lb>tiſſimo figmento reſarciendo, noviſque porro commentis ornando <lb></lb>infelicem operam addicunt. </s></p> <p type="main"> <s>Si corpora Planetarum & Cometarum circa Solem deferantur <lb></lb>a Vorticibus; oportet corpora delata & Vorticum partes proxime <lb></lb>ambientes eadem velocitate eademque curſus determinatione mo<lb></lb>veri, & eandem habere denſitatem vel eandem Vim inertiæ pro <lb></lb>mole materiæ. </s> <s>Conſtat vero Planetas & Cometas, dum verſan<lb></lb>tur in iiſdem regionibus Cælorum, velocitatibus variis variaque <lb></lb>curſus determinatione moveri. </s> <s>Neceſſario itaque ſequitur, ut <lb></lb>Fluidi cæleſtis partes illæ, quæ ſunt ad eaſdem diſtantias a Sole, <lb></lb>revolvantur eodem tempore in plagas diverſas cum diverſis ve<lb></lb>locitatibus: etenim alia opus erit directione & velocitate, ut tran<lb></lb>ſire poſſint Planetæ; alia, ut tranſire poſſint Cometæ. </s> <s>Quod cum <lb></lb>explicari nequeat; vel fatendum erit, univerſa corpora cæleſtia <lb></lb>non deferri a materia Vorticis; vel dicendum erit, eorundem mo<lb></lb>tus repetendos eſle non ab uno eodemque Vortice, ſed a pluribus <lb></lb>qui ab invicem diverſi ſint, idemque ſpatium Soli circumjectum <lb></lb>pervadant. </s></p> <p type="main"> <s>Si plures Vortices in eodem ſpatio contineri, & ſeſe mutuo pe<lb></lb>netrare, motibuſQ.E.D.verſis revolvi ponantur; quoniam hi mo<lb></lb>tus debent eſſe conformes delatorum corporum motibus, qui <pb xlink:href="039/01/021.jpg"></pb>ſunt ſumme regulares, & peraguntur in Sectionibus Conicis, nunc <lb></lb>valde eccentricis, nunc ad Circulorum proxime formam acceden<lb></lb>tibus; jure quærendum erit, qui fieri poſſit, ut iidem integri con<lb></lb>ſerventur, nec ab actionibus materiæ occurſantis per tot ſæcula <lb></lb>quicquam perturbentur. </s> <s>Sane ſi motus hi fictitii ſunt magis com<lb></lb>poſiti & difficilius explicantur, quam veri illi motus Planetarum <lb></lb>& Cometarum; fruſtra mihi videntur in Philoſophiam recipi: <lb></lb>omnis enim Cauſa debet eſſe Effectu ſuo ſimplicior. </s> <s>Conceſſa <lb></lb>Fabularum licentia, affirmaverit aliquis Planetas omnes & Cometas <lb></lb>circumcingi Atmoſphæris, adinſtar Telluris noſtræ; quæ quidem <lb></lb>Hypotheſis rationi magis conſentanea videbitur quam Hypothe<lb></lb>ſis Vorticum. </s> <s>Affirmaverit deinde has Atmoſphæras, ex natura <lb></lb>ſua, circa Solem moveri & Sectiones Conicas deſcribere; qui <lb></lb>ſane motus multo facilius concipi poteſt, quam conſimilis motus <lb></lb>Vorticum ſe invicem permeantium. </s> <s>Denique Planetas ipſos & <lb></lb>Cometas circa Solem deferri ab Atmoſphæris ſuis credendum eſſe <lb></lb>ſtatuat, & ob repertas motuum cæleſtium cauſas triumphum agat. </s> <s><lb></lb>Quiſquis autem hanc Fabulam rejiciendam eſſe putet, idem & alte<lb></lb>ram Fabulam rejiciet: nam ovum non eſt ovo ſimilius, quam Hy<lb></lb>potheſis Atmoſphærarum Hypotheſi Vorticum. </s></p> <p type="main"> <s>Docuit <emph type="italics"></emph>Galilæus,<emph.end type="italics"></emph.end>lapidis projecti & in Parabola moti deflexio<lb></lb>nem a curſu rectilineo oriri a Gravitate lapidis in Terram, ab oc<lb></lb>culta ſcilicet qualitate. </s> <s>Fieri tamen poteſt ut alius aliquis, naſi <lb></lb>acutioris, Philoſophus cauſam aliam comminiſcatur. </s> <s>Finget igi<lb></lb>tur ille materiam quandam ſubtilem, quæ nec viſu, nec tactu, <lb></lb>neque ullo ſenſu percipitur, verſari in regionibus quæ proxime <lb></lb>contingunt Telluris ſuperficiem. </s> <s>Hanc autem materiam, in di<lb></lb>verſas plagas, variis & plerumque contrariis motibus ferri, & li<lb></lb>neas Parabolicas deſcribere contendet. </s> <s>Deinde vero lapidis de<lb></lb>flexionem pulchre ſic expediet, & vulgi plauſum merebitur. </s> <s>La<lb></lb>pis, inquiet, in Fluido illo ſubtili natat; & curſui ejus obſequen<lb></lb>do, non poteſt non eandem una ſemitam deſcribere. </s> <s>Fluidum <lb></lb>vero movetur in lineis Parabolicis; ergo lapidem in Parabola <lb></lb>moveri neceſſe eſt. </s> <s>Quis nunc non mirabitur acutiſſimum hujuſce <lb></lb>Philoſophi ingenium, ex cauſis Mechanicis, materia ſcilicet & <lb></lb>motu, phænomena Naturæ ad vulgi etiam captum præclare de<lb></lb>ducentis? </s> <s>Quis vero non ſubſannabit bonum illum <emph type="italics"></emph>Galilæum,<emph.end type="italics"></emph.end>qui <lb></lb>magno molimine Mathematico qualitates occultas, e Philoſophia <lb></lb>feliciter excluſas, denuo revocare ſuſtinuerit? </s> <s>Sed pudet nugis <lb></lb>diutius immorari. </s></p><pb xlink:href="039/01/022.jpg"></pb> <p type="main"> <s>Summa rei huc tandem redìt: Cometarum ingens eſt numerus; <lb></lb>motus eorum ſunt ſumme regulares, & eaſdem leges cum Plane<lb></lb>tarum motibus obſervant. </s> <s>Moventur in Orbibus Conicis, hi or<lb></lb>bes ſunt valde admodum eccentrici. </s> <s>Feruntur undiQ.E.I. omnes <lb></lb>Cælorum partes, & Planetarum regiones liberrime pertranſeunt, <lb></lb>& ſæpe contra Signorum ordinem incedunt. </s> <s>Hæc Phænomena <lb></lb>certiſſime confirmantur ex Obſervationibus Aſtronomicis: & per <lb></lb>Vortices nequeunt explicari: Imo, ne quidem cum Vorticibus <lb></lb>Planetarum conſiſtere poſſunt. </s> <s>Cometarum motibus omnino lo<lb></lb>cus non erit; niſi materia illa fictitia penitus e Cælis amo<lb></lb>veatur. </s></p> <p type="main"> <s>Si enim Planetæ circum Solem a Vorticibus devehuntur; Vor<lb></lb>ticum partes, quæ proxime ambiunt unumquemque Planetam, ejuſ<lb></lb>dem denſitatis erunt ac Planeta; uti ſupra dictum eſt. </s> <s>Itaque <lb></lb>materia illa omnis quæ contigua eſt Orbis magni perimetro, pa<lb></lb>rem habebit ac Tellus denſitatem: quæ vero jacet intra Orbem <lb></lb>magnum atque Orbem Saturni, vel parem vel majorem habebit. </s> <s><lb></lb>Nam ut conſtitutio Vorticis permanere poſſit, debent partes mi<lb></lb>nus denſæ centrum occupare, magis denſæ longius a centro abire. </s> <s><lb></lb>Cum enim Planetarum tempora periodica ſint in ratione ſeſqui<lb></lb>plicata diſtantiarum a Sole, oportet partium Vorticis periodos <lb></lb>eandem rationem ſervare. </s> <s>Inde vero ſequitur, vires centrifugas <lb></lb>harum partium fore reciproce ut quadrata diſtantiarum. </s> <s>Quæ <lb></lb>igitur majore intervallo diſtant a centro, nituntur ab eodem re<lb></lb>cedere minore vi: unde ſi minus denſæ fuerint, neceſſe eſt ut ce<lb></lb>dant vi majori, qua partes centro propiores aſcendere conantur. </s> <s><lb></lb>Aſcendent ergo denſiores, deſcendent minus denſæ, & loeorum <lb></lb>fiet invicem permutatio; donec ita fuerit diſpoſita atque ordinata <lb></lb>materia fluida totius Vorticis, ut conquieſcere jam poſſit in æqui<lb></lb>librio conſtituta. </s> <s>Si bina Fluida, quorum diverſa eſt denſitas, <lb></lb>in eodem vaſe continentur; utique futurum eſt ut Fluidum, cu<lb></lb>jus major eſt denſitas, majore vi Gravitatis infimum petat locum: <lb></lb>& ratione non abſimili omnino dicendum eſt, denſiores Vorticis <lb></lb>partes majore vi centrifuga petere ſupremum locum. </s> <s>Tota igi<lb></lb>tur illa & multo maxima pars Vorticis, quæ jacet extra Telluris <lb></lb>orbem, denſitatem habebit atque adeo vim inertiæ pro mole ma<lb></lb>teriæ, quæ non minor erit quam denſitas & vis inertiæ Telluris: <lb></lb>inde vero Cometis trajectis orietur ingens reſiſtentia, & valde ad<lb></lb>modum ſenſibilis; ne dicam, quæ motum eorundem penitus ſiſtere <lb></lb>atque abſorbere poſſe merito videatur. </s> <s>Conſtat autem ex motu Co-<pb xlink:href="039/01/023.jpg"></pb>metarum prorſus regulari, nullam ipſos reſiſtentiam pati quæ vel <lb></lb>minimum ſentiri poteſt; atque adeo neutiquam in materiam ul<lb></lb>lam incurſare, cujus aliqua ſit vis reſiſtendi, vel proinde cujus ali<lb></lb>qua ſit denſitas ſeu vis Inertiæ. </s> <s>Nam reſiſtentia Mediorum ori<lb></lb>tur vel ab inertia materiæ fluidæ, vel a defectu lubricitatis. </s> <s>Quæ <lb></lb>oritur a defectu lubricitatis, admodum exigua eſt; & ſane vix <lb></lb>obſervari poteſt in Fluidis vulgo notis, niſi valde tenacia fuerint <lb></lb>adinſtar Olei & Mellis. </s> <s>Reſiſtentia quæ ſentitur in Aere, Aqua, <lb></lb>Hydrargyro, & hujuſmodi Fluidis non tenacibus fere tota eſt <lb></lb>prioris generis; & minui non poteſt per ulteriorem quemcunque <lb></lb>gradum ſubtilitatis, manente Fluidi denſitate vel vi inertiæ, cui <lb></lb>ſemper proportionalis eſt hæc reſiſtentia; quemadmodum clariſ<lb></lb>ſime demonſtratum eſt ab Auctore noſtro in peregregia Reſiſten<lb></lb>tiarum Theoria, quæ paulo nunc accuratius exponitur, hac ſe<lb></lb>cunda vice, & per Experimenta corporum cadentium plenius <lb></lb>confirmatur. </s></p> <p type="main"> <s>Corpora progrediendo motum ſuum Fluido ambienti paulatim <lb></lb>communicant, & communicando amittunt, amittendo autem re<lb></lb>tardantur. </s> <s>Eſt itaque retardatio motui communicato proportio<lb></lb>nalis; motus vero communicatus, ubi datur corporis progredientis <lb></lb>velocitas, eſt ut Fluidi denſitas; ergo retardatio ſeu reſiſtentia <lb></lb>erit ut eadem Fluidi denſitas; neque ullo pacto tolli poteſt, niſi <lb></lb>a Fluido ad partes corporis poſticas recurrente reſtituatur motus <lb></lb>amiſſus. </s> <s>Hoc autem dici non poterit, niſi impreſſio Fluidi in cor<lb></lb>pus ad partes poſticas æqualis fuerit impreſſioni corporis in Flui<lb></lb>dum ad partes anticas, hoc eſt, niſi velocitas relativa qua Flui<lb></lb>dum irruit in corpus a tergo, æqualis fuerit velocitati qua cor<lb></lb>pus irruit in Fluidum, id eſt, niſi velocitas abſoluta Fluidi re<lb></lb>currentis duplo major fuerit quam velocitas abſoluta Fluidi pro<lb></lb>pulſi; quod fieri nequit. </s> <s>Nullo igitur modo tolli poteſt Flui<lb></lb>dorum reſiſtentia, quæ oritur ab corundem denſitate & vi in<lb></lb>ertiæ. </s> <s>Itaque concludendum erit; Fluidi cæleſtis nullam eſſe <lb></lb>vim inertiæ, cum nulla ſit vis reſiſtendi: nullam eſſe vim qua <lb></lb>motus communicetur, cum nulla ſit vis inertiæ: nullam eſſe vim <lb></lb>qua mutatio quælibet vel corporibus ſingulis vel pluribus indu<lb></lb>catur, cum nulla ſit vis qua motus communicetur: nullam eſſe <lb></lb>omnino efficaciam, cum nulla ſit facultas mutationem quamlibet <lb></lb>inducendi. </s> <s>Quidni ergo hanc Hypotheſin, quæ fundamento <lb></lb>plane deſtituitur, quæque naturæ rerum explicandæ ne minimum <lb></lb>quidem inſervit, ineptiſſimam vocare liceat & Philoſopho pror-<pb xlink:href="039/01/024.jpg"></pb>ſus indignam. </s> <s>Qui Cælos materia fluida repletos eſſe volunt, <lb></lb>hanc vero non inertem eſſe ſtatuunt; Hi verbis tollunt Vacuum, <lb></lb>re ponunt. </s> <s>Nam cum hujuſmodi materia fluida ratione nulla <lb></lb>ſecerni poſſit ab inani Spatio; diſputatio tota fit de rerum no<lb></lb>minibus, non de naturis. </s> <s>Quod ſi aliqui ſint adeo uſQ.E.D.<lb></lb>diti Materiæ, ut Spatium a corporibus vacuum nullo pacto ad<lb></lb>mittendum credere velint; videamus quo tandem oporteat illos <lb></lb>pervenire. </s></p> <p type="main"> <s>Vel enim dicent hanc, quam confingunt, Mundi per omnia <lb></lb>pleni conſtitutionem ex voluntate Dei profectam eſſe, propter <lb></lb>eum finem, ut operationibus Naturæ ſubſidium præſens haberi <lb></lb>poſſet ab Æthere ſubtiliſſimo cuncta permeante & implente; <lb></lb>quod tamen dici non poteſt, ſiquidem jam oſtenſum eſt ex Co<lb></lb>metarum phænomenis, nullam eſſe hujus Ætheris efficaciam: vel <lb></lb>dicent ex voluntate Dei profectam eſſe, propter finem aliquem <lb></lb>ignotum; quod neQ.E.D.ci debet, ſiquidem diverſa Mundi con<lb></lb>ſtitutio eodem argumento pariter ſtabiliri poſſet: vel denique <lb></lb>non dicent ex voluntate Dei profectam eſſe, ſed ex neceſſitate <lb></lb>quadam Naturæ. </s> <s>Tandem igitur delabi oportet in fæces ſordi<lb></lb>das Gregis impuriſſimi. </s> <s>Hi ſunt qui ſomniant Fato univerſa <lb></lb>regi, non Providentia; Materiam ex neceſſitate ſua ſemper & ubi<lb></lb>que extitiſſe, infinitam eſſe & æternam. </s> <s>Quibus poſitis, erit <lb></lb>etiam undiquaque uniformis: nam varietas formarum cum neceſ<lb></lb>ſitate omnino pugnat. </s> <s>Erit etiam immota: nam ſi neceſſario <lb></lb>moveatur in plagam aliquam determinatam, cum determinata ali<lb></lb>qua velocitate; pari neceſſitate movebitur in plagam diverſam <lb></lb>cum diverſa velocitate; in plagas autem diverſas, cum diverſis <lb></lb>velocitatibus, moveri non poteſt; oportet igitur immotam eſſe. </s> <s><lb></lb>Neutiquam profecto potuit oriri Mundus, pulcherrima forma<lb></lb>rum & motuum varietate diſtinctus, niſi ex liberrima voluntate <lb></lb>cuncta providentis & gubernantis Dei. </s></p> <p type="main"> <s>Ex hoc igitur fonte promanarunt illæ omnes quæ dicuntur <lb></lb>Naturæ leges: in quibus multa ſane ſapientiſſimi conſilii, nulla <lb></lb>neceſſitatis apparent veſtigia. </s> <s>Has proinde non ab incertis con<lb></lb>jecturis petere, ſed Obſervando atque Experiendo addiſcere de<lb></lb>bemus. </s> <s>Qui veræ Phyſicæ principia Legeſque rerum, ſola men<lb></lb>tis vi & interno rationis lumine fretum, invenire ſe poſſe confi<lb></lb>dit; hunc oportet vel ſtatuere Mundum ex neceſſitate fuiſle, Le<lb></lb>geſque propoſitas ex eadem neceſſitate ſequi; vel ſi per volun<lb></lb>tatem Dei conſtitutus ſit ordo Naturæ, ſe tamen, homuncionem <pb xlink:href="039/01/025.jpg"></pb>miſellum, quid optimum factu ſit perſpectum habere. </s> <s>Sana om<lb></lb>nis & vera Philoſophia fundatur in Phænomenis rerum: quæ ſi <lb></lb>nos vel invitos & reluctantes ad hujuſmodi principia deducunt, <lb></lb>in quibus clariſſime cernuntur Conſilium optimum & Dominium <lb></lb>ſummum ſapientiſſimi & potentiſſimi Entis; non erunt hæc ideo <lb></lb>non admittenda principia, quod quibuſdam forſan hominibus <lb></lb>minus grata ſint futura. </s> <s>His vel Miracula vel Qualitates occultæ <lb></lb>dicantur, quæ diſplicent: verum nomina malitioſe indita non ſunt <lb></lb>ipſis rebus vitio vertenda; niſi illud fateri tandem velint, utique <lb></lb>debere Philoſophiam in Atheiſmo fundari. </s> <s>Horum hominum <lb></lb>gratia non erit labefactanda Philoſophia, ſiquidem rerum ordo <lb></lb>non vult immutari. </s></p> <p type="main"> <s>Obtinebit igitur apud probos & æquos Judices præſtantiſſima <lb></lb>Philoſophandi ratio, quæ fundatur in Experimentis & Obſerva<lb></lb>tionibus. </s> <s>Huic vero, dici vix poterit, quanta lux accedat, quanta <lb></lb>dignitas, ab hoc Opere præclaro Illuſtriſſimi noſtri Auctoris; cujus <lb></lb>eximiam ingenii felicitatem, difficillima quæque Problemata eno<lb></lb>dantis, & ad ea porro pertingentis ad quæ nec ſpes erat humanam <lb></lb>mentem aſſurgere potuiſſe, merito admirantur & ſuſpiciunt qui<lb></lb>cunque paulo profundius in hiſce rebus verſati ſunt. </s> <s>Clauſtris <lb></lb>ergo referatis, aditum Nobis aperuit ad pulcherrima rerum my<lb></lb>ſteria. </s> <s>Syſtematis Mundani compagem elegantiſſimam ita tan<lb></lb>dem patefecit & penitius perſpectandam dedit; ut nec ipſe, ſi <lb></lb>nunc reviviſceret, Rex <emph type="italics"></emph>Alphonſus<emph.end type="italics"></emph.end>vel ſimplicitatem vel harmoniæ <lb></lb>gratiam in ea deſideraret. </s> <s>Itaque Naturæ majeſtatem propius jam <lb></lb>licet intueri, & dulciſſima contemplatione frui, Conditorem vero <lb></lb>ac Dominum Univerſorum impenſius colere & venerari, qui fructus <lb></lb>eſt Philoſophiæ multo uberrimus. </s> <s>Cæcum eſſe oportet, qui ex <lb></lb>optimis & ſapientiſſimis rerum ſtructuris non ſtatim videat Fabri<lb></lb>catoris Omnipotentis infinitam ſapientiam & bonitatem: inſanum, <lb></lb>qui profiteri nolit. </s></p> <p type="main"> <s>Extabit igitur Eximium NEWTONI Opus adverſus Atheorum <lb></lb>impetus munitiſſimum præſidium: neque enim alicunde felicius, <lb></lb>quam ex hac pharetra, contra impiam Catervam tela deprompſeris. </s> <s><lb></lb>Hoc ſenſit pridem, & in pereruditis Concionibus Anglice Latineque <lb></lb>editis, primus egregie demonſtravit Vir in omni Literarum genere <lb></lb>præclarus idemque bonarum Artium fautor eximius RICHARDUS <lb></lb>BENTLEIUS, Sæculi ſui & Academiæ noſtræ magnum Orna<lb></lb>mentum, Collegii noſtri <emph type="italics"></emph>S. Trinitatis<emph.end type="italics"></emph.end>Magiſter digniſſimus & in<lb></lb>tegerrimus. </s> <s>Huic ego me pluribus nominibus obſtrictum fateri <pb xlink:href="039/01/026.jpg"></pb>debeo: Huic & Tuas quæ debentur gratias, Lector benevole, non <lb></lb>denegabis. </s> <s>Is enim, cum a longo tempore Celeberrimi Auctoris <lb></lb>amicitia intima frueretur, (qua etiam apud Poſteros cenſeri non <lb></lb>minoris æſtimat, quam propriis Scriptis quæ literato orbi in de<lb></lb>liciis ſunt inclareſcere) Amici ſimul famæ & ſcientiarum incre<lb></lb>mento conſuluit. </s> <s>Itaque cum Exemplaria prioris Editionis rariſ<lb></lb>ſima admodum & immani pretio coemenda ſupereſſent; ſuaſit Ille <lb></lb>crebris efflagitationibus & tantum non objurgando perpulit deNI<lb></lb>que Virum Præſtantiſſimum, nec modeſtia minus quam eruditi<lb></lb>one ſumma Inſignem, ut novam hanc Operis Editionem, per om<lb></lb>nia elimatam denuo & egregiis inſuper acceſſionibus ditatam, ſuis <lb></lb>ſumptibus & auſpiciis prodire pateretur: Mihi vero, pro jure <lb></lb>ſuo, penſum non ingratum demandavit, ut quam poſſet emendate <lb></lb>id fieri curarem. </s></p> <p type="main"> <s><emph type="italics"></emph>Cantabrigiæ,<emph.end type="italics"></emph.end><lb></lb>Maii 12. 1713. </s></p> <p type="main"> <s>ROGERUS COTES Collegii <emph type="italics"></emph>S. Trinitatis<emph.end type="italics"></emph.end>Socius, <lb></lb>Aſtronomiæ & Philoſophiæ Experimentalis <lb></lb>Profeſſor <emph type="italics"></emph>Plumianus.<emph.end type="italics"></emph.end></s></p></chap><chap><pb xlink:href="039/01/027.jpg"></pb> <p type="main"> <s><emph type="center"></emph>INDEX CAPITUM <lb></lb>TOTIUS OPERIS.<emph.end type="center"></emph.end></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"></emph>DE MOTU CORPORUM LIBER PRIMUS.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>SECT. I. <emph type="italics"></emph>DE Methodo rationum primarum & ultima<lb></lb>rum.<emph.end type="italics"></emph.end>24 </s></p> <p type="main"> <s>SECT. II. <emph type="italics"></emph>De inventione Virium centripetarum.<emph.end type="italics"></emph.end>34 </s></p> <p type="main"> <s>SECT. III. <emph type="italics"></emph>De motu corporum in Conicis ſectionibus eccentri<lb></lb>cis.<emph.end type="italics"></emph.end>48 </s></p> <p type="main"> <s>SECT. IV. <emph type="italics"></emph>De inventione Orbium Elliptieorum, Parabolieorum <lb></lb>& Hyperbolieorum ex Umbilico dato.<emph.end type="italics"></emph.end>59 </s></p> <p type="main"> <s>SECT. V. <emph type="italics"></emph>De inventione Orbium ubi Umbilicus neuter datur.<emph.end type="italics"></emph.end>66 </s></p> <p type="main"> <s>SECT. VI. <emph type="italics"></emph>De inventione Motuum in Orbibus datis.<emph.end type="italics"></emph.end>97 </s></p> <p type="main"> <s>SECT. VII. <emph type="italics"></emph>De corporum Aſcenſu & Deſcenſu rectilineo.<emph.end type="italics"></emph.end>105 </s></p> <p type="main"> <s>SECT. VII. <emph type="italics"></emph>De inventione Orbium in quibus corpora Viribus <lb></lb>quibuſcunque centripetis agitata revolvuntur.<emph.end type="italics"></emph.end>114 </s></p> <p type="main"> <s>SECT. IX. <emph type="italics"></emph>De Motu corporum in Orbibus mobilibus, deque <lb></lb>Motu Apſidum.<emph.end type="italics"></emph.end>121 </s></p> <p type="main"> <s>SECT. X. <emph type="italics"></emph>De Motu corporum in Superficiebus datis, deque <lb></lb>Funependulorum Motu reciproco.<emph.end type="italics"></emph.end>132 </s></p> <p type="main"> <s>SECT. XI. <emph type="italics"></emph>De Motu corporum Viribus centripetis ſe mutuo pe<lb></lb>tentium.<emph.end type="italics"></emph.end>147 </s></p> <p type="main"> <s>SECT. XII. <emph type="italics"></emph>De corporum Sphærieorum Viribus attractivis.<emph.end type="italics"></emph.end>173 </s></p><pb xlink:href="039/01/028.jpg"></pb> <p type="main"> <s>SECT. XIII. <emph type="italics"></emph>De corporum non Sphærieorum Viribus attracti<lb></lb>vis.<emph.end type="italics"></emph.end>192 </s></p> <p type="main"> <s>SECT. XIV. <emph type="italics"></emph>De Motu corporum Minimorum, quæ Veribus cen<lb></lb>tripetis ad ſingulas Magni alicujus corporis partes ten<lb></lb>dentibus agitantur.<emph.end type="italics"></emph.end>203 </s></p> <p type="main"> <s><emph type="center"></emph>DE MOTU CORPORUM LIBER SECUNDUS.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>SECT. I. <emph type="italics"></emph>DE Motu corporum quibus reſiſtitur in ratione <lb></lb>Velocitatis.<emph.end type="italics"></emph.end>211 </s></p> <p type="main"> <s>SECT. II. <emph type="italics"></emph>De Motu corporum quibus reſiſtitur in duplicata ra<lb></lb>tione Velocitatis.<emph.end type="italics"></emph.end>220 </s></p> <p type="main"> <s>SECT. III. <emph type="italics"></emph>De Motu corporum quibus reſiſtitur partim in ratione <lb></lb>Velocitatis, partim in ejuſdem ratione duplicata.<emph.end type="italics"></emph.end>245 </s></p> <p type="main"> <s>SECT. IV. <emph type="italics"></emph>De corporum Circulari motu in Mediis reſiſtentibus.<emph.end type="italics"></emph.end><lb></lb>253 </s></p> <p type="main"> <s>SECT. V. <emph type="italics"></emph>De denſitate & compreſſione Fluidorum, deque Hy<lb></lb>droſtatica.<emph.end type="italics"></emph.end>260 </s></p> <p type="main"> <s>SECT. VI. <emph type="italics"></emph>De Motu & Reſiſtentia corporum Funependulorum.<emph.end type="italics"></emph.end><lb></lb>272 </s></p> <p type="main"> <s>SECT. VII. <emph type="italics"></emph>De motu Fluidorum & reſiſtentia Projectilium.<emph.end type="italics"></emph.end>294 </s></p> <p type="main"> <s>SECT. VIII. <emph type="italics"></emph>De motu per Fluida propagato.<emph.end type="italics"></emph.end>329 </s></p> <p type="main"> <s>SECT. IX. <emph type="italics"></emph>De motu Circulari Fluidorum.<emph.end type="italics"></emph.end>345 </s></p> <p type="main"> <s><emph type="center"></emph>DE MUNDI SYSTEMATE LIBER TERTIUS.<emph.end type="center"></emph.end></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"></pb> <p type="main"> <s><emph type="center"></emph>PHILOSOPHIÆ <lb></lb>NATURALIS <lb></lb>Principia <lb></lb>MATHEMATICA.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIONES.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Quantitas Materiæ eſt menſura ejuſdem orta ex illius Denſitate & <lb></lb>Magnitudine conjunctim.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>AER, denſitate duplicata, in ſpatio etiam duplicato fit qua<lb></lb>druplus; in triplicato ſextuplus. </s> <s>Idem intellige de Nive & <lb></lb>Pulveribus per compreſſionem vel liquefactionem conden<lb></lb>ſatis. </s> <s>Et par eſt ratio corporum omnium, quæ per cauſas quaſcun<lb></lb>Q.E.D.verſimode condenſantur. </s> <s>Medii interea, ſi quod fuerit, in<lb></lb>terſtitia partium libere pervadentis, hic nullam rationem habeo. </s> <s><lb></lb>Hanc autem Quantitatem ſub nomine Corporis vel Maſſæ in ſe<lb></lb>quentibus paſſim intelligo. </s> <s>Innoteſcit ea per corporis cujuſque <lb></lb>Pondus. </s> <s>Nam Ponderi proportionalem eſſe reperi per experi<lb></lb>menta Pendulorum accuratiſſime inſtituta, uti poſthac docebitur. </s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Quantitas Motus eſt menſura ejuſdem orta ex Velocitate & Quan<lb></lb>titate Materiæ conjunctim.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Motus totius eſt ſumma motuum in partibus ſingulis; adeoque <lb></lb>in corpore duplo majore æquali cum velocitate duplus eſt, & du<lb></lb>pla cum velocitate quadruplus. </s></p><pb xlink:href="039/01/030.jpg" pagenum="2"></pb> <p type="main"> <s><emph type="center"></emph>DEFINITIO III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Materiæ Vis Inſita eſt potentia reſiſtendi, qua corpus unumquodque, <lb></lb>quantum in ſe eſt, perſeverat in ſtatu ſuo vel quieſcendi vel <lb></lb>movendi uniformiter in directum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Hæc ſemper proportionalis eſt ſuo corpori, neQ.E.D.ffert quic<lb></lb>quam ab Inertia maſſæ, niſi in modo concipiendi. </s> <s>Per inertiam <lb></lb>materiæ, fit ut corpus omne de ſtatu ſuo vel quieſcendi vel moven<lb></lb>di difficulter deturbetur. </s> <s>Unde etiam vis inſita nomine ſignifican<lb></lb>tiſſimo Vis Inertiæ dici poſſit. </s> <s>Exercet vero corpus hanc vim ſolum<lb></lb>modo in mutatione ſtatus ſui per vim aliam in ſe impreſſam facta; <lb></lb><expan abbr="eſtq;">eſtque</expan> exercitium ejus ſub diverſo reſpectu & Reſiſtentia & Impetus: <lb></lb>reſiſtentia, quatenus corpus ad conſervandum ſtatum ſuum relucta<lb></lb>tur vi impreſſæ; impetus, quatenus corpus idem, vi reſiſtentis ob<lb></lb>ſtaculi difficulter cedendo, conatur ſtatum ejus mutare. </s> <s>Vulgus <lb></lb>reſiſtentiam quieſcentibus & impetum moventibus tribuit: ſed mo<lb></lb>tus & quies, uti vulgo concipiuntur, reſpectu ſolo diſtinguuntur <lb></lb>ab invicem; <expan abbr="neq;">neque</expan> ſemper vere quieſcunt quæ vulgo tanquam quie<lb></lb>ſcentia ſpectantur. </s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Vis Impreſſa eſt actio in corpus exercita, ad mutandum ejus ſtatum <lb></lb>vel quieſcendi vel movendi uniformiter in directum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Conſiſtit hæc vis in actione ſola, neque poſt actionem permanet <lb></lb>in corpore. </s> <s>Perſeverat enim corpus in ſtatu omni novo per ſolam <lb></lb>vim inertiæ. </s> <s>Eſt autem vis impreſſa diverſarum originum, ut ex <lb></lb>Ictu, ex Preſſione, ex vi Centripeta. </s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Vis Centripeta eſt, qua corpora verſus punctum aliquod tanquam ad <lb></lb>Centrum undique trahuntur, impelluntur, vel utcunque tendunt.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hujus generis eſt Gravitas, qua corpora tendunt ad centrum ter<lb></lb>ræ; Vis Magnetica, qua ferrum petit magnetem; & Vis illa, <lb></lb><expan abbr="quæcunq;">quæcunque</expan> ſit, qua Planetæ perpetuo retrahuntur a motibus rectili<lb></lb>neis, & in lineis curvis revolvi coguntur. </s> <s>Lapis, in funda circum-<pb xlink:href="039/01/031.jpg" pagenum="3"></pb>actus, a circumagente manu abire conatur; & conatu ſuo fundam <lb></lb>diſtendit, <expan abbr="eoq;">eoque</expan> fortius quo celerius revolvitur; &, quamprimum di<lb></lb>mittitur, avolat. </s> <s>Vim conatui illi contrariam, qua funda lapidem <lb></lb>in manum perpetuò retrahit & in orbe retinet, quoniam in manum <lb></lb>ceu orbis centrum dirigitur, Centripetam appello. </s> <s>Et par eſt ratio <lb></lb>corporum omnium, quæ in gyrum aguntur. </s> <s>Conantur ea omnia a <lb></lb>centris orbium recedere; & niſi adſit vis aliqua conatui iſti contra<lb></lb>ria, qua cohibeantur & in orbibus retineantur, quamQ.E.I.eò Centri<lb></lb>petam appello, abibunt in rectis lineis uniformi cum motu. </s> <s>Pro<lb></lb>jectile, ſi vi Gravitatis deſtitueretur, non deflecteretur in terram, ſed <lb></lb>in linea recta abiret in cælos; idque uniformi cum motu, ſi modo <lb></lb>aeris reſiſtentia tolleretur. </s> <s>Per gravitatem ſuam retrahitur a curſu <lb></lb>rectilineo & in terram perpetuo flectitur, idque magis vel minus <lb></lb>pro gravitate ſua & velocitate motus. </s> <s>Quo minor erit ejus gravitas pro quantitate materiæ vel major &c. </s> <s><lb></lb>vel major velocitas quacum projicitur, eo minus deviabit a curſu <lb></lb>rectilineo & longius perget. </s> <s>Si Globus plumbeus, data cum velo<lb></lb>citate ſecundum lineam horizontalem a montis alicujus vertice vi <lb></lb>pulveris tormentarii projectus, pergeret in linea curva ad diſtantiam <lb></lb>duorum milliarium, priuſquam in terram decideret: hic dupla cum <lb></lb>velocitate quaſi duplo longius pergeret, & decupla cum velocitate <lb></lb>quaſi decuplo longius: ſi modo aeris reſiſtentia tolleretur. </s> <s>Et augendo <lb></lb>velocitatem augeri poſſet pro lubitu diſtantia in quam projiceretur, <lb></lb>& minui curvatura lineæ quam deſcriberet, ita ut tandem caderet <lb></lb>ad diſtantiam graduum decem vel triginta vel nonaginta; vel etiam <lb></lb>ut terram totam circuiret priuſquam caderet; vel denique ut in <lb></lb>terram nunquam caderet, ſed in cælos abiret & motu abeundi per<lb></lb>geret in infinitum. </s> <s>Et eadem ratione, qua Projectile vi gravitatis <lb></lb>in orbem flecti poſſet & terram totam circuire, poteſt & Luna vel <lb></lb>vi gravitatis, ſi modo gravis ſit, vel alia quacunque vi, qua in ter<lb></lb>ram urgeatur, retrahi ſemper a curſu rectilineo terram verſus, & <lb></lb>in orbem ſuum flecti: & abſque tali vi Luna in orbe ſuo retineri <lb></lb>non poteſt. </s> <s>Hæc vis, ſi juſto minor eſſet, non ſatis flecteret Lunam <lb></lb>de curſu rectilineo: ſi juſto major, plus ſatis flecteret, ac de orbe <lb></lb>terram verſus deduceret. </s> <s>Requiritur quippe, ut ſit juſtæ magnitudinis: <lb></lb>& Mathematieorum eſt invenire Vim, qua corpus in dato quovis <lb></lb>orbe data cum velocitate accurate retineri poſſit; & viciſſim inve<lb></lb>nire Viam curvilineam, in quam corpus e dato quovis loco data <lb></lb>cum velocitate egreſſum a data vi flectatur. </s> <s>Eſt autem vis hujus cen<lb></lb>tripetæ Quantitas trium generum, Abſoluta, Acceleratrix, & Motrix. </s></p><pb xlink:href="039/01/032.jpg" pagenum="4"></pb> <p type="main"> <s><arrow.to.target n="note1"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note1"></margin.target>NI<lb></lb>ES.</s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Vis centripetæ Quantitas Abſoluta eſt menſura ejuſdem major vel minor <lb></lb>pro Efficacia cauſæ eam propagantis a centro per regiones in circuitu.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ut vis Magnetica pro mole magnetis vel intenſione virtutis major <lb></lb>in uno magnete, minor in alio. </s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Vis centripetæ Quantitas Acceleratrix eſt ipſius menſura Velocitati <lb></lb>proportionalis, quam dato tempore generat.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Uti Virtus magnetis ejuſdem major in minori diſtantia, minor <lb></lb>in majori: vel vis Gravitans major in vallibus, minor in cacumiNI<lb></lb>bus præaltorum montium, atque adhuc minor (ut poſthac patebit) <lb></lb>in majoribus diſtantiis a globo terræ; in æqualibus autem diſtan<lb></lb>tiis eadem undique, propterea quod corpora omnia cadentia (gra<lb></lb>via an levia, magna an parva) ſublata Aeris reſiſtentia, æqualiter <lb></lb>accelerat. </s></p> <p type="main"> <s><emph type="center"></emph>DEFINITIO VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Vis centripetæ Quantitas Motrix eſt ipſius menſura proportionalis. </s> <s><lb></lb>Motui, quem dato tempore generat.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Uti Pondus majus in majore corpore, minus in minore; inque <lb></lb>corpore eodem majus prope terram, minus in cælis. </s> <s>Hæc Quantitas <lb></lb>eſt corporis totius centripetentia ſeu propenſio in centrum, & (ut ita <lb></lb>dicam) Pondus; & innoteſcit ſemper per vim ipſi contrariam & æ<lb></lb>qualem, qua deſcenſus corporis impediri poteſt. </s></p> <p type="main"> <s>Haſce virium quantitates brevitatis gratia nominare licet vires <lb></lb>motrices, acceleratrices, & abſolutas; & diſtinctionis gratia referre ad <lb></lb>Corpora, centrum petentia, ad corporum Loca, & ad Centrum virium: <lb></lb>nimirum vim motricem ad Corpus, tanquam conatum & propenſio<lb></lb>nem totius in centrum ex propenſionibus omnium partium compoſi<lb></lb>tam; & vim acceleratricem ad Locum corporis, tanquam efficaciam <lb></lb>quandam, de centro per loca ſingula in circuitu diffuſam, ad movenda <lb></lb>corpora quæ in ipſis ſunt; vim autem abſolutam ad Centrum, tan<lb></lb>quam cauſa aliqua præditum, ſine qua vires motrices non propa<lb></lb>gantur per regiones in circuitu; ſive cauſa illa ſit corpus aliquod <lb></lb>centrale (quale eſt Magnes in centro vis magneticæ, vel Terra in <pb xlink:href="039/01/033.jpg" pagenum="5"></pb>centro vis gravitantis) ſive alia aliqua quæ non apparet. </s> <s>Mathe<lb></lb>maticus duntaxat eſt hic conceptus. </s> <s>Nam virium cauſas & ſedes phy<lb></lb>ſicas jam non expendo. </s></p> <p type="main"> <s>Eſt igitur vis acceleratrix ad vim motricem ut celeritas ad mo<lb></lb>tum. </s> <s>Oritur enim quantitas motus ex celeritate ducta in quanti<lb></lb>tatem materiæ, & vis motrix ex vi acceleratrice ducta in quantita<lb></lb>tem ejuſdem materiæ. </s> <s>Nam ſumma actionum vis acceleratricis in <lb></lb>ſingulas corporis particulas eſt vis motrix totius. </s> <s>Unde juxta <lb></lb>ſuperficiem Terræ, ubi gravitas acceleratrix ſeu vis gravitans in <lb></lb>corporibus univerſis eadem eſt, gravitas motrix ſeu pondus eſt ut <lb></lb>corpus: at ſi in regiones aſcendatur ubi gravitas acceleratrix fit mi<lb></lb>nor, pondus pariter minuetur, eritque ſemper ut corpus in <lb></lb>gravitatem acceleratricem ductum. </s> <s>Sic in regionibus ubi gravitas <lb></lb>acceleratrix duplo minor eſt, pondus corporis duplo vel triplo <lb></lb>minoris erit quadruplo vel ſextuplo minus. </s></p> <p type="main"> <s>Porro attractiones & impulſus eodem ſenſu acceleratrices & mo<lb></lb>trices nomino. </s> <s>Voces autem Attractionis, Impulſus, vel Propen<lb></lb>ſionis cujuſcunQ.E.I. centrum, indifferenter & pro ſe mutuo pro<lb></lb>miſcue uſurpo; has vires non Phyſice ſed Mathematice tantum con<lb></lb>ſiderando. </s> <s>Unde caveat lector, ne per hujuſmodi voces cogitet me <lb></lb>ſpeciem vel modum actionis cauſamve aut rationem Phyſicam ali<lb></lb>cubi definire, vel centris (quæ ſunt puncta Mathematica) vires <lb></lb>vere & Phyſice tribuere; ſi forte aut centra trahere, aut vires cen<lb></lb>trorum eſſe dixero. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hactenus voces minus notas, quo ſenſu in ſequentibus acci<lb></lb>piendæ ſint, explicare viſum eſt. </s> <s>Nam Tempus, Spatium, Locum <lb></lb>& Motum, ut omnibus notiſſima, non definio. </s> <s>Notandum tamen, quod <lb></lb>vulgus quantitates haſce non aliter quam ex relatione ad ſenſibilia <lb></lb>concipiat. </s> <s>Et inde oriuntur præjudicia quædam, quibus tollendis <lb></lb>convenit eaſdem in abſolutas & relativas, veras & apparentes, ma<lb></lb>thematicas & vulgares diſtingui. </s></p> <p type="main"> <s>I. </s> <s>Tempus Abſolutum, verum, & mathematicum, in ſe & natura <lb></lb>ſua <expan abbr="abſq;">abſque</expan> relatione ad externum quodvis, æquabiliter fluit, <expan abbr="alioq;">alioque</expan> <lb></lb>nomine dicitur Duratio: Relativum, apparens, & vulgare eſt ſenſibilis <lb></lb>& externa quævis Durationis per motum menſura (ſeu accurata <lb></lb>ſeu inæquabilis) qua vulgus vice veri temporis utitur; ut Hora, <lb></lb>Dies, Menſis, Annus. </s></p><pb xlink:href="039/01/034.jpg" pagenum="6"></pb><p><s><arrow.to.target n="note2"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note2"></margin.target></s></p> <p type="main"> <s>II. </s> <s>Spatium Abſolutum, natura ſua abſque relatione ad externum <lb></lb>quodvis, ſemper manet ſimilare & immobile: Relativum eſt ſpatii <lb></lb>hujus menſura ſeu dimenſio quælibet mobilis, quæ a ſenſibus noſtris <lb></lb>per ſitum ſuum ad corpora definitur, & a vulgo pro ſpatio immo<lb></lb>bili uſurpatur: uti dimenſio ſpatii ſubterranei, aerei vel cæleſtis <lb></lb>definita per ſitum ſuum ad Terram. </s> <s>Idem ſunt ſpatium abſolutum <lb></lb>& relativum, ſpecie & magnitudine; ſed non permanent idem ſem<lb></lb>per numero. </s> <s>Nam ſi Terra, verbi gratia, movetur; ſpatium Aeris <lb></lb>noſtri, quod relative & reſpectu Terræ ſemper manet idem, nunc <lb></lb>erit una pars ſpatii abſoluti in quam Aer tranſit, nunc alia pars ejus; <lb></lb>& ſic abſolute mutabitur perpetuo. </s></p> <p type="main"> <s>III. </s> <s>Locus eſt pars ſpatii quam corpus occupat, <expan abbr="eſtq;">eſtque</expan> pro ratione <lb></lb>ſpatii vel Abſolutus vel Relativus. </s> <s>Pars, inquam, ſpatii; non Situs <lb></lb>corporis, vel Superficies ambiens. </s> <s>Nam ſolidorum æqualium <lb></lb>æquales ſemper ſunt loci; Superficies autem ob diſſimilitudinem <lb></lb>figurarum ut plurimum inæquales ſunt; Situs vero proprie loquen<lb></lb>do quantitatem non habent, <expan abbr="neq;">neque</expan> tam ſunt loca quam affectiones <lb></lb>loeorum. </s> <s>Motus totius idem eſt cum ſumma motuum partium, <lb></lb>hoc eſt, tranſlatio totius de ſuo loco eadem eſt cum ſumma tranſla<lb></lb>tionum partium de locis ſuis; <expan abbr="adeoq;">adeoque</expan> locus totius idem cum ſumma <lb></lb>loeorum partium, & propterea internus & in corpore toto. </s></p> <p type="main"> <s>IV. </s> <s>Motus Abſolutus eſt tranſlatio corporis de loco abſoluto in <lb></lb>locum abſolutum, Relativus de relativo in relativum. </s> <s>Sic in navi <lb></lb>quæ velis paſſis fertur, relativus corporis Locus eſt navigii regio illa <lb></lb>in qua corpus verſatur, ſeu cavitatis totius pars illa quam corpus <lb></lb>implet, <expan abbr="quæq;">quæque</expan> adeo movetur una cum navi: & Quies relativa eſt <lb></lb>permanſio corporis in eadem illa navis regione vel parte cavita<lb></lb>tis. </s> <s>At quies Vera eſt permanſio corporis in eadem parte ſpatii <lb></lb>illius immoti in qua navis ipſa una cum cavitate ſua & contentis <lb></lb>univerſis movetur. </s> <s>Unde ſi Terra vere quieſcit, corpus quod rela<lb></lb>tive quieſcit in navi, movebitur vere & abſolute ea cum velocitate <lb></lb>qua navis movetur in Terra. </s> <s>Sin Terra etiam movetur; orietur <lb></lb>verus & abſolutus corporis motus, partim ex Terræ motu vero in <lb></lb>ſpatio immoto, partim ex navis motu relativo in Terra: & ſi cor<lb></lb>pus etiam movetur relative in navi; orietur verus ejus motus, par<lb></lb>tim ex vero motu Terræ in ſpatio immoto, partim ex relativis mo<lb></lb>tibus tum navis in Terra, tum corporis in navi; & ex his motibus <lb></lb>relativis orietur corporis motus relativus in Terra. </s> <s>Ut ſi Terræ pars <lb></lb>illa, ubi navis verſatur, moveatur vere in orientem cum velocitate <lb></lb>partium 10010; & velis <expan abbr="ventoq;">ventoque</expan> feratur navis in occidentem cum <lb></lb>velocitate partium decem; Nauta autem ambulet in navi ori-<pb xlink:href="039/01/035.jpg" pagenum="7"></pb>entem verſus cum velocitatis parte una: movebitur Nauta vere & <lb></lb>abſolute in ſpatio immoto cum velocitatis partibus 10001 in o<lb></lb>rientem, & relative in terra occidentem verſus cum velocitatis <lb></lb>partibus novem. </s></p> <p type="main"> <s>Tempus Abſolutum a relativo diſtinguitur in Aſtronomia per Æ<lb></lb>quationem temporis vulgi. </s> <s>Inæquales enim ſunt dies Naturales, <lb></lb>qui vulgo tanquam æquales promenſura temporis habentur. </s> <s>Hanc <lb></lb>inæqualitatem corrigunt Aſtronomi, ut ex veriore tempore </s> <s><lb></lb>motus cæleſtes. </s> <s>Poſſibile eſt, ut nullus ſit motus æquabilis quo <lb></lb>Tempus accurate menſuretur. </s> <s>Accelerari & retardari poſſunt motus <lb></lb>omnes, ſed fluxus temporis Abſoluti mutari nequit. </s> <s>Eadem eſt du<lb></lb>ratio ſeu perſeverantia exiſtentiæ rerum; ſive motus ſint celeres, ſive <lb></lb>tardi, ſive nulli: proinde hæc a menſuris ſuis ſenſibilibus merito <lb></lb>diſtinguitur, & ex iiſdem colligitur per Æquationem Aſtronomi<lb></lb>cam. </s> <s>Hujus autem æquationis in determinandis Phænomenis ne<lb></lb>ceſſitas, tum per experimentum Horologii Oſcillatorii, tum etiam <lb></lb>per eclipſes Satellitum Jovis evincitur. </s></p> <p type="main"> <s>Ut partium Temporis ordo eſt immutabilis, ſic etiam ordo par<lb></lb>tium Spatii. </s> <s>Moveantur hæ de locis ſuis, & movebuntur (ut ita <lb></lb>dicam) de ſeipſis. </s> <s>Nam tempora & ſpatia ſunt ſui ipſorum & <lb></lb>rerum omnium quaſi Loca. </s> <s>In Tempore quoad ordinem ſucceſſi<lb></lb>onis; in Spatio quoad ordinem ſitus locantur univerſa. </s> <s>De illo<lb></lb>rum eſſentia eſt ut ſint Loca: & loca primaria moveri abſurdum <lb></lb>eſt. </s> <s>Hæc ſunt igitur abſoluta Loca; & ſolæ tranſlationes de his lo<lb></lb>cis ſunt abſoluti Motus. </s></p> <p type="main"> <s>Verum quoniam hæ Spatii partes videri nequeunt, & ab invi<lb></lb>cem per ſenſus noſtros diſtingui; earum vice adhibemus menſuras <lb></lb>ſenſibiles. </s> <s>Ex poſitionibus enim & diſtantiis rerum a corpore ali<lb></lb>quo, quod ſpectamus ut immobile, deſinimus loca univerſa: deinde <lb></lb>etiam & omnes motus æſtimamus cum reſpectu ad prædicta loca, <lb></lb>quatenus corpora ab iiſdem transferri concipimus. </s> <s>Sic vice loco<lb></lb>rum & motuum abſolutorum relativis utimur; nec incommode in <lb></lb>rebus humanis: in Philoſophicis autem abſtrahendum eſt a ſenſibus. </s> <s><lb></lb>Fieri etenim poteſt, ut nullum revera quieſcat corpus, ad quod loca <lb></lb>motuſque referantur. </s></p> <p type="main"> <s>Diſtinguuntur autem Quies & Motus abſoluti & relativi ab invi<lb></lb>cem per Proprietates ſuas & Cauſas & Effectus. </s> <s>Quietis proprietas <lb></lb>eſt, quod corpora vere quieſcentia quieſcunt inter ſe. </s> <s>Ideoque <lb></lb>cum poſſibile ſit, ut corpus aliquod in regionibus Fixarum, aut longe <lb></lb>ultra, quieſcat abſolute; ſciri autem non poſſit ex ſitu corporum <lb></lb>ad invicem in regionibus noſtris, horumne aliquod ad longin-</s></p><pb xlink:href="039/01/036.jpg" pagenum="8"></pb> <p type="main"> <s><arrow.to.target n="note3"></arrow.to.target>quum illud datam poſitionem ſervet necne; quies vera ex horum <lb></lb>ſitu inter ſe definiri nequit. </s></p> <p type="margin"> <s><margin.target id="note3"></margin.target></s></p> <p type="main"> <s>Motus proprietas eſt, quod partes, quæ datas ſervant poſitiones <lb></lb>ad tota, participant motus eorundem totorum. </s> <s>Nam Gyrantium <lb></lb>partes omnes conantur recedere ab axe motus, & Progredientium <lb></lb>impetus oritur ex conjuncto impetu partium ſingularum. </s> <s>Motis <lb></lb>igitur corporibus ambientibus, moventur quæ in ambientibus rela<lb></lb>tive quieſcunt. </s> <s>Et propterea motus verus & abſolutus definiri ne<lb></lb>quit per tranſlationem e vicinia corporum, quæ tanquam quieſcen<lb></lb>tia ſpectantur. </s> <s>Debent enim corpora externa non ſolum tanquam qui<lb></lb>eſcentia ſpectari, ſed etiam vere quieſcere. </s> <s>Alioquin incluſa om<lb></lb>nia, præter tranſlationem e vicinia ambientium, participabunt <lb></lb>etiam ambientium motus veros; & ſublata illa tranſlatione non <lb></lb>vere quieſcent, ſed tanquam quieſcentia ſolummodo ſpectabun<lb></lb>tur. </s> <s>Sunt enim ambientia ad incluſa, ut totius pars exterior ad <lb></lb>partem interiorem, vel ut cortex ad nucleum. </s> <s>Moto autem cor<lb></lb>tice, nucleus etiam, <expan abbr="abſq;">abſque</expan> tranſlatione de vicinia corticis, ceu pars <lb></lb>totius movetur. </s></p> <p type="main"> <s>Præcedenti proprietati affinis eſt, quod moto Loco movetur una <lb></lb>Locatum: adeoque corpus, quod de loco moto movetur, participat <lb></lb>etiam loci ſui motum. </s> <s>Motus igitur omnes, qui de locis motis <lb></lb>fiunt, ſunt partes ſolummodo motuum integrorum & abſolutorum: <lb></lb>& motus omnis integer componitur ex motu corporis de loco ſuo <lb></lb>primo, & motu loci hujus de loco ſuo, & ſic deinceps; uſQ.E.D.m <lb></lb>perveniatur ad locum immotum, ut in exemplo Nautæ ſupra me<lb></lb>morato. </s> <s>Unde motus integri & abſoluti non niſi per loca immota <lb></lb>definiri poſſunt: & propterea hos ad loca immota, relativos ad mo<lb></lb>bilia ſupra retuli. </s> <s>Loca autem immota non ſunt, niſi quæ omnia <lb></lb>ab infinito in infinitum datas ſervant poſitiones ad invicem; atque <lb></lb>adeo ſemper manent immota, ſpatiumque conſtituunt quod Immo<lb></lb>bile appello. </s></p> <p type="main"> <s>Cauſæ, quibus motus veri & relativi diſtinguuntur ab invicem, <lb></lb>ſunt Vires in corpora impreſſæ ad motum generandum. </s> <s>Motus <lb></lb>verus nec generatur nec mutatur, niſi per vires in ipſum corpus mo<lb></lb>tum impreſſas: at motus relativus generari & mutari poteſt <expan abbr="abſq;">abſque</expan> <lb></lb>viribus impreſſis in hoc corpus. </s> <s>Sufficit enim ut imprimantur in <lb></lb>alia ſolum corpora ad quæ fit relatio, ut iis cedentibus mutetur <lb></lb>relatio illa in qua hujus quies vel motus relativus conſiſtit. </s> <s>Rur<lb></lb>ſum motus verus a viribus in corpus motum impreſſis ſemper muta<lb></lb>tur; at motus relativus ab his viribus non mutatur neceſſario. </s> <s>Nam <lb></lb>ſi eædem vires in alia etiam corpora, ad quæ ſit relatio, ſic impri-<pb xlink:href="039/01/037.jpg" pagenum="9"></pb>mantur ut ſitus relativus conſervetur, conſervabitur relatio in qua <lb></lb>motus relativus conſiſtit. </s> <s>Mutari igitur poteſt motus omnis relati<lb></lb>vus ubi verus conſervatur, & conſervari ubi verus mutatur; & prop<lb></lb>terea motus verus in ejuſmodi relationibus minime conſiſtit. </s></p> <p type="main"> <s>Effectus quibus motus abſoluti & relativi diſtinguuntur ab invi<lb></lb>cem, ſunt vires recedendi ab axe motus circularis. </s> <s>Nam in motu <lb></lb>circulari nude relativo hæ vires nullæ ſunt, in vero autem & abſo<lb></lb>luto majores vel minores pro quantitate motus. </s> <s>Si pendeat ſitula <lb></lb>a filo prælongo, agaturque perpetuo in orbem, donec filum a con<lb></lb>torſione admodum rigeſcat, dein impleatur aqua, & una cum aqua <lb></lb>quieſcat; tum vi aliqua ſubitanea agatur motu contrario in orbem, <lb></lb>& filo ſe relaxante, diutius perſeveret in hoc motu; ſuperficies a<lb></lb>quæ ſub initio plana erit, quemadmodum ante motum vaſis: at <lb></lb>poſtquam, vi in aquam paulatim impreſſa, effecit vas, ut hæc quoque <lb></lb>ſenſibiliter revolvi incipiat; recedet ipſa paulatim a medio, aſcen<lb></lb>detque ad latera vaſis, figuram concavam induens, (ut ipſe exper<lb></lb>tus ſum) & incitatiore ſemper motu aſcendet magis & magis, do<lb></lb>nec revolutiones in æqualibus cum vaſe temporibus peragendo, <lb></lb>quieſcat in eodem relative. </s> <s>Indicat hic aſcenſus conatum rece<lb></lb>dendi ab axe motus, & per talem conatum innoteſcit & menſura<lb></lb>tur motus aquæ circularis verus & abſolutus, motuique relativo <lb></lb>hic omnino contrarius. </s> <s>Initio, ubi maximus erat aquæ motus rela<lb></lb>tivus in vaſe, motus ille nullum excitabat conatum recedendi ab <lb></lb>axe: aqua non petebat circumferentiam aſcendendo ad latera va<lb></lb>ſis, ſed plana manebat, & propterea motus illius circularis verus <lb></lb>nondum inceperat. </s> <s>Poſtea vero, ubi aquæ motus relativus decre<lb></lb>vit, aſcenſus ejus ad latera vaſis indicabat conatum recedendi ab <lb></lb>axe; atque hic conatus monſtrabat motum illius circularem verum <lb></lb>perpetuo creſcentem, ac tandem maximum factum ubi aqua quie<lb></lb>ſcebat in vaſe relative. </s> <s>Igitur conatus iſte non pendet a tranſla<lb></lb>tione aquæ reſpectu corporum ambientium, & propterea motus cir<lb></lb>cularis verus per tales tranſlationes definiri nequit. </s> <s>Unicus eſt cor<lb></lb>poris cujuſque revolventis motus vere circularis, conatui unico tan<lb></lb>quam proprio & adæquato effectui reſpondens: motus autem rela<lb></lb>tivi pro variis relationibus ad externa innumeri ſunt; & relationum <lb></lb>inſtar, effectibus veris omnino deſtituuntur, niſi quatenus verum <lb></lb>illum & unicum motum participant. </s> <s>Unde & in Syſtemate eorum <lb></lb>qui Cælos noſtros infra Cælos Fixarum in orbem revolvi volunt, <lb></lb>& Planetas ſecum deferre; ſingulæ Cælorum partes, & Planetæ <lb></lb>qui relative quidem in Cælis ſuis proximis quieſcunt, moven-<pb xlink:href="039/01/038.jpg" pagenum="10"></pb><arrow.to.target n="note4"></arrow.to.target>tur vere. </s> <s>Mutant enim poſitiones ſuas ad invicem (ſecus quam fit <lb></lb>in vere quieſcentibus) unaque cum cælis delati participant eorum <lb></lb>motus, & ut partes revolventium totorum, ab eorum axibus rece<lb></lb>dere conantur. </s></p> <p type="margin"> <s><margin.target id="note4"></margin.target>NI<lb></lb>ES.</s></p> <p type="main"> <s>Igitur quantitates relativæ non ſunt eæ ipſæ quantitates, quarum <lb></lb>nomina præ ſe ferunt, ſed earum menſuræ illæ ſenſibiles (veræ an <lb></lb>errantes) quibus vulgus loco quantitatum menſuratarum utitur. </s> <s>At <lb></lb>ſi ex uſu definiendæ ſunt verborum ſignificationes; per nomina il<lb></lb>la Temporis, Spatii, Loci & Motus proprie intelligendæ erunt hæ <lb></lb>menſuræ; & ſermo erit inſolens & pure Mathematicus, ſi quantita<lb></lb>tes menſuratæ hic intelligantur. </s> <s>Proinde vim inferunt Sacris <lb></lb>Literis, qui voces haſce de quantitatibus menſuratis ibi interpre<lb></lb>tantur. </s> <s>Neque minus contaminant Matheſin & Philoſophiam, <lb></lb>qui quantitates veras cum ipſarum relationibus & vulgaribus men<lb></lb>furis confundunt. </s></p> <p type="main"> <s>Motus quidem veros corporum ſingulorum cognoſcere, & ab <lb></lb>apparentibus actu diſcriminare, difficillimum. </s> <s>eſt propterea quod <lb></lb>partes ſpatii illius immobilis, in quo corpora vere moventur, non <lb></lb>incurrunt in ſenſus. </s> <s>Cauſa tamen non eſt prorſus deſperata. </s> <s>Nam <lb></lb>ſuppetunt argumenta, partim ex motibus apparentibus qui ſunt <lb></lb>motuum verorum differentiæ, partim ex viribus quæ ſunt mo<lb></lb>tuum verorum cauſæ & effectus. </s> <s>Ut ſi globi duo, ad datam ab in<lb></lb>vicem diſtantiam filo intercedente connexi, revolverentur circa <lb></lb>commune gravitatis centrum; innoteſceret ex tenſione fili cona<lb></lb>tus globorum recedendi ab axe motus, & inde quantitas motus <lb></lb>circularis computari poſſet. </s> <s>Deinde ſi vires quælibet æquales in <lb></lb>alternas globorum facies ad motum circularem augendum vel mi<lb></lb>nuendum ſimul imprimerentur, innoteſceret ex aucta vel diminuta <lb></lb>fili tenſione augmentum vel decrementum motus; & inde tandem <lb></lb>inveniri poſſent facies globorum in quas vires imprimi deberent, <lb></lb>ut motus maxime augeretur; id eſt, facies poſticæ, ſive quæ in mo<lb></lb>tu circulari ſequuntur. </s> <s>Cognitis autem faciebus quæ ſequuntur, <lb></lb>& faciebus oppoſitis quæ præcedunt, cognoſceretur determinatio <lb></lb>motus. </s> <s>In hunc modum inveniri poſſet & quantitas & determi<lb></lb>natio motus hujus circularis in vacuo quovis immenſo, ubi nihil <lb></lb>extaret externum & ſenſibile quocum globi conferri poſſent. </s> <s>Si <lb></lb>jam conſtituerentur in ſpatio illo corpora aliqua longinqua datam <lb></lb>inter ſe poſitionem ſervantia, qualia ſunt Stellæ Fixæ in regionibus <lb></lb>noſtris: ſciri quidem non poſſet ex relativa globorum tranſlatione <lb></lb>inter corpora, utrum his an illis tribuendus eſſet motus. </s> <s>At ſi <pb xlink:href="039/01/039.jpg" pagenum="11"></pb>attenderetur ad filum, & deprenderetur tenſionem ejus illam ipſam <lb></lb>eſſe quam motus globorum requireret; concludere liceret mo<lb></lb>tum eſſe globorum, & corpora quieſcere; & tum demum ex <lb></lb>tranſlatione globorum inter corpora, determinationem hujus <lb></lb>motus colligere. </s> <s>Motus autem veros ex eorum cauſis, effecti<lb></lb>bus, & apparentibus differentiis colligere; & contra ex motibus <lb></lb>ſeu veris ſeu apparentibus eorum cauſas & effectus, docebitur <lb></lb>fuſius in ſequentibus. </s> <s>Hunc enim in finem Tractatum ſequentem <lb></lb>compoſui. <pb xlink:href="039/01/040.jpg" pagenum="12"></pb><arrow.to.target n="note5"></arrow.to.target></s></p></chap><chap> <p type="margin"> <s><margin.target id="note5"></margin.target>TA,</s></p> <p type="main"> <s><emph type="center"></emph>AXIOMATA, <lb></lb>SIVE <lb></lb>LEGES MOTUS.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="center"></emph>LEX I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corpus omne perſeverare in ſtatu ſuo quieſcendi vel movendi uNI<lb></lb>formiter in directum, niſi quatenus a viribus impreſſis cogitur <lb></lb>ſtatum illum mutare.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>PRojectilia perſeverant in motibus ſuis, niſi quatenus a reſi<lb></lb>ſtentia aeris retardantur, & vi gravitatis impelluntur deorſum. </s> <s><lb></lb>Trochus, cujus partes cohærendo perpetuo retrahunt ſeſe a mo<lb></lb>tibus rectilineis, non ceſſat rotari, niſi quatenus ab aere retardatur. </s> <s><lb></lb>Majora autem Planetarum & Cometarum corpora motus ſuos & <lb></lb>progreſſivos & circulares in ſpatiis minus reſiſtentibus factos con<lb></lb>ſervant diutius. </s></p> <p type="main"> <s><emph type="center"></emph>LEX II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Mutationem motus proportionalem eſſe vi motrici impreſſæ, & fieri <lb></lb>ſecundum lineam rectam qua vis illa imprimitur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si vis aliqua motum quemvis generet; dupla duplum, tripla tri<lb></lb>plum generabit, ſive ſimul & ſemel, ſive gradatim & ſucceſſive im<lb></lb>preſſa fuerit. </s> <s>Et hic motus (quoniam in eandem ſemper plagam <lb></lb>cum vi generatrice determinatur) ſi corpus antea movebatur, mo<lb></lb>tui ejus vel conſpiranti additur, vel contrario ſubducitur, vel obli<lb></lb>quo oblique adjicitur, & cum eo ſecundum utriuſQ.E.D.termina<lb></lb>tionem componitur. </s></p><pb xlink:href="039/01/041.jpg" pagenum="13"></pb> <p type="main"> <s><emph type="center"></emph>LEX III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Actioni contrariam ſemper & æqualem eſſe reactionem: ſive cor<lb></lb>porum duorum actiones in ſe mutuo ſemper eſſe æquales & in par<lb></lb>tes contrarias dirigi.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Quicquid premit vel trahit alterum, tantundem ab eo premitur <lb></lb>vel trahitur. </s> <s>Si quis lapidem digito premit, premitur & hujus <lb></lb>digitus a lapide. </s> <s>Si equus lapidem funi alligatum trahit, retrahe<lb></lb>tur etiam & equus (ut ita dicam) æqualiter in lapidem: nam funis <lb></lb>utrinQ.E.D.ſtentus eodem relaxandi ſe conatu urgebit equum ver<lb></lb>ſus lapidem, ac lapidem verſus equum; tantumQ.E.I.pediet pro<lb></lb>greſſum unius quantum promovet progreſſum alterius. </s> <s>Si corpus <lb></lb>aliquod in corpus aliud impingens, motum ejus vi ſua quomodo<lb></lb>cunque mutaverit, idem quoque viciſſim in motu proprio eandem <lb></lb>mutationem in partem contrariam vi alterius ob æqualitatem preſ<lb></lb>ſionis mutuæ) ſubibit. </s> <s>His actionibus æquales fiunt mutationes, <lb></lb>non velocitatum, ſed motuum; ſcilicet in corporibus non aliunde <lb></lb>impeditis. </s> <s>Mutationes enim velocitatum, in contrarias itidem <lb></lb>partes factæ, quia motus æqualiter mutantur, ſunt corporibus re<lb></lb>ciproce proportionales. </s> <s>Obtinet etiam hæc Lex in Attractionibus, <lb></lb>ut in Scholio proximo probabitur. </s></p> <p type="main"> <s><emph type="center"></emph>COROLLARIUM I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Corpus viribus conjunctis diagonalem parallelogrammi eodem tem<lb></lb>pore deſcribere, quo latera ſeparatis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si corpus dato tempore, vi ſola <lb></lb><figure id="id.039.01.041.1.jpg" xlink:href="039/01/041/1.jpg"></figure><lb></lb><emph type="italics"></emph>M<emph.end type="italics"></emph.end>in loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>impreſſa, ferretur uNI<lb></lb>formi cum motu ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>B<emph.end type="italics"></emph.end>; & vi <lb></lb>ſola <emph type="italics"></emph>N<emph.end type="italics"></emph.end>in eodem loco impreſſa, fer<lb></lb>retur ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>C:<emph.end type="italics"></emph.end>compleatur pa<lb></lb>rallelogrammum <emph type="italics"></emph>ABDC,<emph.end type="italics"></emph.end>& vi utra<lb></lb>que feretur id eodem tempore in diagonali ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>D.<emph.end type="italics"></emph.end>Nam quo<lb></lb>niam vis <emph type="italics"></emph>N<emph.end type="italics"></emph.end>agit ſecundum lineam <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>parallelam, hæc vis per <lb></lb>Legem 11 nihil mutabit velocitatem accedendi ad lineam illam <emph type="italics"></emph>BD<emph.end type="italics"></emph.end><lb></lb>a vi altera genitam. </s> <s>Accedet igitur corpus eodem tempore ad lineam <lb></lb><emph type="italics"></emph>BD,<emph.end type="italics"></emph.end>ſive vis <emph type="italics"></emph>N<emph.end type="italics"></emph.end>imprimatur, ſive non; atque adeo in fine illius tempo<lb></lb>ris reperietur alicubi in linea illa <emph type="italics"></emph>BD.<emph.end type="italics"></emph.end>Eodem argumento in fine tem<lb></lb>poris ejuſdem reperietur alicubi in linea <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end>& idcirco in utriuſque <lb></lb>lineæ concurſu <emph type="italics"></emph>D<emph.end type="italics"></emph.end>reperiri neceſſe eſt. </s> <s>Perget autem motu rectili<lb></lb>neo ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>D<emph.end type="italics"></emph.end>per Legem 1. <pb xlink:href="039/01/042.jpg" pagenum="14"></pb><arrow.to.target n="note6"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note6"></margin.target>TA, <lb></lb>E</s></p> <p type="main"> <s><emph type="center"></emph>COROLLARIUM II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Et hinc patet compoſitio vis directæ<emph.end type="italics"></emph.end>AD <emph type="italics"></emph>ex viribus quibuſvis <lb></lb>obliquis<emph.end type="italics"></emph.end>AB <emph type="italics"></emph>&<emph.end type="italics"></emph.end>BD, <emph type="italics"></emph>& viciſſim reſolutio vis cujuſvis directæ<emph.end type="italics"></emph.end><lb></lb>AD <emph type="italics"></emph>in obliquas quaſcunque<emph.end type="italics"></emph.end>AB <emph type="italics"></emph>&<emph.end type="italics"></emph.end>BD.</s><s> <emph type="italics"></emph>Quæ quidem compoſitio <lb></lb>& reſolutio abunde confirmatur ex Mechanica.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Ut ſi de rotæ alicujus centro <emph type="italics"></emph>O<emph.end type="italics"></emph.end>exeuntes radii inæquales <emph type="italics"></emph>OM, <lb></lb>ON<emph.end type="italics"></emph.end>filis <emph type="italics"></emph>MA, NP<emph.end type="italics"></emph.end>ſuſtineant pondera <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& quærantur vi<lb></lb>res ponderum ad movendam rotam: Per centrum <emph type="italics"></emph>O<emph.end type="italics"></emph.end>agatur recta <lb></lb><emph type="italics"></emph>KOL<emph.end type="italics"></emph.end>filis perpendiculariter occurrens in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>centroque <emph type="italics"></emph>O<emph.end type="italics"></emph.end>& <lb></lb>intervallorum <emph type="italics"></emph>OK, OL<emph.end type="italics"></emph.end>majore <emph type="italics"></emph>OL<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.042.1.jpg" xlink:href="039/01/042/1.jpg"></figure><lb></lb>deſcribatur circulus occurrens filo <lb></lb><emph type="italics"></emph>MA<emph.end type="italics"></emph.end>in <emph type="italics"></emph>D:<emph.end type="italics"></emph.end>& actæ rectæ <emph type="italics"></emph>OD<emph.end type="italics"></emph.end>pa<lb></lb>rallela ſit <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>& perpendicularis <lb></lb><emph type="italics"></emph>DC.<emph.end type="italics"></emph.end>Quoniam nihil refert, utrum <lb></lb>filorum puncta <emph type="italics"></emph>K, L, D<emph.end type="italics"></emph.end>affixa ſint <lb></lb>an non affixa ad planum rotæ; pon<lb></lb>dera idem valebunt, ac ſi ſuſpende<lb></lb>rentur a punctis <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L.<emph.end type="italics"></emph.end><lb></lb>Ponderis autem <emph type="italics"></emph>A<emph.end type="italics"></emph.end>exponatur vis to<lb></lb>ta per lineam <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>& hæc reſolvetur <lb></lb>in vires <emph type="italics"></emph>AC, CD,<emph.end type="italics"></emph.end>quarum <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>trahendo radium <emph type="italics"></emph>OD<emph.end type="italics"></emph.end>directe a cen<lb></lb>tro nihil valet ad movendam rotam; vis autem altera <emph type="italics"></emph>DC,<emph.end type="italics"></emph.end>trahen<lb></lb>do radium <emph type="italics"></emph>DO<emph.end type="italics"></emph.end>perpendiculariter, idem valet ac ſi perpendiculari<lb></lb>ter traheret radium <emph type="italics"></emph>OL<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>OD<emph.end type="italics"></emph.end>æqualem; hoc eſt, idem atque <lb></lb>pondus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſi modo pondus illud ſit ad pondus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ut vis <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>ad <lb></lb>vim <emph type="italics"></emph>DA,<emph.end type="italics"></emph.end>id eſt (ob ſimilia triangula <emph type="italics"></emph>ADC, DOK,<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>OK<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>OD<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>OL.<emph.end type="italics"></emph.end>Pondera igitur <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>quæ ſunt reciproce ut <lb></lb>radii in directum poſiti <emph type="italics"></emph>OK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>OL,<emph.end type="italics"></emph.end>idem pollebunt, & ſic conſi<lb></lb>ſtent in æquilibrio: quæ eſt proprietas notiſſima Libræ, Vectis, & <lb></lb>Axis in Peritrochio. </s> <s>Sin pondus alterutrum ſit majus quam in hac <lb></lb>ratione, erit vis ejus ad movendam rotam tanto major. </s></p> <p type="main"> <s>Quod ſi pondus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>ponderi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>æquale partim ſuſpendatur filo <emph type="italics"></emph>Np,<emph.end type="italics"></emph.end><lb></lb>partim incumbat plano obliquo <emph type="italics"></emph>pG:<emph.end type="italics"></emph.end>agantur <emph type="italics"></emph>pH, NH,<emph.end type="italics"></emph.end>prior ho<lb></lb>rizonti, poſterior plano <emph type="italics"></emph>pG<emph.end type="italics"></emph.end>perpendicularis; & ſi vis ponderis <emph type="italics"></emph>p<emph.end type="italics"></emph.end><lb></lb>deorſum tendens, exponatur per lineam <emph type="italics"></emph>pH,<emph.end type="italics"></emph.end>reſolvi poteſt hæc in <lb></lb>vires <emph type="italics"></emph>pN, HN.<emph.end type="italics"></emph.end>Si filo <emph type="italics"></emph>pN<emph.end type="italics"></emph.end>perpendiculare eſſet planum aliquod <lb></lb><emph type="italics"></emph>pQ,<emph.end type="italics"></emph.end>ſecans planum alterum <emph type="italics"></emph>pG<emph.end type="italics"></emph.end>in linea ad horizontem paral<lb></lb>lela; & pondas <emph type="italics"></emph>p<emph.end type="italics"></emph.end>his planis <emph type="italics"></emph>pQ, pG<emph.end type="italics"></emph.end>ſolummodo incumberet; ur-<pb xlink:href="039/01/043.jpg" pagenum="15"></pb>geret illud hæc plana viribus <emph type="italics"></emph>pN, HN<emph.end type="italics"></emph.end>perpendiculariter, nimirun <lb></lb>planum <emph type="italics"></emph>pQ<emph.end type="italics"></emph.end>vi <emph type="italics"></emph>pN,<emph.end type="italics"></emph.end>& planum <emph type="italics"></emph>pG<emph.end type="italics"></emph.end>vi <emph type="italics"></emph>HN.<emph.end type="italics"></emph.end>Ideoque ſi tollatur pla<lb></lb>num <emph type="italics"></emph>pQ,<emph.end type="italics"></emph.end>ut pondus tendat filum; quoniam filum ſuſtinendo pon<lb></lb>dus jam vicem præſtat plani ſublati, tendetur illud eadem vi <emph type="italics"></emph>pN,<emph.end type="italics"></emph.end><lb></lb>qua planum antea urgebatur. </s> <s>Unde tenſio fili hujus obliqui erit <lb></lb>ad tenſionem ſili alterius perpendicularis <emph type="italics"></emph>PN,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>pN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>pH.<emph.end type="italics"></emph.end>Id. </s> <s><lb></lb>eoque ſi pondus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>ſit ad pondus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in ratione quæ componitur ex<lb></lb>ratione reciproca minimarum diſtantiarum ſuorum ſuorum <emph type="italics"></emph>pN, <lb></lb>AM<emph.end type="italics"></emph.end>a centro rotæ, & ratione directa <emph type="italics"></emph>pH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>pN<emph.end type="italics"></emph.end>; pondera idem <lb></lb>valebunt ad rotam movendam, atque adeo ſe mutuo ſuſtinebunt, <lb></lb>ut quilibet experiri poteſt. </s></p> <p type="main"> <s>Pondus autem <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>planis illis duobus obliquis incumbens, rationem <lb></lb>habet cunei inter corporis fiſſi facies internas: & inde vires cunei <lb></lb>& mallei innoteſcunt: utpote cum vis qua pondus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>urget planum <lb></lb><emph type="italics"></emph>pQ<emph.end type="italics"></emph.end>ſit ad vim, qua idem vel gravitate ſua vel ictu mallei impellitur <lb></lb>ſecundum lineam <emph type="italics"></emph>pH<emph.end type="italics"></emph.end>in plano, &c. </s> <s>ut <emph type="italics"></emph>pN<emph.end type="italics"></emph.end>and <emph type="italics"></emph>pH<emph.end type="italics"></emph.end>; atque ad vim, qua <lb></lb>urget planum alterum <emph type="italics"></emph>pG,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>pN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>NH.<emph.end type="italics"></emph.end>Sed & vis Cochleæ per <lb></lb>ſimilem virium diviſionem colligitur; quippe quæ cuneus eſt a ve<lb></lb>cte impulſus. </s> <s>Uſus igitur Corollarii hujus latiſſime patet, & late <lb></lb>patendo veritatem ſuam evincit; cum pendeat ex jam dictis Mecha<lb></lb>nica tota ab Auctoribus diverſimode demonſtrata. </s> <s>Ex hiſce enim <lb></lb>facile derivantur vires Machinarum, quæ ex Rotis, Tympanis, <lb></lb>Trochleis, Vectibus, nervis tenſis & ponderibus directe vel obli<lb></lb>que aſcendentibus, cæteriſque potentiis Mechanicis componi ſo<lb></lb>lent, ut & vires Tendinum ad animalium oſſa movenda. </s></p> <p type="main"> <s><emph type="center"></emph>COROLLARIUM III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Quantitas motus quæ colligitur capiendo ſummam motuum factorum <lb></lb>ad eandem partem, & differentiam factorum ad contrarias, non <lb></lb>mutatur ab actione corporum inter ſe.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Etenim actio eique contraria reactio æquales ſunt per Legem 111, <lb></lb>adeoque per Legem 11 æquales in motibus efficiunt mutationes ver<lb></lb>ſus contrarias partes. </s> <s>Ergo ſi motus fiunt ad eandem partem; quic<lb></lb>quid additur motui corporis fugientis, ſubducetur motui corporis <lb></lb>inſequentis ſic, ut ſumma maneat eadem quæ prius. </s> <s>Sin corpora ob<lb></lb>viam eant; æqualis erit ſubductio de motu utriuſque, adeoQ.E.D.ffe<lb></lb>rentia motuum factorum in contrarias partes manebit eadem. </s></p> <p type="main"> <s>Ut ſi corpus ſphæricum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſit triplo majus corpore ſphærico <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>ha<lb></lb>beatQ.E.D.as velocitatis partes; & <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ſequatur in eadem recta cum ve-<pb xlink:href="039/01/044.jpg" pagenum="16"></pb><arrow.to.target n="note7"></arrow.to.target>locitatis partibus decem, adeoque motus ipſius <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſit ad motum ipſius <lb></lb><emph type="italics"></emph>B,<emph.end type="italics"></emph.end>ut ſex ad decem: ponantur motus illis eſſe partium ſex & par<lb></lb>tium decem, & ſumma erit partium ſexdecim. </s> <s>In corporum igitur <lb></lb>concurſu, ſi corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>lucretur motus partes tres vel quatuor vel <lb></lb>quinque, corpus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>amittet partes totidem, adeoque perget corpus <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>poſt reflexionem cum partibus novem vel decem vel undecim, <lb></lb>& <emph type="italics"></emph>B<emph.end type="italics"></emph.end>cum partibus ſeptem vel ſex vel quinque, exiſtente ſemper ſum<lb></lb>ma partium ſexdecim ut prius. </s> <s>Si corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>lucretur partes novem <lb></lb>vel decem vel undecim vel duodecim, adeoque progrediatur poſt <lb></lb>concurſum cum partibus quindecim vel ſexdecim vel ſeptendecim <lb></lb>vel octodecim; corpus <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>amittendo tot partes quot <emph type="italics"></emph>A<emph.end type="italics"></emph.end>lucratur, <lb></lb>vel cum una parte progredietur amiſſis partibus novem, vel qui<lb></lb>eſcet amiſſo motu ſuo progreſſivo partium decem, vel cum una par<lb></lb>te regredietur amiſſo motu ſuo & (ut ita dicam) una parte amplius, <lb></lb>vel regredietur cum partibus duabus ob detractum motum progreſ<lb></lb>ſivum partium duodecim. </s> <s>AtQ.E.I.a ſummæ motuum conſpirantium <lb></lb>15+1 vel 16+c, & differentiæ contrariorum 17-1 & 18-2 ſemper <lb></lb>erunt partium ſexdecim, ut ante concurſum & reflexionem. </s> <s>CogNI<lb></lb>tis autem motibus quibuſcum corpora poſt reflexionem pergent, in<lb></lb>venietur cujuſque velocitas, ponendo eam eſſe ad velocitatem ante <lb></lb>reflexionem, ut motus poſt eſt ad motum ante. </s> <s>Ut in caſu ultimo, ubi <lb></lb>corporis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>motus erat partium ſex ante reflexionem & partium octo<lb></lb>decim poſtea, & velocitas partium duarum ante reflexionem; in<lb></lb>venietur ejus velocitas partium ſex poſt reflexionem, dicendo, ut <lb></lb>motus partes ſex ante reflexionem ad motus partes octodecim poſt<lb></lb>ea, ita velocitatis partes duæ ante reflexionem ad velocitatis partes <lb></lb>ſex poſtea. </s></p> <p type="margin"> <s><margin.target id="note7"></margin.target>TA,</s></p> <p type="main"> <s>Quod ſi corpora vel non Sphærica vel diverſis in rectis moventia <lb></lb>incidant in ſe mutuo oblique, & requirantur eorum motus poſt refle<lb></lb>xionem; cognoſcendus eſt ſitus plani a quo corpora concurrentia tan<lb></lb>guntur in puncto concurſus: dein corporis utriuſque motus (per <lb></lb>Corol.11.) diſtinguendus eſt in duos, unum huic plano perpendicu<lb></lb>larem, alterum eidem parallelum: motus autem paralleli, propter<lb></lb>ea quod corpora agant in ſe invicem ſecundum lineam huic plano <lb></lb>perpendicularem, retinendi ſunt iidem poſt reflexionem atque an<lb></lb>tea; & motibus perpendicularibus mutationes æquales in partes con<lb></lb>trarias tribuendæ ſunt ſic, ut ſumma conſpirantium & differentia <lb></lb>contrariorum maneat eadem quæ prius. </s> <s>Ex hujuſmodi reflexio<lb></lb>nibus oriri etiam ſolent motus circulares corporum circa centra pro<lb></lb>pria. </s> <s>Sed hos caſus in ſequentibus non conſidero, & nimis longum <lb></lb>eſſet omnia huc ſpectantia demonſtrare.</s> <pb xlink:href="039/01/045.jpg" pagenum="17"></pb> <s>COROLLARIUM IV.</s></p> <p type="main"> <s><emph type="italics"></emph>Commune gravitas Centrum, corporum duorum vel plurimum, ab actio<lb></lb>nibus corporum inter ſe non mutat ſtatum ſuum vel motus vel quie<lb></lb>tis; & propterea corporum omnium in ſ mutuo agentium (excluſis<lb></lb>actionibus & impedimentis externis) commune Centrum gravitatis<lb></lb>vel quieſcit vel movetur uniformiter in directum.</s></p> <p type="main"> <s>Nam ſi puncta duo progrediantur uniformi cum motu in lineis<lb></lb>rectis, & diſtantia eorum dividatur in ratione data, punctum divi<lb></lb>dens vel quieſcit vel progreditur uniformiter in linea recta. </s> <s>Hoc<lb></lb>poſtea in Lemmate XXIII demonſtratur, ſi corpora quotcunque moventur uNI<lb></lb>formiter in lineis rectis, commune centrum gravitatis duorum quorumvis vel quieſcit vel progreditur uniformiter in linea recta; propterea quod linea, horum corporum centra in recta uniformiter<lb></lb>progredientia jungens, dividitur ab hoc centro communis corporum duo<lb></lb>rum & centri communis tertii in data ratione.</s> <s>Eodem modo &<lb></lb>commune centrum horum trium & quarti cujuſvis vel quieſcit vel<lb></lb>progreditur uniformiter in linea recta; propterea quod ab eo divi<lb></lb>ditur diſtantia inter centrum commune trium & centrum quarti in<lb></lb>data ratione, & ſic in infinitum.</s> <s>Igitur in ſyſtemate corporum quæ<lb></lb>actionibus in ſe invicem aliiſque omnibus in ſe extrinſecus impreſ<lb></lb>ſis omnino vacant, adeoque moventur ſingula uniformiter in rectis<lb></lb>ſingulis, commune omnium centrum gravitatis vel quieſcit vel mo<lb></lb>vetur uniformiter in directum.</s></p> <p type="main"> <s>Porro in ſyſtemate duorum corporum in ſe invicem agentium,<lb></lb>cum distantiæ centrorum utriusque a communi gravitatis centro ſint<lb></lb>reciproce ut corpora; erunt motus relativi corporum eorundem, vel<lb></lb>accedendi ad centrum illud vel ab eodem recedendi, æqualibus mutationibus in<lb></lb>partes contrarias factis, atque adeo ab actionibus horum corpo<lb></lb>rum inter ſe, nec promovetur nec retardatur nec mutationem pa<lb></lb>titur in ſtatu ſuo quoad motum vel quietem.</s> <s>In ſyſtemate autem<lb></lb>corporum plurimum, quoniam duorum quorumvis in ſe mutuo agen<lb></lb>tium commune gravitatis centrum ob actionem illam nullatenus<pb xlink:href="039/01/046.jpg" pagenum="18"></pb><arrow.to.target n="note8"></arrow.to.target>mutat ſtatum ſuum; & reliquorum, quibuſcum actio illa non in<lb></lb>tercedit, commune gravitatis centrum nihil inde patitur; diſtantia <lb></lb>autem horum duorum centrorum dividitur a communi corporum <lb></lb>omnium centro in partes ſummis totalibus corporum quorum <lb></lb>ſunt centra reciproce proportionales; adeoque centris illis duobus <lb></lb>ſtatum ſuum movendi vel quieſcendi ſervantibus, commune omNI<lb></lb>um centrum ſervat etiam ſtatum ſuum: manifeſtum eſt quod com<lb></lb>mune illud omnium centrum ob actiones binorum corporum inter <lb></lb>ſe nunquam mutat ſtatum ſuum quoad motum & quietem. </s> <s>In tali <lb></lb>autem ſyſtemate actiones omnes corporum inter ſe, vel inter bina <lb></lb>ſunt corpora, vel ab actionibus inter bina compoſitæ; & propterea <lb></lb>communi omnium centro mutationem in ſtatu motus ejus vel quie<lb></lb>tis nunquam inducunt. </s> <s>Quare cum centrum illud ubi corpora non <lb></lb>agunt in ſe invicem, vel quieſcit, vel in recta aliqua progreditur uNI<lb></lb>formiter; perget idem, non obſtantibus corporum actionibus inter <lb></lb>ſe, vel ſemper quieſcere, vel ſemper progredi uniformiter in dire<lb></lb>ctum; niſi a viribus in ſyſtema extrinſecus impreſſis deturbetur de hoc <lb></lb>ſtatu. </s> <s>Eſt igitur ſyſtematis corporum plurium Lex eadem quæ cor<lb></lb>poris ſolitarii, quoad perſeverantiam in ſtatu motus vel quietis. </s> <s>Mo<lb></lb>tus enim progreſſivus ſeu corporis ſolitarii ſeu ſyſtematis corporum <lb></lb>ex motu centri gravitatis æſtimari ſemper debet. </s></p> <p type="margin"> <s><margin.target id="note8"></margin.target>IATA, <lb></lb>VF.</s></p> <p type="main"> <s><emph type="center"></emph>COROLLARIUM V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corporum dato ſpatio incluſorum iidem ſunt motus inter ſe, ſive ſpa<lb></lb>tium illud quieſcat, ſive moveatur idem uniformiter in directum <lb></lb>abſque motu circulari.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam differentiæ motuum tendentium ad eandem partem, & ſum<lb></lb>mæ tendentium ad contrarias, eædem ſunt ſub initio in <expan abbr="utroq;">utroque</expan> caſu (ex <lb></lb>hypotheſi) & ex his ſummis vel differentiis oriuntur congreſſus & im<lb></lb>petus quibus corpora ſe mutuo feriunt. </s> <s>Ergo per Legem 11 æquales e<lb></lb>runt congreſſuum effectus in <expan abbr="utroq;">utroque</expan> caſu; & propterea manebunt mo<lb></lb>tus inter ſe in uno caſu æquales motibus inter ſe in altero. </s> <s>Idem com<lb></lb>probatur experimento luculento. </s> <s>Motus omnes eodem modo ſe ha<lb></lb>bent in Navi, ſive ea quieſcat, ſive moveatur uniformiter in directum. </s></p> <p type="main"> <s><emph type="center"></emph>COROLLARIUM VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpora <expan abbr="moveãtur">moveantur</expan> <expan abbr="quomodocunq;">quomodocunque</expan> inter ſe, & a viribus acceler atrici<lb></lb>bus æqualibus ſecundum lineas parallelas urgeantur; pergent omnia <lb></lb>eodem modo moveri inter ſe, ac ſi viribus illis non eſſent incitata.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam vires illæ æqualiter (pro quantitatibus movendorum corpo-<pb xlink:href="039/01/047.jpg" pagenum="19"></pb>rum) & ſecundum lineas parallelas agendo, corpora omnia æquali<lb></lb>ter (quoad velocitatem) movebunt per Legem 11. adeoque nunquam <lb></lb>mutabunt poſitiones & motus eorum inter ſe. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hactenus principia tradidi a Mathematicis recepta & experien<lb></lb>tia multiplici confirmata. </s> <s>Per Leges duas primas & Corollaria duo <lb></lb>prima <emph type="italics"></emph>Galilæus<emph.end type="italics"></emph.end>invenit deſcenſum Gravium eſſe in duplicata ratione <lb></lb>temporis, & motum Projectilium fieri in Parabola; conſpirante ex<lb></lb>perientia, niſi quatenus motus illi per aeris reſiſtentiam aliquantu<lb></lb>lum retardantur. </s> <s>Ab iiſdem Legibus & Corollariis pendent de<lb></lb>monſtrata de temporibus oſcillantium Pendulorum, ſuffragante Ho<lb></lb>rologiorum experientia quotidiana. </s> <s>Ex his iiſdem & Lege tertia <lb></lb><emph type="italics"></emph>Chriſtophorus Wrennus<emph.end type="italics"></emph.end>Eques Auratus, <emph type="italics"></emph>Jobannes Walliſius S.T.D.<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>Chriſtianus Hugenius,<emph.end type="italics"></emph.end>hujus ætatis Geometrarum facile prin<lb></lb>cipes, regulas congreſſuum & reflexionum duorum corporum ſe<lb></lb>orſim invenerunt, & eodem fere tempore cum <emph type="italics"></emph>Societate Regia<emph.end type="italics"></emph.end><lb></lb>communicarunt, inter ſe (quoad has leges) omnino conſpirantes: <lb></lb>& primus quidem <emph type="italics"></emph>Walliſius,<emph.end type="italics"></emph.end>deinde <emph type="italics"></emph>Wrennus<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Hugenius<emph.end type="italics"></emph.end>inven<lb></lb>tum prodiderunt. </s> <s>Sed & veritas comprobata eſt a <emph type="italics"></emph>Wrenno<emph.end type="italics"></emph.end>co<lb></lb>ram <emph type="italics"></emph>Regia Societate<emph.end type="italics"></emph.end>per experimentum Pendulorum: quod etiam <lb></lb><emph type="italics"></emph>Clariſſimus Mariottus<emph.end type="italics"></emph.end>libro integro exponere mox dignatus eſt. </s> <s>Ve<lb></lb>rum, ut hoc experimentum cum Theoriis ad amuſſim congruat, ha<lb></lb>benda eſt ratio cum reſiſtentiæ aeris, tum etiam vis Elaſticæ con<lb></lb>currentium corporum. </s> <s>Pendeant corpora <emph type="italics"></emph>A, B<emph.end type="italics"></emph.end>filis parallelis & <lb></lb>æqualibus <emph type="italics"></emph>AC, BD,<emph.end type="italics"></emph.end>a centris <emph type="italics"></emph>C, D.<emph.end type="italics"></emph.end>His centris & intervallis de<lb></lb>ſcribantur ſemicirculi <emph type="italics"></emph>EAF, GBH<emph.end type="italics"></emph.end>radiis <emph type="italics"></emph>CA, DB<emph.end type="italics"></emph.end>biſecti. </s> <s>Tra<lb></lb>hatur corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad arcus <emph type="italics"></emph>EAF<emph.end type="italics"></emph.end>punctum quodvis <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>& (ſubducto <lb></lb>corpore <emph type="italics"></emph>B<emph.end type="italics"></emph.end>) demittatur inde, redeatque poſt unam oſcillationem <lb></lb>ad punctum <emph type="italics"></emph>V.<emph.end type="italics"></emph.end>Eſt <emph type="italics"></emph>RV<emph.end type="italics"></emph.end>re<lb></lb><figure id="id.039.01.047.1.jpg" xlink:href="039/01/047/1.jpg"></figure><lb></lb>tardatio ex reſiſtentia aeris. </s> <s><lb></lb>Hujus <emph type="italics"></emph>RV<emph.end type="italics"></emph.end>fiat <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>pars quar<lb></lb>ta ſita in medio, ita ſcilicet <lb></lb>ut <emph type="italics"></emph>RS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>TV<emph.end type="italics"></emph.end>æquentur, ſit<lb></lb>que <emph type="italics"></emph>RS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>ut 3 ad 2. <lb></lb>Et iſta <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>exhibebit retarda<lb></lb>tionem in deſcenſu ab <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>A<emph.end type="italics"></emph.end><lb></lb>quam proxime. </s> <s>Reſtituatur <lb></lb>corpus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>in locum ſuum. </s> <s>Cadat corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>de puncto <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>& velo<lb></lb>citas ejus in loco reflexionis <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>abſque errore ſenſibili, tanta erit ae <pb xlink:href="039/01/048.jpg" pagenum="20"></pb>ſi in vacuo cecidiſſet de loco <emph type="italics"></emph>T.<emph.end type="italics"></emph.end>Exponatur igitur hæc velocitas <lb></lb><arrow.to.target n="note9"></arrow.to.target>per chordam arcus <emph type="italics"></emph>TA.<emph.end type="italics"></emph.end>Nam velocitatem Penduli in puncto in<lb></lb>fimo eſſe ut chordam arcus quem cadendo deſcripſit, Propoſitio eſt <lb></lb>eſt Geometris notiſſima. </s> <s>Poſt reflexionem perveniat corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <lb></lb>locum <emph type="italics"></emph>s,<emph.end type="italics"></emph.end>& corpus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad locum <emph type="italics"></emph>k.<emph.end type="italics"></emph.end>Tollatur corpus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& invenia<lb></lb>tur locus <emph type="italics"></emph>v<emph.end type="italics"></emph.end>; a quo ſi corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>demittatur & poſt unam oſcillatio<lb></lb>nem redeat ad locum <emph type="italics"></emph>r,<emph.end type="italics"></emph.end>ſit <emph type="italics"></emph>st<emph.end type="italics"></emph.end>pars quarta ipſius <emph type="italics"></emph>rv<emph.end type="italics"></emph.end>ſita in medio, <lb></lb>ita videlicet ut <emph type="italics"></emph>rs<emph.end type="italics"></emph.end>& <emph type="italics"></emph>tu<emph.end type="italics"></emph.end>æquentur; & per chordam arcus <emph type="italics"></emph>tA<emph.end type="italics"></emph.end>ex<lb></lb>ponatur velocitas quam corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>proxime poſt reflexionem habuit <lb></lb>in loco <emph type="italics"></emph>A.<emph.end type="italics"></emph.end>Nam <emph type="italics"></emph>t<emph.end type="italics"></emph.end>erit locus ille verus & correctus, ad quem cor<lb></lb>pus <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ſublata aeris reſiſtentia, aſcendere debuiſſet: Simili me<lb></lb>thodo corrigendus erit locus <emph type="italics"></emph>k,<emph.end type="italics"></emph.end>ad quem corpus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſcendit, & in<lb></lb>veniendus locus <emph type="italics"></emph>l,<emph.end type="italics"></emph.end>ad quem corpus illud aſcendere debuiſſet in va<lb></lb>cuo. </s> <s>Hoc pacto experiri licet omnia perinde ac ſi in vacuo con<lb></lb>ſtituti eſſemus. </s> <s>Tandem ducendum erit corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in chordam ar<lb></lb>cus <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>(quæ velocitatem ejus exhibet) ut habeatur motus ejus in <lb></lb>loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>proxime ante reflexionem; deinde in chordam arcus <emph type="italics"></emph>tA,<emph.end type="italics"></emph.end>ut <lb></lb>habeatur motus ejus in loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>proxime poſt reflexionem. </s> <s>Et ſic <lb></lb>corpus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ducendum erit in chordam arcus <emph type="italics"></emph>Bb,<emph.end type="italics"></emph.end>ut habeatur motus <lb></lb>ejus proxime poſt reflexionem. </s> <s>Et ſimili methodo, ubi corpora duo <lb></lb>ſimul demittuntur de locis diverſis, inveniendi ſunt motus <expan abbr="utriuſq;">utriuſque</expan> <lb></lb>tam ante, quam poſt reflexionem; & tum demum conferendi ſunt <lb></lb>motus inter ſe & colligendi effectus reflexionis. </s> <s>Hoc modo in <lb></lb>Pendulis pedum decem rem tentando, idQ.E.I. corporibus tam <lb></lb>inæqualibus quam æqualibus, & faciendo ut corpora de intervallis <lb></lb>ampliſſimis, puta pedum octo vel duodecim vel ſexdecim, concurre<lb></lb>rent; reperi ſemper ſine errore trium digitorum in menſuris, ubi <lb></lb>corpora ſibi mutuo directe occurrebant, quod æquales erant muta<lb></lb>tiones motuum corporibus in partes contrarias illatæ, atque adeo <lb></lb>quod actio & reactio ſemper <lb></lb><figure id="id.039.01.048.1.jpg" xlink:href="039/01/048/1.jpg"></figure><lb></lb>erant æquales. </s> <s>Ut ſi corpus <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>incidebat in corpus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>cum <lb></lb>novem partibus motus, & a<lb></lb>miſſis ſeptem partibus perge<lb></lb>bat poſt reflexionem cum du<lb></lb>abus; corpus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>reſiliebat cum <lb></lb>partibus iſtis ſeptem. </s> <s>Si cor<lb></lb>pora obviam ibant <emph type="italics"></emph>A<emph.end type="italics"></emph.end>cum <lb></lb>duodecim partibus & <emph type="italics"></emph>B<emph.end type="italics"></emph.end>cum ſex, & redibat <emph type="italics"></emph>A<emph.end type="italics"></emph.end>cum duabus; redi<lb></lb>bat <emph type="italics"></emph>B<emph.end type="italics"></emph.end>cum octo, facta detractione partium quatuordecim utrin<lb></lb>que. </s> <s>De motu ipſius <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſubducantur partes duodecim, & reſtabit <pb xlink:href="039/01/049.jpg" pagenum="21"></pb>nihil: ſubducantur aliæ partes duæ, & fiet motus duarum partium <lb></lb>in plagam contrariam: & ſic de motu corporis <emph type="italics"></emph>B<emph.end type="italics"></emph.end>partium ſex ſub<lb></lb>ducendo partes quatuordecim, fient partes octo in plagam contra<lb></lb>riam. </s> <s>Quod ſi corpora ibant ad eandam plagam, <emph type="italics"></emph>A<emph.end type="italics"></emph.end>velocius cum <lb></lb>partibus quatuordecim, & <emph type="italics"></emph>B<emph.end type="italics"></emph.end>tardius cum partibus quinque, & poſt <lb></lb>reflexionem pergebat <emph type="italics"></emph>A<emph.end type="italics"></emph.end>cum quinque partibus; pergebat <emph type="italics"></emph>B<emph.end type="italics"></emph.end>cum qua<lb></lb>tuordecim, facta tranſlatione partium novem de <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in <emph type="italics"></emph>B.<emph.end type="italics"></emph.end>Et ſic <lb></lb>in reliquis. </s> <s>A congreſſu & colliſione corporum nunquam muta<lb></lb>batur quantitas motus, quæ ex ſumma motuum conſpirantium & <lb></lb>differentia contrariorum colligebatur. </s> <s>Nam errorem digiti unius <lb></lb>& alterius in menſuris tribuerim difficultati peragendi ſingula <lb></lb>ſatis accurate. </s> <s>Difficile erat, tum pendula ſimul demittere fic, ut <lb></lb>corpora in ſe mutuo impingerent in loco infimo <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; tum loca <emph type="italics"></emph>s, <lb></lb>k<emph.end type="italics"></emph.end>notare, ad quæ corpora aſcendebant poſt concurſum. </s> <s>Sed & in <lb></lb>ipſis pilis inæqualis partium denſitas, & textura aliis de cauſis irre<lb></lb>gularis, errores inducebant. </s></p> <p type="margin"> <s><margin.target id="note9"></margin.target>LEGES<lb></lb>MOTUS</s></p> <p type="main"> <s>Porro nequis objiciat Regulam, ad quam probandam inventum <lb></lb>eſt hoc experimentum, præſupponere corpora vel abſolute dura <lb></lb>eſſe, vel ſaltem perfecte elaſtica, cujuſmodi nulla reperiuntur in <lb></lb>compoſitionibus naturalibus; addo quod Experimenta jam deſcrip<lb></lb>ta ſuccedunt in corporibus mollibus æque ac in duris, nimirum a <lb></lb>conditione duritiei neutiquam pendentia. </s> <s>Nam ſi Regula illa in <lb></lb>corporibus non perfecte duris tentanda eſt, debebit ſolummodo <lb></lb>reflexio minui in certa proportione pro quantitate vis Elaſticæ. </s> <s>In <lb></lb>Theoria <emph type="italics"></emph>Wrenni<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Hugenii<emph.end type="italics"></emph.end>corpora abſolute dura redeunt ab invi<lb></lb>cem cum velocitate congreſſus. </s> <s>Certius id affirmabitur de perfecte <lb></lb>Elaſticis. </s> <s>In imperfecte Elaſticis velocitas reditus minuenda eſt ſi<lb></lb>mul cum vi Elaſtica; propterea quod vis illa; (niſi ubi partes cor<lb></lb>porum ex congreſſu læduntur, vel extenſionem aliqualem quaſi ſub <lb></lb>malleo patiuntur,) certa ac determinata ſit (quantum ſentio) faci<lb></lb>atque corpora redire ab invicem cum velocitate relativa, quæ ſit ad <lb></lb>relativam velocitatem concurſus in data ratione. </s> <s>Id in pilis ex lana <lb></lb>arcte conglomerata & fortiter conſtricta ſic tentavi. </s> <s>Primum demit<lb></lb>tendo Pendula & menſurando reflexionem, inveni quantitatem vis <lb></lb>Elaſticæ; deinde per hanc vim determinavi reflexiones in aliis ca<lb></lb>ſibus concurſuum, & reſpondebant Experimenta. </s> <s>Redibant ſemper <lb></lb>pilæ ab invicem cum velocitate relativa, quæ eſſet ad velocitatem <lb></lb>relativam concurſus ut 5 ad 9 circiter. </s> <s>Eadem fere cum velocitate <lb></lb>redibant pilæ ex chalybe: aliæ ex ſubere cum paulo minore: in vi<lb></lb>treis autem proportio erat 15 ad 16 circiter. </s> <s>Atque hoc pacto Lex <lb></lb>tertia quoad ictus & reflexiones per Theoriam comprobata eſt, quæ <lb></lb>cum experientia plane congruit. <pb xlink:href="039/01/050.jpg" pagenum="22"></pb><arrow.to.target n="note10"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note10"></margin.target>AXIOMATA <lb></lb>SIVE</s></p> <p type="main"> <s>In Attractionibus rem ſic breviter oſtendo. </s> <s>Corporibus duobus <lb></lb>quibuſvis <emph type="italics"></emph>A, B<emph.end type="italics"></emph.end>ſe mutuo trahentibus, concipe obſtaculum quodvis <lb></lb>interponi quo congreſſus eorum impediatur. </s> <s>Si corpus alterutrum <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>magis trahitur verſus corpus alterum <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>quam illud alterum <emph type="italics"></emph>B<emph.end type="italics"></emph.end><lb></lb>in prius <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>obſtaculum magis urgebitur preſſione corporis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>quam <lb></lb>preſſione corporis <emph type="italics"></emph>B<emph.end type="italics"></emph.end>; proindeque non manebit in æquilibrio. </s> <s>Præ<lb></lb>valebit preſſio fortior, facietque ut ſyſtema corporum duorum & <lb></lb>obſtaculi moveatur in directum in partes verſus <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>motuQ.E.I. ſpatiis <lb></lb>liberis ſemper accelerato abeat in infinitum. </s> <s>Quod eſt abſurdum & <lb></lb>Legi primæ contrarium. </s> <s>Nam per Legem primam debebit ſyſtema <lb></lb>perſeverare in ſtatu ſuo quieſcendi vel movendi uniformiter in di<lb></lb>rectum, proindeque corpora æqualiter urgebunt obſtaculum, & id<lb></lb>circo æqualiter trahentur in invicem. </s> <s>Tentavi hoc in Magnete & <lb></lb>Ferro. </s> <s>Si hæc in vaſculis propriis ſeſe contingentibus ſeorſim po<lb></lb>ſita, in aqua ſtagnante juxta fluitent; neutrum propellet alterum, <lb></lb>ſed æqualitate attractionis utrinque ſuſtinebunt conatus in ſe mu<lb></lb>tuos, ac tandem in æquilibrio conſtituta quieſcent. </s></p> <p type="main"> <s>Sic etiam gravitas inter Terram & ejus partes, mutua eſt. </s> <s>Se<lb></lb>cetur Terra <emph type="italics"></emph>FI<emph.end type="italics"></emph.end>plano quovis <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>in partes duas <emph type="italics"></emph>EGF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EGI:<emph.end type="italics"></emph.end><lb></lb>& æqualia erunt harum pondera in ſe mu<lb></lb><figure id="id.039.01.050.1.jpg" xlink:href="039/01/050/1.jpg"></figure><lb></lb>tuo. </s> <s>Nam ſi plano alio <emph type="italics"></emph>HK<emph.end type="italics"></emph.end>quod priori <lb></lb><emph type="italics"></emph>EG<emph.end type="italics"></emph.end>parallelum ſit, pars major <emph type="italics"></emph>EGI<emph.end type="italics"></emph.end>ſe<lb></lb>cetur in partes duas <emph type="italics"></emph>EGKH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>HKI,<emph.end type="italics"></emph.end><lb></lb>quarum <emph type="italics"></emph>HKI<emph.end type="italics"></emph.end>æqualis ſit parti prius ab<lb></lb>ſciſſæ <emph type="italics"></emph>EFG:<emph.end type="italics"></emph.end>manifeſtum eſt quod pars <lb></lb>media <emph type="italics"></emph>EGKH<emph.end type="italics"></emph.end>pondere proprio in neu<lb></lb>tram partium extremarum propendebit, <lb></lb>ſed inter utramQ.E.I. æquilibrio, ut ita <lb></lb>dicam, ſuſpendetur, & quieſcet. </s> <s>Pars autem extrema <emph type="italics"></emph>HKI<emph.end type="italics"></emph.end>toto <lb></lb>ſuo pondere incumbet in partem mediam, & urgebit illam in <lb></lb>partem alteram extremam <emph type="italics"></emph>EGF<emph.end type="italics"></emph.end>; ideoque vis qua partium <lb></lb><emph type="italics"></emph>HKI<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EGKH<emph.end type="italics"></emph.end>ſumma <emph type="italics"></emph>EGI<emph.end type="italics"></emph.end>tendit verſus partem tertiam <lb></lb><emph type="italics"></emph>EGF,<emph.end type="italics"></emph.end>æqualis eſt ponderi partis <emph type="italics"></emph>HKI,<emph.end type="italics"></emph.end>id eſt ponderi partis ter<lb></lb>tiæ <emph type="italics"></emph>EGF.<emph.end type="italics"></emph.end>Et propterea pondera partium duarum <emph type="italics"></emph>EGI, EGF<emph.end type="italics"></emph.end><lb></lb>in ſe mutuo ſunt æqualia, uti volui oſtendere. </s> <s>Et niſi pondera illa <lb></lb>æqualia eſſent, Terra tota in libero æthere fluitans ponderi majori <lb></lb>cederet, & ab eo fugiendo abiret in infinitum. </s></p> <p type="main"> <s>Ut corpora in concurſu & reflexione idem pollent, quorum ve<lb></lb>locitates ſunt reciproce ut vires inſitæ: ſic in movendis Inſtru<lb></lb>mentis Mechanicis agentia idem pollent & conatibus contrariis ſe <lb></lb>mutuo ſuſtinent, quorum velocitates ſecundum determinationem <pb xlink:href="039/01/051.jpg" pagenum="23"></pb>virium æſtimatæ, ſunt reciproce ut vires. </s> <s>Sie pondera æquipollent <lb></lb>ad movenda brachia Libræ, quæ oſcillante Libra ſunt reciproce ut <lb></lb>eorum velocitates ſurſum & deorſum: hoc eſt, pondera, ſi recta <lb></lb>aſcendunt & deſcendunt, æquipollent, quæ ſunt reciproce ut pun<lb></lb>ctorum a quibus ſuſpenduntur diſtantiæ ab axe Libræ; ſin planis <lb></lb>obliquis aliiſve admotis obſtaculis impedita aſcendunt vel deſcen<lb></lb>dunt oblique, æquipollent quæ ſunt reciproce ut aſcenſus & deſcen<lb></lb>ſus, quatenus facti ſecundum perpendiculum: id adeo ob determi<lb></lb>nationem gravitatis deorſum. </s> <s>Similiter in Trochlea ſeu Polyſpaſto <lb></lb>vis manus funem directe trahentis, quæ ſit ad pondus vel directe <lb></lb>vel oblique aſcendens ut velocitas aſcenſus perpendicularis ad ve<lb></lb>locitatem manus funem trahentis, ſuſtinebit pondus. </s> <s>In Horolo<lb></lb>giis & ſimilibus inſtrumentis, quæ ex rotulis commiſſis conſtructa <lb></lb>ſunt, vires contrariæ ad motum rotularum promovendum & impe<lb></lb>diendum, ſi ſunt reciproce ut velocitates partium rotularum in quas <lb></lb>imprimuntur, ſuſtinebunt ſe mutuo. </s> <s>Vis Cochleæ ad premendum <lb></lb>corpus eſt ad vim manus manubrium circumagentis, ut circularis <lb></lb>velocitas manubrii ea in parte ubi a manu urgetur, ad velocitatem <lb></lb>progreſſivam cochleæ verſus corpus preſſum. </s> <s>Vires quibus Cu<lb></lb>neus urget partes duas ligni fiſſi ſunt ad vim mallei in cuneum, ut <lb></lb>progreſſus cunei ſecundum determinationem vis a malleo in ipſum <lb></lb>impreſſæ, ad velocitatem qua partes ligni cedunt cuneo, ſecundum <lb></lb>lineas faciebus cunei perpendiculares. </s> <s>Et par eſt ratio Machina<lb></lb>rum omnium. </s></p> <p type="main"> <s>Harum efficacia & uſus in eo ſolo conſiſtit, ut diminuendo velo<lb></lb>citatem augeamus vim, & contra: Unde ſolvitur in omni aptorum <lb></lb>inſtrumentorum genere Problema, <emph type="italics"></emph>Datum pondus data vi moven<lb></lb>di,<emph.end type="italics"></emph.end>aliamve datam reſiſtentiam vi data ſuperandi. </s> <s>Nam ſi Ma<lb></lb>chinæ ita formentur, ut velocitates Agentis & Reſiſtentis ſine reci<lb></lb>proce ut vires; Agens reſiſtentiam ſuſtinebit: & majori cum veloci<lb></lb>tatum diſparitate eandem vincet. </s> <s>Certe ſi tanta ſic velocitatum <lb></lb>diſparitas, ut vincatur etiam reſiſtentia omnis, quæ tam ex conti<lb></lb>guorum & inter ſe labentium corporum attritione, quam ex con<lb></lb>tinuorum & ab invicem ſeparandorum cohæſione & elevandorum <lb></lb>ponderibus orirj ſolet; ſuperata omni ea reſiſtentia, vis redun<lb></lb>dans accelerationem motus ſibi proportionalem, partim in parti<lb></lb>bus machinæ, partim in corpore reſiſtente producet. </s> <s>Ceterum <lb></lb>Mechanicam tractare non eſt hujus inſtituti. </s> <s>Hiſce volui tan<lb></lb>tum oſtendere, quam late pateat quamque certa ſit Lex tertia <lb></lb>Motus. </s> <s>Nam ſi æſtimetur Agentis actio ex ejus vi & veloci-</s></p><pb xlink:href="039/01/052.jpg" pagenum="24"></pb> <p type="main"> <s><arrow.to.target n="note11"></arrow.to.target>tate conjunctim; & ſimiliter Reſiſtentis reactio æſtimetur conjun<lb></lb>ctim ex ejus partium ſingularum velocitatibus & viribus reſiſtendi <lb></lb>ab earum attritione, cohæſione, pondere, & acceleratione ori<lb></lb>undis; erunt actio & reactio, in omni inſtrumentorum uſu, <lb></lb>ſibi invicem ſemper æquales. </s> <s>Et quatenus actio propagatur per <lb></lb>inſtrumentum & ultimo imprimitur in corpus omne reſiſtens, <lb></lb>ejus ultima determinatio determinationi reactionis ſemper erit <lb></lb>contraria. <lb></lb></s></p> <p type="margin"> <s><margin.target id="note11"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p></chap><chap><subchap1><subchap2> <p type="main"> <s><emph type="center"></emph>DE <lb></lb>MOTU CORPORUM <lb></lb>LIBER PRIMUS.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="center"></emph>SECTIO I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Methodo Rationum primarum & ultimarum, cujus ope ſequentia <lb></lb>demonſtrantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>QUantitates, ut & quantitatum rationes, quæ ad æqualitatem <lb></lb>tempore quovis finito conſtanter tendunt, & ante finem tempo<lb></lb>ris illius propius ad invicem accedunt quam pro data quavis diffe<lb></lb>tia, fiunt ultimo æquales.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Si negas; fiant ultimò inequales, & ſit earum ultima differentia <lb></lb><emph type="italics"></emph>D.<emph.end type="italics"></emph.end>Ergo nequeunt propius ad æqualitatem accedere quam pro <lb></lb>data differentia <emph type="italics"></emph>D:<emph.end type="italics"></emph.end>contra hypotheſin. </s></p><pb xlink:href="039/01/053.jpg" pagenum="25"></pb> <p type="main"> <s><emph type="center"></emph>LEMMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si in Figura quavis<emph.end type="italics"></emph.end>AacE, <emph type="italics"></emph>rectis<emph.end type="italics"></emph.end>Aa, AE <emph type="italics"></emph>& curva<emph.end type="italics"></emph.end>acE <emph type="italics"></emph>com <lb></lb>prehenſa, inſcribantur parallelogramma quotcunque<emph.end type="italics"></emph.end>Ab, Bc, Cd <lb></lb>&c. <emph type="italics"></emph>ſub baſibus<emph.end type="italics"></emph.end>AB, BC, CD, &c. <emph type="italics"></emph>æqualibus, & lateribuſ<emph.end type="italics"></emph.end><lb></lb>Bb, Cc, Dd, &c. <emph type="italics"></emph>Figuræ lateri<emph.end type="italics"></emph.end>Aa <emph type="italics"></emph>pa<lb></lb>rallelis contenta; & compleantur paral-<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.053.1.jpg" xlink:href="039/01/053/1.jpg"></figure><lb></lb><emph type="italics"></emph>lelogramma<emph.end type="italics"></emph.end>aKbl, bLcm, cMdn, &c. <lb></lb><emph type="italics"></emph>Dein horum parallelogrammorum lati<lb></lb>tudo minuatur, & numerus augeatur <lb></lb>in infinitum: dico quod ultimæ rationes, <lb></lb>quas habent ad ſe invicem Figura in<lb></lb>ſcripta<emph.end type="italics"></emph.end>AKbLcMdD, <emph type="italics"></emph>circumſcripta<emph.end type="italics"></emph.end><lb></lb>AalbmcndoE, <emph type="italics"></emph>& curvilinea<emph.end type="italics"></emph.end>AbcdE, <lb></lb><emph type="italics"></emph>ſunt rationes æqualitatis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam Figuræ inſcriptæ & circumſcriptæ differentia eſt ſumma pa<lb></lb>rallelogrammorum <emph type="italics"></emph>Kl, Lm, Mn, Do,<emph.end type="italics"></emph.end>hoc eſt (ob æquales om<lb></lb>nium baſes) rectangulum ſub unius baſi <emph type="italics"></emph>Kb<emph.end type="italics"></emph.end>& altitudinum ſumma <lb></lb><emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end>id eſt, rectangulum <emph type="italics"></emph>ABla.<emph.end type="italics"></emph.end>Sed hoc rectangulum, eo quod <lb></lb>latitudo ejus <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in infinitum minuitur, fit minus quovis dato. </s> <s>Er<lb></lb>go (per Lemma 1) Figura inſcripta & circumſcripta & multo magis <lb></lb>Figura curvilinea intermedia fiunt ultimo æquales. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Eædem rationes ultimæ ſunt etiam rationes æqualitatis, ubi paral<lb></lb>lelogrammorum latitudines<emph.end type="italics"></emph.end>AB, BC, CD, &c. <emph type="italics"></emph>ſunt inæquales, <lb></lb>& omnes minuuntur in infinitum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit enim <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>æqualis latitudini maximæ, & compleatur paralle<lb></lb>logrammum <emph type="italics"></emph>FAaf.<emph.end type="italics"></emph.end>Hoc erit majus quam differentia Figuræ in<lb></lb>ſcriptæ & Figuræ circumſcriptæ; at latitudine ſua <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>in infinitum <lb></lb>diminuta, minus fiet quam datum quodvis rectangulum. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſumma ultima parallelogrammorum evaneſcentium <lb></lb>coincidit omni ex parte cum Figura curvilinea. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et multo magis Figura rectilinea, quæ chordis evaneſ-<pb xlink:href="039/01/054.jpg" pagenum="26"></pb><arrow.to.target n="note12"></arrow.to.target>centium arcuum <emph type="italics"></emph>ab, bc, cd, &c.<emph.end type="italics"></emph.end>comprehenditur, coincidit ultimo <lb></lb>cum Figura curvilinea. </s></p> <p type="margin"> <s><margin.target id="note12"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Ut & Figura rectilinea circumſcripta quæ tangentibus <lb></lb>eorundem arcuum comprehenditur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Et propterea hæ Figuræ ultimæ (quoad perimetros <emph type="italics"></emph>acE,<emph.end type="italics"></emph.end>) <lb></lb>non ſunt rectilineæ, ſed rectilinearum limites curvilinei. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si in duabus Figuris<emph.end type="italics"></emph.end>AacE, PprT, <emph type="italics"></emph>inſcribantur (ut ſupra) duæ <lb></lb>parallelogrammorum ſeries, ſitQ.E.I.em amborum numerus, & ubi <lb></lb>latitudines in infinitum diminuuntur, rationes ultimæ parallelo<lb></lb>grammorum in una Figura ad parallelogramma in altera, ſingulorum <lb></lb>ad fingula, ſint eædem; dico quod Figuræ duæ<emph.end type="italics"></emph.end>AacE, PprT, <lb></lb><emph type="italics"></emph>ſunt ad invicem in eadem illa ratione.<emph.end type="italics"></emph.end></s></p><figure id="id.039.01.054.1.jpg" xlink:href="039/01/054/1.jpg"></figure> <p type="main"> <s>Etenim ut ſunt parallelogramma ſingula ad ſingula, ita (compo<lb></lb>nendo) fit ſumma omnium ad ſummam omnium, & ita Figura ad <lb></lb>Figuram; exiſtente nimirum Figura priore (per Lemma 111) ad ſum<lb></lb>mam priorem, & Figura poſteriore ad ſummam poſteriorem in ra<lb></lb>tione æqualitatis. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc ſi duæ cujuſcunque generis quantitates in eundem <lb></lb>partium numerum utcunQ.E.D.vidantur; & partes illæ, ubi numerus <lb></lb>earum augetur & magnitudo diminuitur in infinitum, datam obti<lb></lb>neant rationem ad invicem, prima ad primam, ſecunda ad ſecundam, <lb></lb>cæteræque ſuo ordine ad cæteras: erunt tota ad invicem in eadem <lb></lb>illa data ratione. </s> <s>Nam ſi in Lemmatis hujus Figuris ſumantur pa-<pb xlink:href="039/01/055.jpg" pagenum="27"></pb>rallelogramma inter ſe ut partes, ſummæ partium ſemper erunt ut <lb></lb>ſummæ parallelogrammorum; atque adeo, ubi partium & paralle<lb></lb>logrammorum numerus augetur & magnitudo diminuitur in infiNI<lb></lb>tum, in ultima ratione parallelogrammi ad parallelogrammum, id <lb></lb>eſt (per hypotheſin) in ultima ratione partis ad partem. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Similium Figurarum latera omnia, quæ ſibi mutuo reſpondent, ſunt <lb></lb>proportionalia, tam curvilinea quam rectilinea; & areæ ſunt in <lb></lb>duplicata ratione laterum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si arcus quilibet poſitione datus<emph.end type="italics"></emph.end>AB <emph type="italics"></emph>ſub-<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.055.1.jpg" xlink:href="039/01/055/1.jpg"></figure><lb></lb><emph type="italics"></emph>tendatur chorda<emph.end type="italics"></emph.end>AB, <emph type="italics"></emph>& in puncto <lb></lb>aliquo<emph.end type="italics"></emph.end>A, <emph type="italics"></emph>in medio curvaturæ continuæ, <lb></lb>tangatur a recta utrinque producta<emph.end type="italics"></emph.end><lb></lb>AD; <emph type="italics"></emph>dein puncta<emph.end type="italics"></emph.end>A, B <emph type="italics"></emph>ad invicem <lb></lb>accedant & coëant; dico quod angulus<emph.end type="italics"></emph.end><lb></lb>BAD, <emph type="italics"></emph>ſub chorda & tangente conten<lb></lb>tus, minuetur in infinitum & ultimo e<lb></lb>vaneſcet.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi angulus ille non evaneſcit, continebit arcus <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>cum tan<lb></lb>gente <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>angulum rectilineo æqualem, & propterea curvatura ad <lb></lb>ad punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>non erit continua, contra hypotheſin. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis; dico quod ultima ratio arcus, chordæ, & tangentis <lb></lb>ad invicem est ratio æqualitatis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam dum punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>accedit, intelligantur ſemper <lb></lb><emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>ad puncta longinqua <emph type="italics"></emph>b<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>d<emph.end type="italics"></emph.end>produci, & ſecanti <emph type="italics"></emph>BD<emph.end type="italics"></emph.end><lb></lb>parallela agatur <emph type="italics"></emph>bd.<emph.end type="italics"></emph.end>Sitque arcus <emph type="italics"></emph>Ab<emph.end type="italics"></emph.end>ſemper ſimilis arcui <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end><lb></lb>Et punctis <emph type="italics"></emph>A, B<emph.end type="italics"></emph.end>coeuntibus, angulus <emph type="italics"></emph>dAb,<emph.end type="italics"></emph.end>per Lemma ſuperius, <lb></lb>evaneſcet; adeoque rectæ ſemper ſinitæ <emph type="italics"></emph>Ab, Ad<emph.end type="italics"></emph.end>& arcus interme<lb></lb>dius <emph type="italics"></emph>Ab<emph.end type="italics"></emph.end>coincident, & propterea æquales erunt. </s> <s>Unde & hiſce <lb></lb>ſemper proportionales rectæ <emph type="italics"></emph>AB, AD,<emph.end type="italics"></emph.end>& arcus intermedius <emph type="italics"></emph>AB<emph.end type="italics"></emph.end><pb xlink:href="039/01/056.jpg" pagenum="28"></pb><arrow.to.target n="note13"></arrow.to.target>evaneſcent, & rationem ultimam habebunt æqualitatis. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note13"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Unde ſi per <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ducatur tangenti parallela <emph type="italics"></emph>BF,<emph.end type="italics"></emph.end>rectam <lb></lb>quamvis <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>per <emph type="italics"></emph>A<emph.end type="italics"></emph.end>tranſe<lb></lb><figure id="id.039.01.056.1.jpg" xlink:href="039/01/056/1.jpg"></figure><lb></lb>untem perpetuo ſecans in <emph type="italics"></emph>F,<emph.end type="italics"></emph.end><lb></lb>hæc <emph type="italics"></emph>BF<emph.end type="italics"></emph.end>ultimo ad arcum e<lb></lb>vaneſcentem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>rationem <lb></lb>habebit æqualitatis, eo quod <lb></lb>completo parallelogrammo <emph type="italics"></emph>AFBD<emph.end type="italics"></emph.end>rationem ſemper habet æqua<lb></lb>litatis ad <emph type="italics"></emph>AD.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et ſi per <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ducantur plures rectæ <emph type="italics"></emph>BE, BD, AF, <lb></lb>AG,<emph.end type="italics"></emph.end>ſecantes tangentem <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>& ipſius parallelam <emph type="italics"></emph>BF<emph.end type="italics"></emph.end>; ratio ulti<lb></lb>ma abſciſſarum omnium <emph type="italics"></emph>AD, AE, BF, BG,<emph.end type="italics"></emph.end>chordæque & ar<lb></lb>cus <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad invicem erit ratio æqualitatis. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et propterea hæ omnes lineæ, in omni de rationibus ul<lb></lb>timis argumentatione, pro ſe invicem uſurpari poſſunt. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si rectæ datæ<emph.end type="italics"></emph.end>AR, BR <emph type="italics"></emph>cum arcu<emph.end type="italics"></emph.end>AB, <emph type="italics"></emph>chorda<emph.end type="italics"></emph.end>AB <emph type="italics"></emph>& tangente<emph.end type="italics"></emph.end><lb></lb>AD, <emph type="italics"></emph>triangula tria<emph.end type="italics"></emph.end>ARB, ARB, ARD <emph type="italics"></emph>conſtituunt, dein <lb></lb>puncta<emph.end type="italics"></emph.end>A, B <emph type="italics"></emph>accedunt ad invicem: dico quod ultima forma <lb></lb>triangulorum evaneſcentium est ſimilitudinis, & ultima ratio <lb></lb>æqualitatis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam dum punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.056.2.jpg" xlink:href="039/01/056/2.jpg"></figure><lb></lb>accedit, <expan abbr="intelligãtur">intelligantur</expan> ſemper <emph type="italics"></emph>AB, AD, AR<emph.end type="italics"></emph.end><lb></lb>ad puncta longinqua <emph type="italics"></emph>b, d<emph.end type="italics"></emph.end>& <emph type="italics"></emph>r<emph.end type="italics"></emph.end>produci, <lb></lb>ipſique <emph type="italics"></emph>RD<emph.end type="italics"></emph.end>parallela agi <emph type="italics"></emph>rbd,<emph.end type="italics"></emph.end>& arcui <lb></lb><emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ſimilis ſemper ſit arcus <emph type="italics"></emph>Ab.<emph.end type="italics"></emph.end>Et coe<lb></lb>untibus punctis <emph type="italics"></emph>A, B,<emph.end type="italics"></emph.end>angulus <emph type="italics"></emph>bAd<emph.end type="italics"></emph.end>eva<lb></lb>neſcet, & propterea triangula tria ſemper <lb></lb>finita <emph type="italics"></emph>rAb, rAb, rAd<emph.end type="italics"></emph.end>coincident, ſunt<lb></lb>que eo nomine ſimilia & æqualia. </s> <s>Unde <lb></lb>& hiſce ſemper ſimilia & proportionalia <lb></lb><emph type="italics"></emph>RAB, RAB, RAD<emph.end type="italics"></emph.end>ſient ultimo ſibi <lb></lb>invicem ſimilia & æqualia. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Et hinc triangula illa, in omni de rationibus ultimis argu<lb></lb>mentatione, pro ſe invicem uſurpari poſſunt. </s></p><pb xlink:href="039/01/057.jpg" pagenum="29"></pb> <p type="main"> <s><emph type="center"></emph>LEMMA IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si recta<emph.end type="italics"></emph.end>AE <emph type="italics"></emph>& curva<emph.end type="italics"></emph.end>ABC <emph type="italics"></emph>poſitione datæ ſe mutuo ſecent in <lb></lb>angulo dato<emph.end type="italics"></emph.end>A, <emph type="italics"></emph>& ad rectam illam in alio dato angulo ordina<lb></lb>tim applicentur<emph.end type="italics"></emph.end>BD, CE, <emph type="italics"></emph>curvæ occurrentes in<emph.end type="italics"></emph.end>B, C; <emph type="italics"></emph>dein <lb></lb>puncta<emph.end type="italics"></emph.end>B, C <emph type="italics"></emph>ſimul accedant ad punctum<emph.end type="italics"></emph.end>A: <emph type="italics"></emph>dico quod areæ tri<lb></lb>angulorum<emph.end type="italics"></emph.end>ABD, ACE <emph type="italics"></emph>erunt ultimo ad invicem in duplicata <lb></lb>ratione laterum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Etenim dum puncta <emph type="italics"></emph>B, C<emph.end type="italics"></emph.end>acce<lb></lb><figure id="id.039.01.057.1.jpg" xlink:href="039/01/057/1.jpg"></figure><lb></lb>dunt ad punctum <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>intelligatur <lb></lb>ſemper <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>produci ad puncta lon<lb></lb>ginqua <emph type="italics"></emph>d<emph.end type="italics"></emph.end>& <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>ut ſint <emph type="italics"></emph>Ad, Ae<emph.end type="italics"></emph.end>ip<lb></lb>ſis <emph type="italics"></emph>AD, AE<emph.end type="italics"></emph.end>proportionales, & e<lb></lb>rigantur ordinatæ <emph type="italics"></emph>db, ec<emph.end type="italics"></emph.end>ordina<lb></lb>tis <emph type="italics"></emph>DB, EC<emph.end type="italics"></emph.end>parallelæ quæ occur<lb></lb>rant ipſis <emph type="italics"></emph>AB, AC<emph.end type="italics"></emph.end>productis in <lb></lb><emph type="italics"></emph>b<emph.end type="italics"></emph.end>& <emph type="italics"></emph>c.<emph.end type="italics"></emph.end>Duci intelligatur, tum curva <lb></lb><emph type="italics"></emph>Abc<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>ABC<emph.end type="italics"></emph.end>ſimilis, tum recta <lb></lb><emph type="italics"></emph>Ag,<emph.end type="italics"></emph.end>quæ tangat curvam utramque <lb></lb>in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>& ſecet ordinatim applica<lb></lb>tas <emph type="italics"></emph>DB, EC, db, ec<emph.end type="italics"></emph.end>in <emph type="italics"></emph>F, G, f, g.<emph.end type="italics"></emph.end><lb></lb>Tum manente longitudine <emph type="italics"></emph>Ae<emph.end type="italics"></emph.end>coeant puncta <emph type="italics"></emph>B, C<emph.end type="italics"></emph.end>cum puncto <emph type="italics"></emph>A<emph.end type="italics"></emph.end>; <lb></lb>& angulo <emph type="italics"></emph>cAg<emph.end type="italics"></emph.end>evaneſcente, coincident areæ curvilineæ <emph type="italics"></emph>Abd, Ace<emph.end type="italics"></emph.end><lb></lb>cum rectilineis <emph type="italics"></emph>Afd, Age:<emph.end type="italics"></emph.end>adeoque (per Lemma v) erunt in dupli<lb></lb>cata ratione laterum <emph type="italics"></emph>Ad, Ae:<emph.end type="italics"></emph.end>Sed his areis proportionales ſemper <lb></lb>ſunt areæ <emph type="italics"></emph>ABD, ACE,<emph.end type="italics"></emph.end>& his lateribus latera <emph type="italics"></emph>AD, AE.<emph.end type="italics"></emph.end>Ergo & <lb></lb>areæ <emph type="italics"></emph>ABD, ACE<emph.end type="italics"></emph.end>ſunt ultimo in duplicata ratione laterum <emph type="italics"></emph>AD, <lb></lb>AE. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Spatia quæ corpus urgente quacunque Vi finita deſcribit, five Vis <lb></lb>illa determinata & immutabilis ſit, five eadem continuo auge<lb></lb>atur vel continuo diminuatur, ſunt ipſo motus initio in duplica<lb></lb>ta ratione Temporum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Exponantur tempora per lineas <emph type="italics"></emph>AD, AE,<emph.end type="italics"></emph.end>& velocitates genitæ <lb></lb>per ordinatas <emph type="italics"></emph>DB, EC<emph.end type="italics"></emph.end>; & ſpatia his velocitatibus deſcripta, erunt <lb></lb>ut areæ <emph type="italics"></emph>ABD, ACE<emph.end type="italics"></emph.end>his ordinatis deſcriptæ, hoc eſt, ipſo motus <lb></lb>initio (per Lemma IX) in duplicata ratione temporum <emph type="italics"></emph>AD, AE. <lb></lb><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/058.jpg" pagenum="30"></pb><arrow.to.target n="note14"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note14"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Et hinc facile colligitur, quod corporum ſimiles ſimi<lb></lb>lium Figurarum partes temporibus proportionalibus deſcribentium <lb></lb>Errores, qui viribus quibuſvis æqualibus ad corpora ſimiliter ap<lb></lb>plicatis generantur, & menſurantur per diſtantias corporum a Fi<lb></lb>gurarum ſimilium locis illis ad quæ corpora eadem temporibus iiſ<lb></lb>dem proportionalibus abſque viribus iſtis pervenirent, ſunt ut qua<lb></lb>drata temporum in quibus generantur quam proxime. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Errores autem qui viribus proportionalibus ad ſimiles <lb></lb>Figurarum ſimilium partes ſimiliter applicatis generantur, ſunt ut <lb></lb>vires & quadrata temporum conjunctim. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Idem intelligendum eſt de ſpatiis quibuſvis quæ corpo<lb></lb>ra urgentibus diverſis viribus deſcribunt. </s> <s>Hæc ſunt, ipſo motus iNI<lb></lb>tio, ut vires & quadrata temporum conjunctim. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Ideoque vires ſunt ut ſpatia, ipſo motus initio, deſcripta <lb></lb>directe & quadrata temporum inverſe. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et quadrata temporum ſunt ut deſcripta ſpatia directe <lb></lb>& vires inverſe. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si quantitates indeterminatæ diverſorum generum conferantur <lb></lb>inter ſe, & earum aliqua dicatur eſſe ut eſt alia quævis directe vel <lb></lb>inverſe: ſenſus eſt, quod prior augetur vel diminuitur in eadem <lb></lb>ratione cum poſteriore, vel cum ejus reciproca. </s> <s>Et ſi earum aliqua <lb></lb>dicatur eſſe ut ſunt aliæ duæ vel plures directe vel inverſe: ſenſus <lb></lb>eſt, quod prima augetur vel diminuitur in ratione quæ componitur <lb></lb>ex rationibus in quibus aliæ vel aliarum reciprocæ augentur vel di<lb></lb>minuuntur. </s> <s>Ut ſi A dicatur eſſe ut B directe & C directe & D in<lb></lb>verſe: ſenſus eſt, quod A augetur vel diminuitur in eadem ratione <lb></lb>cum BXCX1/D, hoc eſt, quod A & (BC/D) ſunt ad invicem in ratio<lb></lb>ne data. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Subtenſa evaneſcens anguli contactus, in curvis omnibus curvatu<lb></lb>ram finitam ad punctum contactus habentibus, est ultimo in ra<lb></lb>tione duplicata ſubtenſæ arcus contermini.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>1. Sit arcus ille <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>tangens ejus <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>ſubtenſa anguli con<lb></lb>tactus ad tangentem perpendicularis <emph type="italics"></emph>BD,<emph.end type="italics"></emph.end>ſubtenſa arcus <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Huic <lb></lb>ſubtenſæ <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& tangenti <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>perpendiculares erigantur <emph type="italics"></emph>AG, BG,<emph.end type="italics"></emph.end><pb xlink:href="039/01/059.jpg" pagenum="31"></pb>concurrentes in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>; dein accedant puncta <emph type="italics"></emph>D, B, G,<emph.end type="italics"></emph.end>ad puncta <emph type="italics"></emph>d, b, g,<emph.end type="italics"></emph.end><lb></lb>ſitque <emph type="italics"></emph>J<emph.end type="italics"></emph.end>interſectio linearum <emph type="italics"></emph>BG, AG<emph.end type="italics"></emph.end>ultimo facta ubi puncta <emph type="italics"></emph>D, B<emph.end type="italics"></emph.end><lb></lb>accedunt uſque ad <emph type="italics"></emph>A.<emph.end type="italics"></emph.end>Manifeſtum eſt quod diſtantia <emph type="italics"></emph>GJ<emph.end type="italics"></emph.end>minor <lb></lb>eſſe poteſt quam aſſignata quævis. </s> <s>Eſt autem (ex natura circulorum <lb></lb>per puncta <emph type="italics"></emph>ABG, Abg<emph.end type="italics"></emph.end>tranſeuntium) <emph type="italics"></emph>ABquad.<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.059.1.jpg" xlink:href="039/01/059/1.jpg"></figure><lb></lb>æquale <emph type="italics"></emph>AGXBD,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Ab quad.<emph.end type="italics"></emph.end>æquale <emph type="italics"></emph>AgXbd,<emph.end type="italics"></emph.end><lb></lb>adeoque ratio <emph type="italics"></emph>AB quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ab quad.<emph.end type="italics"></emph.end>compo<lb></lb>nitur ex rationibus <emph type="italics"></emph>AG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ag<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>bd.<emph.end type="italics"></emph.end><lb></lb>Sed quoniam <emph type="italics"></emph>GJ<emph.end type="italics"></emph.end>aſſumi poteſt minor longitu<lb></lb>dine quavis aſſignata, fieri poteſt ut ratio <emph type="italics"></emph>AG<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>Ag<emph.end type="italics"></emph.end>minus differat a ratione æqualitatis quam <lb></lb>pro differentia quavis aſſignata, adeoque ut ra<lb></lb>tio <emph type="italics"></emph>AB quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ab quad.<emph.end type="italics"></emph.end>minus differat a ra<lb></lb>tione <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>bd<emph.end type="italics"></emph.end>quam pro differentia quavis <lb></lb>aſſignata. </s> <s>Eſt ergo, per Lemma 1, ratio ultima <lb></lb><emph type="italics"></emph>AB quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ab quad.<emph.end type="italics"></emph.end>æqualis rationi ultimæ <lb></lb><emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>bd. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Inclinetur jam <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>in angulo <lb></lb>quovis dato, & eadem ſemper erit ratio ultima <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>bd<emph.end type="italics"></emph.end>quæ <lb></lb>prius, adeoque eadem ae <emph type="italics"></emph>AB quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ab quad. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Et quamvis angulus <emph type="italics"></emph>D<emph.end type="italics"></emph.end>non detur, ſed recta <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad da<lb></lb>tum punctum convergente, vel alia quacunque lege conſtituatur; <lb></lb>tamen anguli <emph type="italics"></emph>D, d<emph.end type="italics"></emph.end>communi lege conſtituti ad æqualitatem ſemper <lb></lb>vergent & propius accedent ad invicem quam pro differentia qua<lb></lb>vis aſſignata, adeoque ultimo æquales erunt, per Lem. I & prop<lb></lb>terea lineæ <emph type="italics"></emph>BD, bd<emph.end type="italics"></emph.end>ſunt in eadem ratione ad invicem ac prius. <lb></lb><emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Unde eum tangentes <emph type="italics"></emph>AD, Ad,<emph.end type="italics"></emph.end>arcus <emph type="italics"></emph>AB, Ab,<emph.end type="italics"></emph.end>& eo<lb></lb>rum ſinus <emph type="italics"></emph>BC, bc<emph.end type="italics"></emph.end>fiant ultimo chordis <emph type="italics"></emph>AB, Ab<emph.end type="italics"></emph.end>æquales; erunt <lb></lb>etiam illorum quadrata ultimo ut ſubtenſæ <emph type="italics"></emph>BD, bd.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Eorundem quadrata ſunt etiam ultimo ut ſunt arcuum <lb></lb>ſagittæ quæ chordas biſecant & ad datum punctum convergunt. </s> <s><lb></lb>Nam ſagittæ illæ ſunt ut ſubtenſæ <emph type="italics"></emph>BD, bd.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Ideoque ſagitta eſt in duplicata ratione temporis quo <lb></lb>corpus data velocitate deſcribit arcum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Triangula rectilinea <emph type="italics"></emph>ADB, Adb<emph.end type="italics"></emph.end>ſunt ultimo in tripli<lb></lb>cata ratione laterum <emph type="italics"></emph>AD, Ad,<emph.end type="italics"></emph.end>inque ſeſquiplicata laterum <emph type="italics"></emph>DB, <lb></lb>db<emph.end type="italics"></emph.end>; utpote in compoſita ratione laterum <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DB, Ad<emph.end type="italics"></emph.end>& <emph type="italics"></emph>db<emph.end type="italics"></emph.end><lb></lb>exiſtentia. </s> <s>Sic & triangula <emph type="italics"></emph>ABC, Abc<emph.end type="italics"></emph.end>ſunt ultimo in triplicata <lb></lb>ratione laterum <emph type="italics"></emph>BC, bc.<emph.end type="italics"></emph.end>Rationem vero Seſquiplicatam voco tri<lb></lb>plicatæ ſubduplicatam, quæ nempe ex ſimplici & ſubduplicata com<lb></lb>ponitur, quamque alias Seſquialteram dicunt. </s></p><pb xlink:href="039/01/060.jpg" pagenum="32"></pb> <p type="main"> <s><arrow.to.target n="note15"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note15"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et quoniam <emph type="italics"></emph>DB, db<emph.end type="italics"></emph.end>ſunt ultimo parallelæ & in dupli<lb></lb>cata ratione ipſarum <emph type="italics"></emph>AD, Ad:<emph.end type="italics"></emph.end>erunt areæ ultimæ curvilineæ <emph type="italics"></emph>ADB, <lb></lb>Adb<emph.end type="italics"></emph.end>(ex natura Parabolæ) duæ tertiæ partes triangulorum rectili<lb></lb>neorum <emph type="italics"></emph>ADB, Adb<emph.end type="italics"></emph.end>; & ſegmenta <emph type="italics"></emph>AB, Ab<emph.end type="italics"></emph.end>partes tertiæ eo<lb></lb>rundem triangulorum. </s> <s>Et inde hæ areæ & hæc ſegmenta erunt in <lb></lb>triplicata ratione tum tangentium <emph type="italics"></emph>AD, Ad<emph.end type="italics"></emph.end>; tum chordarum & <lb></lb>arcuum <emph type="italics"></emph>AB, Ab.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cæterum in his omnibus ſupponimus angulum contactus nec in<lb></lb>finite majorem eſſe angulis contactuum, quos Circuli continent cum <lb></lb>tangentibus ſuis, nec iiſdem infinite minorem; hoc eſt, curvaturam <lb></lb>ad punctum <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>nec infinite parvam eſſe nec infinite magnam, ſeu <lb></lb>intervallum <emph type="italics"></emph>AJ<emph.end type="italics"></emph.end>finitæ eſſe magnitudinis. </s> <s>Capi enim poteſt <emph type="italics"></emph>DB<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>AD<emph type="sup"></emph>3<emph.end type="sup"></emph.end>:<emph.end type="italics"></emph.end>quo in caſu Circulus nullus per punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>inter tangen<lb></lb>tem <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>& curvam <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>duci poteſt, proindeque angulus contactus <lb></lb>erit infinite minor Circularibus. </s> <s>Et ſimili argumento ſi fiat <emph type="italics"></emph>DB<emph.end type="italics"></emph.end><lb></lb>ſucceſſive ut <emph type="italics"></emph>AD<emph.end type="italics"></emph.end><emph type="sup"></emph>4<emph.end type="sup"></emph.end>, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end><emph type="sup"></emph>5<emph.end type="sup"></emph.end>, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end><emph type="sup"></emph>6<emph.end type="sup"></emph.end>, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end><emph type="sup"></emph>7<emph.end type="sup"></emph.end>, &c. </s> <s>habebitur ſeries an<lb></lb>gulorum contactus pergens in infinitum, quorum quilibet poſte<lb></lb>rior eſt infinite minor priore. </s> <s>Et ſi fiat <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>ſucceſſive ut <emph type="italics"></emph>AD<emph.end type="italics"></emph.end><emph type="sup"></emph>2<emph.end type="sup"></emph.end>, <lb></lb><emph type="italics"></emph>AD<emph.end type="italics"></emph.end>3/2, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>4/3, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>5/4, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>6/5, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>7/6, &c. </s> <s>habebitur alia ſeries infinita <lb></lb>angulorum contactus, quorum primus eſt ejuſdem generis cum Cir<lb></lb>cularibus, ſecundus infinite major, & quilibet poſterior infinite ma<lb></lb>jor priore. </s> <s>Sed & inter duos quoſvis ex his angulis poteſt ſeries <lb></lb>utrinQ.E.I. infinitum pergens angulorum intermediorum inſeri, <lb></lb>quorum quilibet poſterior erit infinite major minorve priore. </s> <s>Ut <lb></lb>ſi inter terminos <emph type="italics"></emph>AD<emph.end type="italics"></emph.end><emph type="sup"></emph>2<emph.end type="sup"></emph.end> & <emph type="italics"></emph>AD<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end> inſeratur ſeries <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>(13/6), <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>(1<gap></gap>/5), <lb></lb><emph type="italics"></emph>AD<emph.end type="italics"></emph.end>9/4, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>7/3, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>5/2, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>8/3, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>(11/4), <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>(14/5), <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>(17/6), &c. </s> <s>Et rur<lb></lb>ſus inter binos quoſvis angulos hujus ſeriei inſeri poteſt ſeries no<lb></lb>va angulorum intermediorum ab invicem infinitis intervallis diffe<lb></lb>rentium. </s> <s>Neque novit natura limitem. </s></p> <p type="main"> <s>Quæ de curvis lineis deque ſuperficiebus comprehenſis demon<lb></lb>ſtrata ſunt, facile applicantur ad ſolidorum ſuperficies curvas & <lb></lb>contenta. </s> <s>Præmiſi vero hæc Lemmata, ut effugerem tædium dedu<lb></lb>cendi perplexas demonſtrationes, more veterum Geometrarum, ad <lb></lb>abſurdum. </s> <s>Contractiores enim redduntur demonſtrationes per me<lb></lb>thodum Indiviſibilium. </s> <s>Sed quoniam durior eſt Indiviſibilium hy<lb></lb>potheſis, & propterea methodus illa minus Geometrica cenſetur; <lb></lb>malui demonſtrationes rerum ſequentium ad ultimas quantitatum <pb xlink:href="039/01/061.jpg" pagenum="33"></pb>evaneſcentium ſummas & rationes, primaſque naſcentium, id eſt, <lb></lb>ad limites ſummarum & rationum deducere; & propterea limitum <lb></lb>illorum demonſtrationes qua potui brevitate præmittere. </s> <s>His enim <lb></lb>idem præſtatur quod per methodum Indiviſibilium; & principiis de<lb></lb>monſtratis jam tutius utemur. </s> <s>Proinde in ſequentibus, ſiquando <lb></lb>quantitates tanquam ex particulis conſtantes conſideravero, vel ſi <lb></lb>pro rectis uſurpavero lineolas curvas; nolim indiviſibilia, ſed eva<lb></lb>neſcentia diviſibilia, non ſummas & rationes partium determinata<lb></lb>rum, ſed ſummarum & rationum limites ſemper intelligi; vimque <lb></lb>talium demonſtrationum ad methodum præcedentium Lemmatum <lb></lb>ſemper revocari. </s></p> <p type="main"> <s>Objectio eſt, quod quantitatum evaneſcentium nulla ſit ultima <lb></lb>proportio; quippe quæ, antequam evanuerunt, non eſt ultima, ubi <lb></lb>evanuerunt, nulla eſt. </s> <s>Sed & eodem argumento æque contendi poſſet <lb></lb>nullam eſſe corporis ad certum locum pervenientis velocitatem ul<lb></lb>timam: hanc enim, antequam corpus attingit locum, non eſſe ulti<lb></lb>mam, ubi attingit, nullam eſſe. </s> <s>Et reſponſio facilis eſt: Per velocita<lb></lb>tem ultimam intelligi eam, qua corpus movetur neque antequam <lb></lb>attingit locum ultimum & motus ceſſat, neque poſtea, ſed tunc <lb></lb>cum attingit; id eſt, illam ipſam velocitatem quacum corpus attin<lb></lb>git locum ultimum & quacum motus ceſſat. </s> <s>Et ſimiliter per ulti<lb></lb>mam rationem quantitatum evaneſcentium, intelligendam eſſe ratio<lb></lb>nem quantitatum non antequam evaneſcunt, non poſtea, ſed qua<lb></lb>cum evaneſcunt. </s> <s>Pariter & ratio prima naſcentium eſt ratio qua<lb></lb>cum naſcuntur. </s> <s>Et ſumma prima & ultima eſt quacum eſſe (vel <lb></lb>augeri & minui) incipiunt & ceſſant. </s> <s>Extat limes quem velocitas <lb></lb>in fine motus attingere poteſt, non autem tranſgredi. </s> <s>Hæc eſt <lb></lb>velocitas ultima. </s> <s>Et par eſt ratio limitis quantitatum & propor<lb></lb>tionum omnium incipientium & ceſſantium. </s> <s>Cumque hic limes <lb></lb>ſit certus & definitus, Problema eſt vere Geometricum eundem de<lb></lb>terminare. </s> <s>Geometrica vero omnia in aliis Geometricis determi<lb></lb>nandis ac demonſtrandis legitime uſurpantur. </s></p> <p type="main"> <s>Contendi etiam poteſt, quod ſi dentur ultimæ quantitatum eva<lb></lb>neſcentium rationes, dabuntur & ultimæ magnitudines: & ſic quan<lb></lb>titas omnis conſtabit ex Indiviſibilibus, contra quam <emph type="italics"></emph>Euclides<emph.end type="italics"></emph.end>de <lb></lb>Incommenſurabilibus, in libro decimo Elementorum, demonſtravit. </s> <s><lb></lb>Verum hæc Objectio falſæ innititur hypotheſi. </s> <s>Ultimæ rationes <lb></lb>illæ quibuſcum quantitates evaneſcunt, revera non ſunt rationes <lb></lb>quantitatum ultimarum, ſed limites ad quos quantitatum ſine limi<lb></lb>te decreſcentium rationes ſemper appropinquant; & quas propius <lb></lb>aſſequi poſſunt quam pro data quavis differentia, nunquam vero </s></p><pb xlink:href="039/01/062.jpg" pagenum="34"></pb> <p type="main"> <s><arrow.to.target n="note16"></arrow.to.target>tranſgredi, neque prius attingere quam quantitates diminuuntur in <lb></lb>infinitum. </s> <s>Res clarius intelligetur in infinite magnis. </s> <s>Si quantitates <lb></lb>duæ quarum data eſt differentia auges ntur in infinitum, dabitur <lb></lb>harum ultima ratio, nimirum ratio æqualitatis, nec tamen ideo da<lb></lb>buntur quantitates ultimæ ſeu maximæ quarum iſta eſt ratio. </s> <s>Igitur <lb></lb>in ſequentibus, ſiquando facili rerum conceptui conſulens dixero <lb></lb>quantitates quam minimas, vel evaneſcentes, vel ultimas; cave in<lb></lb>telligas quantitates magnitudine determinatas, ſed cogita ſemper <lb></lb>diminuendas ſine limite. </s></p> <p type="margin"> <s><margin.target id="note16"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"></emph>SECTIO II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Inventione Virium Centripetarum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO I. THEOREMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Areas, quas corpora in gyros acta radiis ad immobile centrum virium <lb></lb>ductis deſcribunt, & in planis immobilibus conſiſtere, & eſſe tem<lb></lb>poribus proportionales.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Dividatur tempus in partes æquales, & prima temporis parte de<lb></lb>ſcribat corpus vi inſita rectam <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Idem ſecunda temporis parte, ſi <lb></lb>nil impediret, recta <lb></lb><figure id="id.039.01.062.1.jpg" xlink:href="039/01/062/1.jpg"></figure><lb></lb>pergeret ad <emph type="italics"></emph>c,<emph.end type="italics"></emph.end>(per <lb></lb>Leg. </s> <s>1.) deſcribens <lb></lb>lineam <emph type="italics"></emph>Bc<emph.end type="italics"></emph.end>æqualem <lb></lb>ipſi <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; adeo ut ra<lb></lb>diis <emph type="italics"></emph>AS, BS, cS<emph.end type="italics"></emph.end>ad <lb></lb>centrum actis, con<lb></lb>fectæ forent æqua<lb></lb>les areæ <emph type="italics"></emph>ASB, BSc.<emph.end type="italics"></emph.end><lb></lb>Verum ubi corpus <lb></lb>venit ad <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>agat vis <lb></lb>centripeta impul<lb></lb>ſu unico ſed mag<lb></lb>no, efficiatque ut <lb></lb>corpus de recta <emph type="italics"></emph>Bc<emph.end type="italics"></emph.end><lb></lb>declinet & pergat <lb></lb>in recta <emph type="italics"></emph>BC.<emph.end type="italics"></emph.end>Ipſi <lb></lb><emph type="italics"></emph>BS<emph.end type="italics"></emph.end>parallela agatur <emph type="italics"></emph>cC,<emph.end type="italics"></emph.end>occurens <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>in <emph type="italics"></emph>C<emph.end type="italics"></emph.end>; & completa ſecunda <lb></lb>temporis parte, corpus (per Legum Corol. </s> <s>1.) reperietur in <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>in <pb xlink:href="039/01/063.jpg" pagenum="35"></pb>eodem plano cum triangulo <emph type="italics"></emph>ASB.<emph.end type="italics"></emph.end>Junge <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>; & triangulum <emph type="italics"></emph>SBC,<emph.end type="italics"></emph.end><lb></lb>ob parallelas <emph type="italics"></emph>SB, Cc,<emph.end type="italics"></emph.end>æquale erit triangulo <emph type="italics"></emph>SBc,<emph.end type="italics"></emph.end>atque adeo etiam <lb></lb>triangulo <emph type="italics"></emph>SAB.<emph.end type="italics"></emph.end>Simili argumento ſi vis centripeta ſucceſſive agat <lb></lb>in <emph type="italics"></emph>C, D, E,<emph.end type="italics"></emph.end>&c. </s> <s>faciens ut corpus ſingulis temporis particulis ſin<lb></lb>gulas deſeribat rectas <emph type="italics"></emph>CD, DE, EF,<emph.end type="italics"></emph.end>&c. </s> <s>jacebunt hæ omnes in <lb></lb>eodem plano; & triangulum <emph type="italics"></emph>SCD<emph.end type="italics"></emph.end>triangulo <emph type="italics"></emph>SBC,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SDE<emph.end type="italics"></emph.end>ipſi <lb></lb><emph type="italics"></emph>SCD,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SEF<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>SDE<emph.end type="italics"></emph.end>æquale erit. </s> <s>Æqualibus igitur tempori<lb></lb>bus æquales areæ in plano immoto deſcribuntur: & componendo, <lb></lb>ſunt arearum ſummæ quævis <emph type="italics"></emph>SADS, SAFS<emph.end type="italics"></emph.end>inter ſe, ut ſunt tem<lb></lb>pora deſcriptionum. </s> <s>Augeatur jam numerus & minuatur latitudo <lb></lb>triangulorum in infinitum; & eorum ultima perimeter <emph type="italics"></emph>ADF,<emph.end type="italics"></emph.end>(per <lb></lb>Corollarium quartum Lemmatis tertii) erit linea curva: adeoque vis <lb></lb>centripeta, qua corpus a tangente hujus curvæ perpetuo retrahitur, <lb></lb>aget indeſinenter; areæ vero quævis deſcriptæ <emph type="italics"></emph>SADS, SAFS<emph.end type="italics"></emph.end><lb></lb>temporibus deſcriptionum ſemper proportionales, erunt iiſdem tem<lb></lb>poribus in hoc caſu proportionales. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Velocitas corporis in centrum immobile attracti eſt in <lb></lb>ſpatiis non reſiſtentibus reciproce ut perpendiculum a centro illo in <lb></lb>Orbis tangentem rectilineam demiſſum. </s> <s>Eſt enim velocitas in locis <lb></lb>illis <emph type="italics"></emph>A, B, C, D, E,<emph.end type="italics"></emph.end>ut ſunt baſes æqualium triangulorum <emph type="italics"></emph>AB, BC, <lb></lb>CD, DE, EF<emph.end type="italics"></emph.end>; & hæ baſes ſunt reciproce ut perpendicula in ipſas <lb></lb>demiſſa. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si arcuum duorum æqualibus temporibus in ſpatiis non <lb></lb>reſiſtentibus ab eodem corpore ſucceſſive deſcriptorum chordæ <emph type="italics"></emph>AB, <lb></lb>BC<emph.end type="italics"></emph.end>compleantur in parallelogrammum <emph type="italics"></emph>ABCU,<emph.end type="italics"></emph.end>& hujus diagona<lb></lb>lis <emph type="italics"></emph>BU<emph.end type="italics"></emph.end>in ea poſitione quam ultimo habet ubi arcus illi in infiNI<lb></lb>tum diminuuntur, producatur utrinque; tranſibit eadem per cen<lb></lb>trum virium. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si arcuum æqualibus temporibus in ſpatiis non reſiſten<lb></lb>tibus deſcriptorum chordæ <emph type="italics"></emph>AB, BC<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>DE, EF<emph.end type="italics"></emph.end>compleantur in <lb></lb>parallelogramma <emph type="italics"></emph>ABCU, DEFZ<emph.end type="italics"></emph.end>; vires in <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>E<emph.end type="italics"></emph.end>ſunt ad invi<lb></lb>cem in ultima ratione diagonalium <emph type="italics"></emph>BU, EZ,<emph.end type="italics"></emph.end>ubi arcus iſti in infi<lb></lb>nitum diminuuntur. </s> <s>Nam corporis motus <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>componun<lb></lb>tur (per Legum Corol. </s> <s>1.) ex motibus <emph type="italics"></emph>Bc, BU<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Ef, EZ:<emph.end type="italics"></emph.end>at<lb></lb>qui <emph type="italics"></emph>BU<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EZ,<emph.end type="italics"></emph.end>ipſis <emph type="italics"></emph>Cc<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end>æquales, in Demonſtratione Pro<lb></lb>poſitionis hujus generabantur ab impulſibus vis centripetæ in B & <lb></lb><emph type="italics"></emph>E,<emph.end type="italics"></emph.end>ideoque ſunt his impulſibus proportionales. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Vires quibus corpora quælibet in ſpatiis non reſiſtenti<lb></lb>bus a motibus rectilineis retrahuntur ac detorquentur in orbes cur<lb></lb>vos ſunt inter ſe ut arcuum æqualibus temporibus deſcriptorum ſa<lb></lb>gittæ illæ quæ convergunt ad centrum virium, & chordas biſecant <pb xlink:href="039/01/064.jpg" pagenum="36"></pb><arrow.to.target n="note17"></arrow.to.target>ubi arcus illi in infinitum diminuuntur. </s> <s>Nam hæ ſagittæ ſunt ſe<lb></lb>miſſes diagonalium de quibus egimus in Corollario tertio. </s></p> <p type="margin"> <s><margin.target id="note17"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Ideoque vires eædem ſunt ad vim gravitatis, ut hæ ſa<lb></lb>gittæ ad ſagittas horizonti perpendiculares arcuum Parabolieorum <lb></lb>quos projectilia eodem tempore deſcribunt. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Eadem omnia obtinent per Legum Corol. </s> <s>IV, ubi plana <lb></lb>in quibus corpora moventur, una cum centris virium quæ in ipſis <lb></lb>fita ſunt, non quieſcunt, ſed moventur uniformiter in directum. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO II. THEOREMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corpus omne, quod movetur in linea aliqua curva in plano de<lb></lb>ſcripta, & radio ducto ad punctum vel immobile, vel motu rectili<lb></lb>neo uniformiter progrediens, deſcribit areas circa punctum illud <lb></lb>temporibus proportionales, urgetur a vi centripeta tendente ad idem <lb></lb>punctum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Nam corpus omne quod movetur in linea curva, detor<lb></lb>quetur de curſu rectilineo per vim aliquam in ipſum agentem (per <lb></lb>Leg. </s> <s>1.) Et vis illa qua corpus de curſu rectilineo detorquetur, & <lb></lb>cogitur triangula quam minima <emph type="italics"></emph>SAB, SBC, SCD,<emph.end type="italics"></emph.end>&c. </s> <s>circa <lb></lb>punctum immobile <emph type="italics"></emph>S<emph.end type="italics"></emph.end>temporibus æqualibus æqualia deſcribere, a<lb></lb>git in loco <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ſecundum lineam parallelam ipſi <emph type="italics"></emph>cC<emph.end type="italics"></emph.end>(per Prop. </s> <s>XL, <lb></lb>Lib. </s> <s>1 Elem. </s> <s>& Leg. </s> <s>11.) hoc eſt, ſecundum lineam <emph type="italics"></emph>BS<emph.end type="italics"></emph.end>; & in loco <lb></lb><emph type="italics"></emph>C<emph.end type="italics"></emph.end>ſecundum lineam ipſi <emph type="italics"></emph>dD<emph.end type="italics"></emph.end>parallelam, hoc eſt, ſecundum lineam <lb></lb><emph type="italics"></emph>SC,<emph.end type="italics"></emph.end>&c. </s> <s>Agit ergo ſemper ſecundum lineas tendentes ad punctum <lb></lb>illud immobile <emph type="italics"></emph>S. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Et, per Legum Corollarium quintum, perinde eſt ſive <lb></lb>quieſcat ſuperficies in qua corpus deſcribit figuram curvilineam, <lb></lb>ſive moveatur eadem una cum corpore, figura deſcripta, & puncto <lb></lb>ſuo <emph type="italics"></emph>S<emph.end type="italics"></emph.end>uniformiter in directum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. In Spatiis vel Mediis non reſiſtentibus, ſi areæ non ſunt <lb></lb>temporibus proportionales, vires non tendunt ad concurſum radio<lb></lb>rum; ſed inde declinant in conſequentia ſeu verſus plagam in quam <lb></lb>fit motus, ſi modo arearum deſcriptio acceleratur: ſin retardatur, de<lb></lb>clinant in antecedentia. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. In Mediis etiam reſiſtentibus, ſi arearum deſcriptio accele<lb></lb>ratur, virium directiones declinant a concurſu radiorum verſus plagam <lb></lb>in quam ſit motus. </s></p><pb xlink:href="039/01/065.jpg" pagenum="37"></pb> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Urgeri poteſt corpus a vi centripeta compoſita ex pluribus viri<lb></lb>bus. </s> <s>In hoc caſu ſenſus Propoſitionis eſt, quod vis illa quæ ex om<lb></lb>nibus componitur, tendit ad punctum <emph type="italics"></emph>S.<emph.end type="italics"></emph.end>Porro ſi vis aliqua agat <lb></lb>perpetuo ſecundum lineam ſuperficiei deſcriptæ perpendicularem; <lb></lb>hæc faciet ut corpus deflectatur a plano ſui motus: ſed quantita<lb></lb>tem ſuperficiei deſcriptæ nec augebit nec minuet, & propterea in <lb></lb>compoſitione virium negligenda eſt. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO III. THEOREMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corpus omne, quod radio ad centrum corporis alterius utcunque moti <lb></lb>ducto deſcribit areas circa centrum illud temporibus proportiona<lb></lb>les, urgetur vi compoſita ex vi centripeta tendente ad corpus il<lb></lb>lud alterum, & ex vi omni acceleratrice qua corpus illud alterum <lb></lb>urgetur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit corpus primum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>& corpus alterum <emph type="italics"></emph>T:<emph.end type="italics"></emph.end>& (per Legum Corol. </s> <s><lb></lb>VI.) ſi vi nova, quæ æqualis & contraria ſit illi qua corpus alterum <lb></lb><emph type="italics"></emph>T<emph.end type="italics"></emph.end>urgetur, urgeatur corpus utrumque ſecundum lineas parallelas; <lb></lb>perget corpus primum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>deſcribere circa corpus alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>areas <lb></lb>eaſdem ac prius: vis autem, qua corpus alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>urgebatur, jam <lb></lb>deſtruetur per vim ſibi æqualem & contrariam; & propterea (per <lb></lb>Leg. </s> <s>1.) corpus illud alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ſibimet ipſi jam relictum vel qui<lb></lb>eſcet vel movebitur uniformiter in directum: & corpus primum <emph type="italics"></emph>L<emph.end type="italics"></emph.end><lb></lb>urgente differentia virium, id eſt, urgente vi reliqua perget areas <lb></lb>temporibus proportionales circa corpus alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>deſcribere. </s> <s>Ten<lb></lb>dit igitur (per Theor. </s> <s>11.) differentia virium ad corpus illud alte<lb></lb>rum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ut centrum. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi corpus unum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>radio ad alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ducto de<lb></lb>ſcribit areas temporibus proportionales; atQ.E.D. vi tota (ſive ſim<lb></lb>plici, ſive ex viribus pluribus, juxta Legum Corollarium ſecundum, <lb></lb>compoſita,) qua corpus prius <emph type="italics"></emph>L<emph.end type="italics"></emph.end>urgetur, ſubducatur (per idem Le<lb></lb>gum Corollarium) vis tota acceleratrix qua corpus alterum urgetur: <lb></lb>vis omnis reliqua qua corpus prius urgetur tendet ad corpus alte<lb></lb>rum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ut centrum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et, ſi areæ illæ ſunt temporibus quamproxime propor<lb></lb>tionales, vis reliqua tendet ad corpus alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>quamproxime. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et vice verſa, ſi vis reliqua tendit quamproxime ad <pb xlink:href="039/01/066.jpg" pagenum="38"></pb><arrow.to.target n="note18"></arrow.to.target>corpus alterum <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>erunt areæ illæ temporibus quamproxime pro<lb></lb>portionales. </s></p> <p type="margin"> <s><margin.target id="note18"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Si corpus <emph type="italics"></emph>L<emph.end type="italics"></emph.end>radio ad alterum corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ducto deſcri<lb></lb>bit areas quæ, cum temporibus collatæ, ſunt valde inæquales; & <lb></lb>corpus illud alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>vel quieſcit vel movetur uniformiter in di<lb></lb>rectum: actio vis centripetæ ad corpus illud alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>tendentis, <lb></lb>vel nulla eſt, vel miſcetur & componitur cum actionibus admodum <lb></lb>potentibus aliarum virium: Viſque tota ex omnibus, ſi plures ſunt <lb></lb>vires, compoſita, ad aliud (ſive immobile ſive mobile) centrum <lb></lb>dirigitur. </s> <s>Idem obtinet, ubi corpus alterum motu quocunque mo<lb></lb>vetur; ſi modo vis centripeta ſumatur, quæ reſtat poſt ſubductio<lb></lb>nem vis totius in corpus illud alterum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>agentis. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Quoniam æquabilis arearum deſcriptio Index eſt Centri, quod <lb></lb>vis illa reſpicit qua corpus maxime afficitur, quaque retrahitur a mo<lb></lb>tu rectilineo & in orbita ſua retinetur: quidni uſurpemus in ſequen<lb></lb>tibus æquabilem arearum deſcriptionem, ut Indicem Centri circum <lb></lb>quod motus omnis circularis in ſpatiis liberis peragitur? </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO IV. THEOREMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corporum, quæ diverſos circulos æquabili motu deſcribunt, vires cen<lb></lb>tripetas ad centra eorundem circulorum tendere; & eſſe inter ſe, <lb></lb>ut ſunt arcuum ſimul deſcriptorum quadrata applicata ad circulo<lb></lb>rum radios.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Tendunt hæ vires ad centra circulorum per Prop.II. & Corol. </s> <s>II. <lb></lb>Prop. </s> <s>1; & ſunt inter ſe ut arcuum æqualibus temporibus quam miNI<lb></lb>mis deſcriptorum ſinus verſi per Corol. </s> <s>IV. Prop. </s> <s>I; hoc eſt, ut qua<lb></lb>drata arcuum eorundem ad diametros circulorum applicata per <lb></lb>Lem. </s> <s>VII: & propterea, cum hi arcus ſint ut arcus temporibus <lb></lb>quibuſvis æqualibus deſcripti, & diametri ſint ut eorum radii; vi<lb></lb>res erunt ut arcuum quorumvis ſimul deſcriptorum quadrata ap<lb></lb>plicata ad radios circulorum. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur, cum arcus illi ſint ut velocitates corporum, vi<lb></lb>res centripetæ ſunt ut velocitatum quadrata applicata ad radios <lb></lb>circulorum: hoc eſt, ut cum Geometris loquar, vires ſunt in ra<lb></lb>tione compoſita ex duplicata ratione velocitatum directe & ratione <lb></lb>ſimplici radiorum inverſe. </s></p><pb xlink:href="039/01/067.jpg" pagenum="39"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et, cum tempora periodica ſint in ratione compoſita ex <lb></lb>ratione radiorum directe & ratione velocitatum inverſe, vires cen<lb></lb>tripetæ ſunt reciproce ut quadrata temporum periodieorum appli<lb></lb>cata ad circulorum radios; hoc eſt, in ratione compoſita ex ratione <lb></lb>radiorum directe & ratione duplicata temporum periodieorum in<lb></lb>verſe. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Unde, ſi tempora periodica æquentur & propterea ve<lb></lb>locitates ſint ut radii; erunt etiam vires centripetæ ut radii: & <lb></lb>contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end>4. Si & tempora periodica & velocitates ſint in ratione ſub<lb></lb>duplicata radiorum; æquales erunt vires centripetæ inter ſe: & <lb></lb>contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Si tempora periodica ſint ut radii & propterea veloci<lb></lb>tates æquales; vires centriperæ erunt reciproce ut radii: & contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Si tempora periodica ſint in ratione ſeſquiplicata radio<lb></lb>rum & propterea velocitates reciproce in radiorum ratione ſubdu<lb></lb>plicata; vires centripetæ erunt reciproce ut quadrata radiorum: <lb></lb>& contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Et univerſaliter, ſi tempus periodicum ſit ut Radii <emph type="italics"></emph>R<emph.end type="italics"></emph.end><lb></lb>poteſtas quælibet <emph type="italics"></emph>R<emph type="sup"></emph>n<emph.end type="sup"></emph.end>,<emph.end type="italics"></emph.end>& propterea velocitas reciproce ut Radii <lb></lb>poteſtas <emph type="italics"></emph>R<emph type="sup"></emph>n-1<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>; erit vis centripeta reciproce ut Radii poteſtas <emph type="italics"></emph>R<emph type="sup"></emph>2n-1<emph.end type="sup"></emph.end>:<emph.end type="italics"></emph.end><lb></lb>& contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Eadem omnia de temporibus, velocitatibus, & viribus, qui<lb></lb>bus corpora ſimiles figurarum quarumcunque ſimilium, centraque <lb></lb>in figuris illis ſimiliter poſita habentium, partes deſcribunt, conſe<lb></lb>quuntur ex Demonſtratione præcedentium ad hoſce caſus applicata. </s> <s><lb></lb>Applicatur autem ſubſtituendo æquabilem arearum deſcriptionem <lb></lb>pro æquabili motu, & diſtantias corporum a centris pro radiis uſur<lb></lb>pando. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>9. Ex eadem demonſtratione conſequitur etiam; quod ar<lb></lb>cus, quem corpus in circulo data vi centripeta uniformiter revolven<lb></lb>do tempore quovis deſcribit, medius eſt proportionalis inter dia<lb></lb>metrum circuli, & deſcenſum corporis eadem data vi eodem que tem<lb></lb>pore cadendo confectum. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Caſus Corollarii ſexti obtinet in corporibus cæleſtibus, (ut ſeor<lb></lb>ſum collegerunt etiam noſtrates <emph type="italics"></emph>Wrennus, Hookius<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Hallæus<emph.end type="italics"></emph.end>) & <lb></lb>propterea quæ ſpectant ad vim centripetam decreſcentem in dupli<lb></lb>cata ratione diſtantiarum a centris, decrevi fuſius in ſequentibus <lb></lb>exponere. <pb xlink:href="039/01/068.jpg" pagenum="40"></pb><arrow.to.target n="note19"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note19"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Porro præcedentis propoſitionis & corollariorum ejus beneficio, <lb></lb>colligitur etiam proportio vis centripetæ ad vim quamlibet notam, <lb></lb>qualis eſt ea Gravitatis. </s> <s>Nam ſi corpus in circulo Terræ concen<lb></lb>trico vi gravitatis ſuæ revolvatur, hæc gravitas eſt ipſius vis centri<lb></lb>peta. </s> <s>Datur autem, ex deſcenſu gravium, & tempus revolutionis <lb></lb>unius, & arcus dato quovis tempore deſcriptus, per hujus Corol. </s> <s><lb></lb>IX. </s> <s>Et hujuſmodi propoſitionibus <emph type="italics"></emph>Hugenius,<emph.end type="italics"></emph.end>in eximio ſuo Tracta<lb></lb>tu <emph type="italics"></emph>de Horologio Oſcillatorio,<emph.end type="italics"></emph.end>vim gravitatis cum revolventium vi<lb></lb>ribus centrifugis contulit. </s></p> <p type="main"> <s>Demonſtrari etiam poſſunt præcedentia in hunc modum. </s> <s>In cir<lb></lb>culo quovis deſcribi intelligatur Polygonum laterum quotcunque. </s> <s><lb></lb>Et ſi corpus, in polygoni lateribus data cum velocitate movendo, <lb></lb>ad ejus angulos ſingulos a circulo reflectatur; vis qua ſingulis re<lb></lb>flexionibus impingit in circulum erit ut ejus velocitas: adeoque <lb></lb>ſumma virium in dato tempore erit ut velocitas illa & numerus re<lb></lb>flexionum conjunctim: hoc eſt (ſi polygonum detur ſpecie) ut longi<lb></lb>tudo dato illo tempore deſcripta & longitudo eadem applicata ad <lb></lb>Radium circuli; id eſt, ut quadratum longitudinis illius applicatum <lb></lb>ad Radium: adeoque, ſi polygonum lateribus infinite diminutis co<lb></lb>incidat cum circulo, ut quadratum arcus dato tempore deſcripti ap<lb></lb>plicatum ad radium. </s> <s>Hæc eſt vis centrifuga, qua corpus urget cir<lb></lb>culum: & huic æqualis eſt vis contraria, qua circulus continuo re<lb></lb>pellit corpus centrum verſus. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO. V. PROBLEMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Data quibuſcunQ.E.I. locis velocitate, qua corpus figuram datam vi<lb></lb>ribus ad commune aliquod centrum tendentibus deſcribit, centrum <lb></lb>illud invenire.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Figuram deſcriptam tangant rectæ tres <emph type="italics"></emph>PT, TQV, VR<emph.end type="italics"></emph.end>in <lb></lb>punctis totidem <emph type="italics"></emph>P, Q, R,<emph.end type="italics"></emph.end>concurrentes in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>V.<emph.end type="italics"></emph.end>Ad tangentes <lb></lb>erigantur perpendicula <emph type="italics"></emph>PA, QB, RC,<emph.end type="italics"></emph.end>velocitatibus corporis in <lb></lb>punctis illis <emph type="italics"></emph>P, Q, R<emph.end type="italics"></emph.end>a quibus eriguntur reciproce proportionalia; <lb></lb>id eſt, ita ut ſit <emph type="italics"></emph>PA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>QB<emph.end type="italics"></emph.end>ut velocitas in <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>ad velocitatem in <lb></lb><emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>QB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>RC<emph.end type="italics"></emph.end>ut velocitas in <emph type="italics"></emph>R<emph.end type="italics"></emph.end>ad velocitatem in <emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end>Per <lb></lb>perpendiculorum terminos <emph type="italics"></emph>A, B, C<emph.end type="italics"></emph.end>ad angulos rectos ducantur <emph type="italics"></emph>AD, <lb></lb>DBE, EC<emph.end type="italics"></emph.end>concurrentes in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>E:<emph.end type="italics"></emph.end>Et actæ <emph type="italics"></emph>TD, VE<emph.end type="italics"></emph.end>concur<lb></lb>rent in centro qæſito <emph type="italics"></emph>S.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/069.jpg" pagenum="41"></pb><figure id="id.039.01.069.1.jpg" xlink:href="039/01/069/1.jpg"></figure> <p type="main"> <s>Nam perpendicula a centro <emph type="italics"></emph>S<emph.end type="italics"></emph.end><lb></lb>in tangentes <emph type="italics"></emph>PT, QT<emph.end type="italics"></emph.end>demiſſa (per <lb></lb>Corol. </s> <s>1. Prop.I.) ſunt reciproce <lb></lb>ut velocitates corporis in punctis <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>V<emph.end type="italics"></emph.end>; &c. </s> <s>adeoque per conſtructio<lb></lb>nem ut perpendicula <emph type="italics"></emph>AP, BQ<emph.end type="italics"></emph.end>di<lb></lb>recte, id eſt ut perpendicula a pun<lb></lb>cto <emph type="italics"></emph>D<emph.end type="italics"></emph.end>in tangentes demiſſa. </s> <s>Un<lb></lb>de facile colligitur quod puncta <lb></lb><emph type="italics"></emph>S, D, T,<emph.end type="italics"></emph.end>ſunt in una recta. </s> <s>Et ſimili <lb></lb>argumento puncta <emph type="italics"></emph>S, E, V<emph.end type="italics"></emph.end>ſunt eti<lb></lb>am in una recta; & propterea centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>in concurſu rectarum <emph type="italics"></emph>TD, VE<emph.end type="italics"></emph.end><lb></lb>verſatur. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO VI. THEOREMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpus in ſpatio non reſiſtente circa centrum immobile in Orbe quocun<lb></lb>que revolvatur, & arcum quemvis jamjam naſcentem tempore quàm <lb></lb>minimo deſcribat, & ſagitta arcus duci intelligatur quæ chordam bi<lb></lb>ſecet, & producta tranſeat per centrum virium: erit vis centripeta <lb></lb>in medio arcus, ut ſagitta directe & tempus bis inverſe.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſagitta dato tempore eſt ut vis (per Corol.4 Prop.I,) & augen<lb></lb>do tempus in ratione quavis, ob auctum arcum in eadem ratione ſa<lb></lb>gitta augetur in ratione illa duplicata (per Corol. </s> <s>2 & 3, Lem. </s> <s>XI,) ad<lb></lb>eoque eſt ut vis ſemel & tempus bis. </s> <s>Subducatur duplicata ratio tempo<lb></lb>ris utrinque, & fiet vis ut ſagitta directe & tempus bis inverſe. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Idem facile demonſtratur etiam per Corol. </s> <s>4 Lem. </s> <s>X. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Si corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>revolvendo <lb></lb><figure id="id.039.01.069.2.jpg" xlink:href="039/01/069/2.jpg"></figure><lb></lb>circa centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>deſcribat lineam <lb></lb>curvam <emph type="italics"></emph>APQ,<emph.end type="italics"></emph.end>tangat verò recta <lb></lb><emph type="italics"></emph>ZPR<emph.end type="italics"></emph.end>curvam illam in puncto <lb></lb>quovis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& ad tangentem ab alio <lb></lb>quovis Curvæ puncto <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>agatur <lb></lb><emph type="italics"></emph>QR<emph.end type="italics"></emph.end>diſtantiæ <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>parallela, ac <lb></lb>demittatur <emph type="italics"></emph>QT<emph.end type="italics"></emph.end>perpendicularis <lb></lb>ad diſtantiam illam <emph type="italics"></emph>SP:<emph.end type="italics"></emph.end>vis cen<lb></lb>tripeta erit reciproce ut ſolidum <lb></lb>(<emph type="italics"></emph>SP quad.XQT quad./QR<emph.end type="italics"></emph.end>) ſi modo ſolidi illius ea ſemper ſumatur quan<lb></lb>titas, quæ ultimò fit ubi coeunt puncta <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end>Nam <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>æqualis </s></p><pb xlink:href="039/01/070.jpg" pagenum="42"></pb> <p type="main"> <s><arrow.to.target n="note20"></arrow.to.target>eſt ſagittæ dupli arcus <emph type="italics"></emph>QP,<emph.end type="italics"></emph.end>in cujus medio eſt <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& duplum trian<lb></lb>guli <emph type="italics"></emph>SQP<emph.end type="italics"></emph.end>ſive <emph type="italics"></emph>SPXQT,<emph.end type="italics"></emph.end>tempori quo arcus iſte duplus deſcribitur <lb></lb>proportionale eſt, ideoque pro temporis exponente ſcribi poteſt. </s></p> <p type="margin"> <s><margin.target id="note20"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Eodem argumento vis centripeta eſt reciprocè ut ſolidum <lb></lb>(<emph type="italics"></emph>SYqXQPq/QR<emph.end type="italics"></emph.end>), ſi modo <emph type="italics"></emph>SY<emph.end type="italics"></emph.end>perpendiculum ſit a centro virium in Or<lb></lb>bis tangentem <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>demiſſum. </s> <s>Nam rectangula <emph type="italics"></emph>SYXQP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SPXQT<emph.end type="italics"></emph.end><lb></lb>æquantur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si Orbis vel circulus eſt, vel angulum contactus cum cir<lb></lb>culo quam minimum continet, eandem habens curvaturam eundem<lb></lb>que radium curvaturæ ad punctum contactus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>; & ſi <emph type="italics"></emph>PV<emph.end type="italics"></emph.end>chorda <lb></lb>ſit circuli hujus a corpore per centrum virium acta: erit vis centri<lb></lb>peta reciproce ut ſolidum <emph type="italics"></emph>SYqXPV.<emph.end type="italics"></emph.end>Nam <emph type="italics"></emph>PV<emph.end type="italics"></emph.end>eſt (<emph type="italics"></emph>QPq/QR<emph.end type="italics"></emph.end>). </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Iiſdem poſitis, eſt vis centripeta ut velocitas bis directe, <lb></lb>& chorda illa inverſe. </s> <s>Nam velocitas eſt reciproce ut perpendicu<lb></lb>lum <emph type="italics"></emph>SY<emph.end type="italics"></emph.end>per Corol. </s> <s>I Prop. </s> <s>I. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Hinc ſi detur figura quævis curvilinea <emph type="italics"></emph>APQ,<emph.end type="italics"></emph.end>& in ea <lb></lb>detur etiam punctum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad quod vis centripeta perpetuo dirigitur, <lb></lb>inveniri poteſt lex vis centripetæ, qua corpus quodvis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>a curſu <lb></lb>rectilineo perpetuò retractum in figuræ illius perimetro detinebitur <lb></lb>eamque revolvendo deſcribet. </s> <s>Nimirum computandum eſt vel ſo<lb></lb>lidum (<emph type="italics"></emph>SPqXQTq/QR<emph.end type="italics"></emph.end>) vel ſolidum <emph type="italics"></emph>SYqXPV<emph.end type="italics"></emph.end>huic vi reciproce pro<lb></lb>portionale. </s> <s>Ejus rei dabimus exempla in Problematis ſequentibus. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO VII. PROBLEMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Gyretur corpus in circumferentia Circuli, requiritur Lex vis centri<lb></lb>petæ tendentis ad punctum quodcunQ.E.D.tum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Eſto Circuli circumferentia <lb></lb><figure id="id.039.01.070.1.jpg" xlink:href="039/01/070/1.jpg"></figure><lb></lb><emph type="italics"></emph>VQPA,<emph.end type="italics"></emph.end>punctum datum ad <lb></lb>quod vis ceu ad <expan abbr="centrũ">centrum</expan> <expan abbr="ſuũ">ſuum</expan> ten<lb></lb>dit <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>corpus in circumferentia <lb></lb>latum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>locus proximus in quem <lb></lb>movebitur <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end>& circuli tangens <lb></lb>ad locum priorem <emph type="italics"></emph>PRZ.<emph.end type="italics"></emph.end>Per <lb></lb>punctum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ducatur chorda <emph type="italics"></emph>PV,<emph.end type="italics"></emph.end><lb></lb>& acta circuli diametro <emph type="italics"></emph>VA<emph.end type="italics"></emph.end>jun<lb></lb>gatur <emph type="italics"></emph>AP,<emph.end type="italics"></emph.end>& ad <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>demittatur <lb></lb>perpendiculum <emph type="italics"></emph>QT,<emph.end type="italics"></emph.end>quod productum occurrat tangenti <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Z,<emph.end type="italics"></emph.end><pb xlink:href="039/01/071.jpg" pagenum="43"></pb>ac denique per punctum <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>agatur <emph type="italics"></emph>LR<emph.end type="italics"></emph.end>quæ ipſi <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>parallela <lb></lb>ſit & occurrat tum circulo in <emph type="italics"></emph>L<emph.end type="italics"></emph.end>tum tangenti <emph type="italics"></emph>PZ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>R.<emph.end type="italics"></emph.end>Et <lb></lb>ob ſimilia triangula <emph type="italics"></emph>ZQR, ZTP, VPA<emph.end type="italics"></emph.end>; erit <emph type="italics"></emph>RP quad.<emph.end type="italics"></emph.end>hoc <lb></lb>eſt <emph type="italics"></emph>QRL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>QT quad.<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AV quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PV quad.<emph.end type="italics"></emph.end>Ideoque <lb></lb>(<emph type="italics"></emph>QRLXPV quad./AV quad.<emph.end type="italics"></emph.end>) æquatur <emph type="italics"></emph>QT quad.<emph.end type="italics"></emph.end>Ducantur hæc æqualia in <lb></lb>(<emph type="italics"></emph>SP quad./QR<emph.end type="italics"></emph.end>) &, punctis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>coeuntibus, ſcribatur <emph type="italics"></emph>PV<emph.end type="italics"></emph.end>pro <emph type="italics"></emph>RL.<emph.end type="italics"></emph.end><lb></lb>Sic fiet (<emph type="italics"></emph>SP quad.XPV cub./AV quad.<emph.end type="italics"></emph.end>) æquale (<emph type="italics"></emph>SP quad.XQT quad./QR<emph.end type="italics"></emph.end>) Ergo (per <lb></lb>Corol.1 & 5 Prop.VI.) vis centripeta eſt reciproce ut (<emph type="italics"></emph>SPqXPV cub./AV quad<emph.end type="italics"></emph.end>) <lb></lb>id eſt, (ob datum <emph type="italics"></emph>AV quad.<emph.end type="italics"></emph.end>) reciproce ut quadratum diſtantiæ ſeu <lb></lb>altitudinis <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>& cubus chordæ <emph type="italics"></emph>PV<emph.end type="italics"></emph.end>conjunctim. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Idem aliter.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ad tangentem <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>productam demittatur perpendiculum <emph type="italics"></emph>SY,<emph.end type="italics"></emph.end><lb></lb>& ob ſimilia triangula <emph type="italics"></emph>SYP, VPA<emph.end type="italics"></emph.end>; erit <emph type="italics"></emph>AV<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PV<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>SY,<emph.end type="italics"></emph.end>ideoque (<emph type="italics"></emph>SPXPV/AV<emph.end type="italics"></emph.end>) æquale <emph type="italics"></emph>SY,<emph.end type="italics"></emph.end>& (<emph type="italics"></emph>SP quad.XPV cub./AV quad.<emph.end type="italics"></emph.end>) æquale <lb></lb><emph type="italics"></emph>SY quad.XPV.<emph.end type="italics"></emph.end>Et propterea (per Corol.3 & 5 Prop.VI.) vis centri<lb></lb>peta eſt reciproce ut (<emph type="italics"></emph>SPqXPV cub./AVq<emph.end type="italics"></emph.end>) hoc eſt, ob datam <emph type="italics"></emph>AV,<emph.end type="italics"></emph.end>reci<lb></lb>proce ut <emph type="italics"></emph>SPqXPV cub. </s> <s><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi punctum datum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad quod vis centripeta ſem<lb></lb>per tendit, locetur in circumferentia hujus circuli, puta ad <emph type="italics"></emph>V<emph.end type="italics"></emph.end>; erit <lb></lb>vis centripeta reciproce ut quadrato cubus altitudinis <emph type="italics"></emph>SP.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Vis qua corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in cir<lb></lb><figure id="id.039.01.071.1.jpg" xlink:href="039/01/071/1.jpg"></figure><lb></lb>culo <emph type="italics"></emph>APTV<emph.end type="italics"></emph.end>circum virium centrum <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end>revolvitur, eſt ad vim qua corpus <lb></lb>idem <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in eodem circulo & eodem <lb></lb>tempore periodico circum aliud quod<lb></lb>vis virium centrum <emph type="italics"></emph>R<emph.end type="italics"></emph.end>revolvi poteſt, <lb></lb>ut <emph type="italics"></emph>RP quad.XSP<emph.end type="italics"></emph.end>ad cubum rectæ <emph type="italics"></emph>SG<emph.end type="italics"></emph.end><lb></lb>quæ a primo virium centro <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad or<lb></lb>bis tangentem <emph type="italics"></emph>PG<emph.end type="italics"></emph.end>ducitur, & diſtan<lb></lb>tiæ corporis a ſecundo virium centro <lb></lb>parallela eſt. </s> <s>Nam, per conſtructionem hujus Propoſitionis, vis <lb></lb>prior eſt ad vim poſteriorem, ut <emph type="italics"></emph>RPqXPT cub.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SPqXPV cub.<emph.end type="italics"></emph.end><pb xlink:href="039/01/072.jpg" pagenum="44"></pb><arrow.to.target n="note21"></arrow.to.target>id eſt, ut <emph type="italics"></emph>SPXRPq<emph.end type="italics"></emph.end>ad (<emph type="italics"></emph>SP cub.XPV cub/PT cub.<emph.end type="italics"></emph.end>) ſive (ob ſimilia <lb></lb>triangula <emph type="italics"></emph>PSG, TPV<emph.end type="italics"></emph.end>) ad <emph type="italics"></emph>SG cub.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note21"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Vis, qua corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Orbe quocunque circum virium <lb></lb>centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>revolvitur, eſt ad vim qua corpus idem <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in eodem <lb></lb>orbe eodemque tempore periodico circum aliud quodvis virium <lb></lb>centrum <emph type="italics"></emph>R<emph.end type="italics"></emph.end>revolvi poteſt, ut <emph type="italics"></emph>SPXRPq<emph.end type="italics"></emph.end>contentum utique ſub di<lb></lb>ſtantia corporis a primo virium centro <emph type="italics"></emph>S<emph.end type="italics"></emph.end>& quadrato diſtantiæ ejus <lb></lb>a ſecundo virium centro <emph type="italics"></emph>R<emph.end type="italics"></emph.end>ad cubum rectæ <emph type="italics"></emph>SG<emph.end type="italics"></emph.end>quæ a primo vi<lb></lb>rium centro <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad orbis tangentem <emph type="italics"></emph>PG<emph.end type="italics"></emph.end>ducitur, & corporis a ſe<lb></lb>cundo virium centro diſtantiæ <emph type="italics"></emph>RP<emph.end type="italics"></emph.end>parallela eſt. </s> <s>Nam vires in <lb></lb>hoc Orbe, ad ejus punctum quodvis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>eædem ſunt ac in Circulo <lb></lb>ejuſdem curvaturæ. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO. VIII. PROBLEMA. III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Moveatur corpus in Circulo<emph.end type="italics"></emph.end>PQA: <emph type="italics"></emph>ad hunc effectum requiritur Lex <lb></lb>vis centripetæ tendentis ad punctum adeo longinquum<emph.end type="italics"></emph.end>S, <emph type="italics"></emph>ut lineæ <lb></lb>omnes<emph.end type="italics"></emph.end>PS, RS <emph type="italics"></emph>ad id ductæ, pro parallelis haberi poſſint.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>A Circuli centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>agatur ſemidiameter <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>parallelas iſtas <lb></lb>perpendiculariter ſecans in <emph type="italics"></emph>M<emph.end type="italics"></emph.end>& <lb></lb><figure id="id.039.01.072.1.jpg" xlink:href="039/01/072/1.jpg"></figure><lb></lb><emph type="italics"></emph>N,<emph.end type="italics"></emph.end>& jungatur <emph type="italics"></emph>CP.<emph.end type="italics"></emph.end>Ob ſimilia <lb></lb>triangula <emph type="italics"></emph>CPM, PZT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>RZQ<emph.end type="italics"></emph.end><lb></lb>eſt <emph type="italics"></emph>CPq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PMq<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PRq<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>QTq<emph.end type="italics"></emph.end>& ex natura Circuli <emph type="italics"></emph>PRq<emph.end type="italics"></emph.end><lb></lb>æquale eſt rectangulo <emph type="italics"></emph>QRX√RN+QN<emph.end type="italics"></emph.end>&c. <lb></lb></s> <s>ſive coeuntibus punctis <emph type="italics"></emph>P, Q<emph.end type="italics"></emph.end>rect<lb></lb>angulo <emph type="italics"></emph>QRX2PM.<emph.end type="italics"></emph.end>Ergo eſt <lb></lb><emph type="italics"></emph>CPq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PM quad.<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>QRX2PM<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>QT quad.<emph.end type="italics"></emph.end>adeoque (<emph type="italics"></emph>QT quad./QR<emph.end type="italics"></emph.end>) <lb></lb>æquale (2<emph type="italics"></emph>PM cub./CP quad.<emph.end type="italics"></emph.end>), & (<emph type="italics"></emph>QT quad.XSP quad./QR<emph.end type="italics"></emph.end>) æquale (2<emph type="italics"></emph>PM cub.XSP qu./CP quad.<emph.end type="italics"></emph.end>) <lb></lb>Eſt ergo (per Corol. </s> <s>1 & 5 Prop. </s> <s>VI.) vis centripeta reciproce ut <lb></lb>(2<emph type="italics"></emph>PMcub.XSP quad./CP quad.<emph.end type="italics"></emph.end>) hoc eſt (neglecta ratione determinata (2<emph type="italics"></emph>SP quad./CP quad.<emph.end type="italics"></emph.end>)) <lb></lb>reciproce ut <emph type="italics"></emph>PM cub. </s> <s><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Idem facile colligitur etiam ex Propoſitione præcedente. </s></p><pb xlink:href="039/01/073.jpg" pagenum="45"></pb> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Et ſimili argumento corpus movebitur in Ellipſi vel etiam in <lb></lb>Hyperbola vel Parabola, vi centripeta quæ ſit reciproce ut cu<lb></lb>bus ordinatim applicatæ ad centrum virium maxime longinquum <lb></lb>tendentis. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO IX. PROBLEMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Gyretur corpus in Spirali<emph.end type="italics"></emph.end>PQS <emph type="italics"></emph>ſecante radios omnes<emph.end type="italics"></emph.end>SP, SQ, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.073.1.jpg" xlink:href="039/01/073/1.jpg"></figure><lb></lb><emph type="italics"></emph>in angulo dato: requiritur Lex <lb></lb>vis centripetæ tendentis ad <lb></lb>centrum Spiralis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Detur angulus indefinite par<lb></lb>vus <emph type="italics"></emph>PSQ,<emph.end type="italics"></emph.end>& ob datos omnes <lb></lb>angulos dabitur ſpecie figura <emph type="italics"></emph>SPQRT.<emph.end type="italics"></emph.end>Ergo datur ratio (<emph type="italics"></emph>QT/QR<emph.end type="italics"></emph.end>), eſtque <lb></lb>(<emph type="italics"></emph>QT quad./QR<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>QT,<emph.end type="italics"></emph.end>hoc eſt ut <emph type="italics"></emph>SP.<emph.end type="italics"></emph.end>Mutetur jam uteunque angulus <emph type="italics"></emph>PSQ,<emph.end type="italics"></emph.end><lb></lb>& recta <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>angulum contactus <emph type="italics"></emph>QPR<emph.end type="italics"></emph.end>ſubtendens mutabitur (per <lb></lb>Lemma XI.) in duplicata ratione ipſius <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>QT.<emph.end type="italics"></emph.end>Ergo manebit <lb></lb>(<emph type="italics"></emph>QT quad./QR<emph.end type="italics"></emph.end>) eadem quæ prius, hoc eſt ut <emph type="italics"></emph>SP.<emph.end type="italics"></emph.end>Quare (<emph type="italics"></emph>QTq.XSPq/QR<emph.end type="italics"></emph.end>) <lb></lb>eſt ut <emph type="italics"></emph>SP cub.<emph.end type="italics"></emph.end>adeoque (per Corol. </s> <s>1 & 5 Prop. </s> <s>VI.) vis centripeta eſt <lb></lb>reciproce ut cubus diſtantiæ <emph type="italics"></emph>SP. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Idem aliter.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Perpendiculum <emph type="italics"></emph>SY<emph.end type="italics"></emph.end>in tangentem demiſſum, & circuli Spiralem <lb></lb>tangentis chorda <emph type="italics"></emph>PV<emph.end type="italics"></emph.end>ſunt ad altitudinem <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>in datis rationibus; <lb></lb>ideoque <emph type="italics"></emph>SP cub.<emph.end type="italics"></emph.end>eſt ut <emph type="italics"></emph>SYqXPV,<emph.end type="italics"></emph.end>hoc eſt (per Corol. </s> <s>3 & 5 Prop.VI.) <lb></lb>reciproce ut vis centripeta. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Parallelogramma omnia, circa datæ Ellipſeos vel Hyperbolæ diametros <lb></lb>quaſvis conjugatas deſcripta, eſſe inter ſe æqualia.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Conſtat ex Conicis. <pb xlink:href="039/01/074.jpg" pagenum="46"></pb><arrow.to.target n="note22"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note22"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO X. PROBLEMA. V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Gyretur corpus in Ellipſi: requiritur lex vis centripetæ tendentis ad <lb></lb>centrum Ellipſeos.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sunto <emph type="italics"></emph>CA, CB<emph.end type="italics"></emph.end>ſemiaxes Ellipſeos; <emph type="italics"></emph>GP, DK<emph.end type="italics"></emph.end>diametri conju<lb></lb>gatæ; <emph type="italics"></emph>PF, Qt<emph.end type="italics"></emph.end>perpendicula ad diametros; <emph type="italics"></emph>Qv<emph.end type="italics"></emph.end>ordinatim appli<lb></lb>cata ad diametrum <lb></lb><figure id="id.039.01.074.1.jpg" xlink:href="039/01/074/1.jpg"></figure><lb></lb><emph type="italics"></emph>GP<emph.end type="italics"></emph.end>; & ſi compleatur <lb></lb>parallelogrammum <lb></lb><emph type="italics"></emph>QvPR,<emph.end type="italics"></emph.end>erit (ex CoNI<lb></lb>cis) <emph type="italics"></emph>PvG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>PC quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CD <lb></lb>quad.<emph.end type="italics"></emph.end>& (ob ſimilia <lb></lb>triangula <emph type="italics"></emph>Qvt, PCF<emph.end type="italics"></emph.end>) <lb></lb><emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>Qt <lb></lb>quad.<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PC quad.<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>PF quad.<emph.end type="italics"></emph.end>& conjun<lb></lb>ctis rationibus, <emph type="italics"></emph>PvG<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>Qt quad.<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PC <lb></lb>quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CD quad.<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>PC quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PF <lb></lb>quad.<emph.end type="italics"></emph.end>id eſt, <emph type="italics"></emph>vG<emph.end type="italics"></emph.end>ad <lb></lb>(<emph type="italics"></emph>Qt quad./Pv<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>PC quad.<emph.end type="italics"></emph.end><lb></lb>ad (<emph type="italics"></emph>CDqXPFq/PCq<emph.end type="italics"></emph.end>). Scribe <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>pro <emph type="italics"></emph>Pv,<emph.end type="italics"></emph.end>& (per Lemma XII.) <emph type="italics"></emph>BCXCA<emph.end type="italics"></emph.end><lb></lb>pro <emph type="italics"></emph>CDXPF,<emph.end type="italics"></emph.end>nec non, punctis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>coeuntibus, 2<emph type="italics"></emph>PC<emph.end type="italics"></emph.end>pro <lb></lb><emph type="italics"></emph>vG,<emph.end type="italics"></emph.end>& ductis extremis & mediis in ſe mutuo, fiet (<emph type="italics"></emph>Qt quad.XPCq/QR<emph.end type="italics"></emph.end>) <lb></lb>æquale (2<emph type="italics"></emph>BCqXCAq/PC<emph.end type="italics"></emph.end>). Eſt ergo (per Corol. </s> <s>5 Prop. </s> <s>VI.) vis centri<lb></lb>peta reciproce ut (2<emph type="italics"></emph>BCqXGAq;/PC<emph.end type="italics"></emph.end>) id eſt (ob datum 2<emph type="italics"></emph>BCqXCAq<emph.end type="italics"></emph.end>) <lb></lb>reciproce ut (1/<emph type="italics"></emph>PC<emph.end type="italics"></emph.end>); hoc eſt, directe ut diſtantia <emph type="italics"></emph>PC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Idem aliter.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In <emph type="italics"></emph>PG<emph.end type="italics"></emph.end>ab altera parte puncti <emph type="italics"></emph>t<emph.end type="italics"></emph.end>poſita intelligatur <emph type="italics"></emph>tu<emph.end type="italics"></emph.end>æqualis ipſi <lb></lb><emph type="italics"></emph>tv<emph.end type="italics"></emph.end>; deinde cape <emph type="italics"></emph>uV<emph.end type="italics"></emph.end>quæ ſit ad <emph type="italics"></emph>vG<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>DC quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PC quad.<emph.end type="italics"></emph.end><lb></lb>Et quoniam ex Conicis est <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PvG,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DC quad.<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>PC quad:<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>æquale <emph type="italics"></emph>PvXuV.<emph.end type="italics"></emph.end>Unde quadratum chor-<pb xlink:href="039/01/075.jpg" pagenum="47"></pb>dæ arcus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>erit æquale rectangulo <emph type="italics"></emph>VPv<emph.end type="italics"></emph.end>; adeoque Circulus qui <lb></lb><arrow.to.target n="note23"></arrow.to.target>tangit Sectionem Conicam in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& tranſit per punctum <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end>tranſibit <lb></lb>etiam per punctum <emph type="italics"></emph>V.<emph.end type="italics"></emph.end>Coeant puncta <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end>& hic circulus <lb></lb>ejuſdem erit curvaturæ cum ſectione conica in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PV<emph.end type="italics"></emph.end>æqualis erit <lb></lb>(2<emph type="italics"></emph>DCq/PC<emph.end type="italics"></emph.end>). Proinde vis qua corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Ellipſi revolvitur, erit reci<lb></lb>proce ut (2<emph type="italics"></emph>DCq/PC<emph.end type="italics"></emph.end>) in <emph type="italics"></emph>PFq<emph.end type="italics"></emph.end>(per Corol. </s> <s>3 Prop. </s> <s>VI.) hoc eſt (ob <lb></lb>datum 2<emph type="italics"></emph>DCq<emph.end type="italics"></emph.end>in <emph type="italics"></emph>PFq<emph.end type="italics"></emph.end>) directe ut <emph type="italics"></emph>PC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note23"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Eſt igitur vis ut diſtantia corporis a centro Ellipſeos: & <lb></lb>viciſſim, ſi vis ſit ut diſtantia, movebitur corpus in Ellipſi centrum <lb></lb>habente in centro virium, aut forte in Circulo, in quem utique <lb></lb>Ellipſis migrare poteſt. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et æqualia erunt revolutionum in Ellipſibus univerſis cir<lb></lb>cum centrum idem factarum periodica tempora. </s> <s>Nam tempora <lb></lb>illa in Ellipſibus ſimilibus æqualia ſunt per Corol. </s> <s>3 & 8, Prop. </s> <s>IV: <lb></lb>in Ellipſibus autem communem habentibus axem majorem, ſunt ad <lb></lb>invicem ut Ellipſeon areæ totæ directe & arearum particulæ ſimul <lb></lb>deſcriptæ inverſe; id eſt, ut axes minores directe & corporum ve<lb></lb>locitates in verticibus principalibus inverſe; hoc eſt, ut axes illi mi<lb></lb>nores directe & ordinatim applicatæ ad axes alteros inverſe; & prop<lb></lb>terea (ob æqualitatem rationum directarum & inverſarum) in ra<lb></lb>tione æqualitatis. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si Ellipſis, centro in infinitum abeunte vertatur in Parabolam, <lb></lb>corpus movebitur in hac Parabola; & vis ad centrum infinite di<lb></lb>ſtans jam tendens evadet æquabilis. </s> <s>Hoc eſt Theorema <emph type="italics"></emph>Galilæi.<emph.end type="italics"></emph.end><lb></lb>Et ſi coni ſectio Parabolica, inclinatione plani ad conum ſectum <lb></lb>mutata, vertatur in Hyperbolam, movebitur corpus in hujus pe<lb></lb>rimetro, vi centripeta in centrifugam verſa. </s> <s>Et quemadmo<lb></lb>dum in Circulo vel Ellipſi, ſi vires tendunt ad centrum figuræ <lb></lb>in Abſciſſa poſitum, hæ vires augendo vel diminuendo Ordinatas in <lb></lb>ratione quacunQ.E.D.ta, vel etiam mutando angulum inclinationis <lb></lb>Ordinatarum ad Abſciſſam, ſemper augentur vel diminuuntur in <lb></lb>ratione diſtantiarum a centro, ſi modo tempora periodica maneant <lb></lb>æqualia: ſic etiam in figuris univerſis, ſi Ordinatæ augeantur vel di<lb></lb>minuantur in ratione quacunQ.E.D.ta, vel angulus ordinationis ut<lb></lb>cunque mutetur, manente tempore periodico; vires ad centrum <lb></lb>quodcunQ.E.I. Abſciſſa poſitum tendentes a binis quibuſvis figurarum locis, ad quæ termi<lb></lb>nantur Ordinatæ correſpondentibus Abſciſſarum punctis inſiſtentes, augentur vel &c. </s> <s>augentur vel diminuun<lb></lb>tur in ratione diſtantiarum a centro. <pb xlink:href="039/01/076.jpg" pagenum="48"></pb><arrow.to.target n="note24"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note24"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De motu Corporum in Conicis Sectionibus excentricis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XI. PROBLEMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Revolvatur corpus in Ellipſi: requiritur Lex vis centripetæ tenden<lb></lb>tis ad umbilicum Ellipſeos.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Eſto Ellipſeos umbilicus <emph type="italics"></emph>S.<emph.end type="italics"></emph.end>Agatur <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ſecans Ellipſeos <lb></lb>tum diametrum <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>tum ordinatim applicatam <emph type="italics"></emph>Qv<emph.end type="italics"></emph.end>in <lb></lb><emph type="italics"></emph>x,<emph.end type="italics"></emph.end>& compleatur parallelogrammum <emph type="italics"></emph>QxPR.<emph.end type="italics"></emph.end>Patet <emph type="italics"></emph>EP<emph.end type="italics"></emph.end>æqua<lb></lb>lem eſſe ſemiaxi ma<lb></lb><figure id="id.039.01.076.1.jpg" xlink:href="039/01/076/1.jpg"></figure><lb></lb>jori <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>eo quod <lb></lb>acta ab altero Ellip<lb></lb>ſeos umbilico <emph type="italics"></emph>H<emph.end type="italics"></emph.end>li<lb></lb>nea <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>EC<emph.end type="italics"></emph.end>pa<lb></lb>rallela, (ob æquales <lb></lb><emph type="italics"></emph>CS, CH<emph.end type="italics"></emph.end>) æquentur <lb></lb><emph type="italics"></emph>ES, EI,<emph.end type="italics"></emph.end>adeo ut <emph type="italics"></emph>EP<emph.end type="italics"></emph.end><lb></lb>ſemiſumma ſit ipſa<lb></lb>rum <emph type="italics"></emph>PS, PI,<emph.end type="italics"></emph.end>id eſt <lb></lb>(ob parallelas <emph type="italics"></emph>HI, <lb></lb>PR<emph.end type="italics"></emph.end>& angulos æqua<lb></lb>les <emph type="italics"></emph>IPR, HPZ<emph.end type="italics"></emph.end>) <lb></lb>ipſarum <emph type="italics"></emph>PS, PH,<emph.end type="italics"></emph.end><lb></lb>quæ <expan abbr="cõjunctim">conjunctim</expan> axem <lb></lb>totum 2<emph type="italics"></emph>AC<emph.end type="italics"></emph.end>adæ<lb></lb>quant. </s> <s>Ad <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>de<lb></lb>mittatur perpendicularis <emph type="italics"></emph>QT,<emph.end type="italics"></emph.end>& Ellipſeos latere recto principali <lb></lb>(ſeu (2<emph type="italics"></emph>BC quad./AC<emph.end type="italics"></emph.end>)) dicto <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>LXQR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>LXPv<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>Pv,<emph.end type="italics"></emph.end>id eſt ut <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>LXPv<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GvP<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>Gv<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>GvP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PC quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CD quad<emph.end type="italics"></emph.end>; & (per Corol. </s> <s><lb></lb>2 Lem. </s> <s>VII.) <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Qx quad,<emph.end type="italics"></emph.end>punctis <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>coeuntibus, <lb></lb>eſt ratio æqualitatis; & <emph type="italics"></emph>Qx quad.<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>QT quad.<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>EP quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PF quad,<emph.end type="italics"></emph.end>id eſt ut <emph type="italics"></emph>CA quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PF quad.<emph.end type="italics"></emph.end>ſive (per <lb></lb>Lem XII.) ut <emph type="italics"></emph>CD quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CB quad.<emph.end type="italics"></emph.end>Et conjunctis his omnibus ratio<lb></lb>nibus, <emph type="italics"></emph>LXQR<emph.end type="italics"></emph.end>fit ad <emph type="italics"></emph>QT quad.<emph.end type="italics"></emph.end>ut <emph type="italics"></emph><expan abbr="ACXLXPCq.XCDq.">ACXLXPCq.XCDque</expan><emph.end type="italics"></emph.end>ſeu 2<emph type="italics"></emph><expan abbr="CBq.">CBque</expan> <lb></lb><expan abbr="XPCq.XCDq.">XPCq.XCDque</expan><emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="PCXGvXCDq.XCBq.">PCXGvXCDq.XCBque</expan><emph.end type="italics"></emph.end>ſive ut 2<emph type="italics"></emph>PC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Gv.<emph.end type="italics"></emph.end><pb xlink:href="039/01/077.jpg" pagenum="49"></pb>Sed, punctis <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>coeuntibus, <expan abbr="æquãtur">æquantur</expan> 2<emph type="italics"></emph>PC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Gv.<emph.end type="italics"></emph.end>Ergo & his pro<lb></lb><arrow.to.target n="note25"></arrow.to.target>portionalia <emph type="italics"></emph>LXQR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>QT quad.<emph.end type="italics"></emph.end>æquantur. </s> <s>Ducantur hæc æqualia in <lb></lb>(<emph type="italics"></emph>SPq/QR<emph.end type="italics"></emph.end>) & fiet <emph type="italics"></emph><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"></emph.end>æquale (<emph type="italics"></emph>SPq.XQTq/QR<emph.end type="italics"></emph.end>). Ergo (per Corol. </s> <s>1 <lb></lb>& 5 Prop. </s> <s>VI.) vis centripeta reciproce eſt ut <emph type="italics"></emph><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"></emph.end>id eſt, reci<lb></lb>proce in ratione duplicata diſtantiæ <emph type="italics"></emph>SP. Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note25"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Idem aliter.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cum vis ad centrum Ellipſeos tendens, qua corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Ellipſi <lb></lb>illa revolvi poteſt, ſit (per Corol. </s> <s>I Prop. </s> <s>X) ut <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>diſtantia cor<lb></lb>poris ab Ellipſeos centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>; ducatur <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>parallela Ellipſeos tan<lb></lb>genti <emph type="italics"></emph>PR:<emph.end type="italics"></emph.end>& vis qua corpus idem <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>circum aliud quodvis Ellip<lb></lb>ſeos punctum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>revolvi poteſt, ſi <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>concurrant in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>erit ut <lb></lb>(<emph type="italics"></emph>PE cub./SPq<emph.end type="italics"></emph.end>) (per Corol. </s> <s>3 Prop. </s> <s>VII,) hoc eſt, ſi punctum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ſit umbili<lb></lb>cus Ellipſeos, adeoque <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>detur, ut <emph type="italics"></emph>SPq<emph.end type="italics"></emph.end>reciproce. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Eadem brevitate qua traduximus Problema quintum ad Parabo<lb></lb>lam, & Hyperbolam, liceret idem hic facere: verum ob dignita<lb></lb>tem Problematis & uſum ejus in ſequentibus, non pigebit caſus ce<lb></lb>teros demonſtratione confirmare. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XII. PROBLEMA. VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Moveatur corpus in Hyperbola: requiritur Lex vis centripetæ ten<lb></lb>dentis ad umbilicum figuræ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sunto <emph type="italics"></emph>CA, CB<emph.end type="italics"></emph.end>ſemi-axes Hyperbolæ; <emph type="italics"></emph>PG, KD<emph.end type="italics"></emph.end>diametri con<lb></lb>jugatæ; <emph type="italics"></emph>PF, Qt<emph.end type="italics"></emph.end>perpendicula ad diametros; & <emph type="italics"></emph>Qv<emph.end type="italics"></emph.end>ordinatim <lb></lb>applicata ad diametrum <emph type="italics"></emph>GP.<emph.end type="italics"></emph.end>Agatur <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ſecans cum diametrum <lb></lb><emph type="italics"></emph>DK<emph.end type="italics"></emph.end>in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>tum ordinatim applicatam <emph type="italics"></emph>Qv<emph.end type="italics"></emph.end>in <emph type="italics"></emph>x,<emph.end type="italics"></emph.end>& compleatur pa<lb></lb>rallelogrammum <emph type="italics"></emph>QRPx.<emph.end type="italics"></emph.end>Patet <emph type="italics"></emph>EP<emph.end type="italics"></emph.end>æqualem eſſe ſemiaxi tranſ<lb></lb>verſo <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>eo quod, acta ab altero Hyperbolæ umbilico <emph type="italics"></emph>H<emph.end type="italics"></emph.end>linea <lb></lb><emph type="italics"></emph>HI<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>EC<emph.end type="italics"></emph.end>parallela, ob æquales <emph type="italics"></emph>CS, CH,<emph.end type="italics"></emph.end>æquentur <emph type="italics"></emph>ES, EI<emph.end type="italics"></emph.end>; <lb></lb>adeo ut <emph type="italics"></emph>EP<emph.end type="italics"></emph.end>ſemidifferentia ſit ipſarum <emph type="italics"></emph>PS, PI,<emph.end type="italics"></emph.end>id eſt (ob pa<lb></lb>rallelas <emph type="italics"></emph>IH, PR<emph.end type="italics"></emph.end>& angulos æquales <emph type="italics"></emph>IPR, HPZ<emph.end type="italics"></emph.end>) ipſarum <emph type="italics"></emph>PS, <lb></lb>PH,<emph.end type="italics"></emph.end>quarum differentia axem totum 2<emph type="italics"></emph>AC<emph.end type="italics"></emph.end>adæquat. </s> <s>Ad <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>de<lb></lb>mittatur perpendicularis <emph type="italics"></emph>QT.<emph.end type="italics"></emph.end>Et Hyperbolæ latere recto princi<lb></lb>pali (ſeu (2<emph type="italics"></emph>BCq/AC<emph.end type="italics"></emph.end>)) dicto <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>LXQR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>LXPv<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Pv,<emph.end type="italics"></emph.end><lb></lb>id eſt, ut <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>; Et <emph type="italics"></emph>LXPv<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GvP<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ad <pb xlink:href="039/01/078.jpg" pagenum="50"></pb><arrow.to.target n="note26"></arrow.to.target><emph type="italics"></emph>Gv<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>GvP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>ut <emph type="italics"></emph><expan abbr="PCq.">PCque</expan><emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CDq<emph.end type="italics"></emph.end>; & (per Corol. </s> <s>2. <lb></lb>Lem. </s> <s>VII.) <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Qx quad.<emph.end type="italics"></emph.end>punctis <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>coeuntibus fit <lb></lb>ratio æqualitatis; & <emph type="italics"></emph>Qx quad.<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>Qv quad.<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>QTq,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>EPq,<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>PFq,<emph.end type="italics"></emph.end>id eſt ut <emph type="italics"></emph>CAq,<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PFq,<emph.end type="italics"></emph.end>ſive (per Lem. </s> <s>XII.) ut <emph type="italics"></emph>CDq,<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>CBq:<emph.end type="italics"></emph.end>& conjunctis his omnibus rationibus <emph type="italics"></emph>LXQR<emph.end type="italics"></emph.end>fit ad <lb></lb><emph type="italics"></emph><expan abbr="QTq.">QTque</expan><emph.end type="italics"></emph.end>ut <emph type="italics"></emph>ACXLXPCqXCDq<emph.end type="italics"></emph.end>ſeu 2<emph type="italics"></emph>CBqXPCqXCDq<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>PCXGvXCDqXCB quad.<emph.end type="italics"></emph.end>ſive ut 2<emph type="italics"></emph>PC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Gv.<emph.end type="italics"></emph.end>Sed punctis <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>coeuntibus æquantur 2<emph type="italics"></emph>PC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Gv.<emph.end type="italics"></emph.end>Ergo & his propor<lb></lb>tionalia <emph type="italics"></emph>LXQR<emph.end type="italics"></emph.end>& <emph type="italics"></emph><expan abbr="QTq.">QTque</expan><emph.end type="italics"></emph.end>æquantur. </s> <s>Ducantur hæc æqualia in <lb></lb>(<emph type="italics"></emph>SPq/QR<emph.end type="italics"></emph.end>). & fiet <emph type="italics"></emph><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"></emph.end>æquale (<emph type="italics"></emph>SPqXQTq/QR<emph.end type="italics"></emph.end>). Ergo (per Corol. </s> <s>I <lb></lb><figure id="id.039.01.078.1.jpg" xlink:href="039/01/078/1.jpg"></figure><lb></lb>& 5 Prop. </s> <s>VI.) vis centripeta reciproce eſt ut <emph type="italics"></emph>LXSPq,<emph.end type="italics"></emph.end>id eſt <lb></lb>reciproce in ratione duplicata diſtantiæ <emph type="italics"></emph>SP. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end><pb xlink:href="039/01/079.jpg" pagenum="51"></pb><arrow.to.target n="note27"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note26"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="margin"> <s><margin.target id="note27"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Idem aliter.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Inveniatur vis quæ tendit ab Hyperbolæ centro <emph type="italics"></emph>C.<emph.end type="italics"></emph.end>Prodibit hæc <lb></lb>diſtantiæ <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>proportionalis. </s> <s>Inde vero (per Corol. </s> <s>3 Prop. </s> <s>VII.) <lb></lb>vis ad umbilicum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>tendens erit ut (<emph type="italics"></emph>PEcub/SPq<emph.end type="italics"></emph.end>), hoc eſt, ob datam <emph type="italics"></emph>PE,<emph.end type="italics"></emph.end><lb></lb>reciproce ut <emph type="italics"></emph><expan abbr="SPq.">SPque</expan> Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Eodem modo demonſtratur quod corpus, hac vi centripeta in <lb></lb>centrifugam verſa, movebitur in Hyperbola conjugata. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Latus rectum Parabolæ ad verticem quemvis pertinens, eſt quadru<lb></lb>plum diſtantiæ verticis illius ab umbilico figuræ.<emph.end type="italics"></emph.end>Patet ex Conicis. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Perpendiculum quod ab umbilico Parabolæ ad tangentem ejus demitti<lb></lb>tur, medium eſt proportionale inter diſtantias umbilici a puncto con<lb></lb>tactus & a vertice principali figuræ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit enim <emph type="italics"></emph>AQP<emph.end type="italics"></emph.end>Parabola, <emph type="italics"></emph>S<emph.end type="italics"></emph.end>umbilicus ejus, <emph type="italics"></emph>A<emph.end type="italics"></emph.end>vertex principa<lb></lb>lis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>punctum <lb></lb><figure id="id.039.01.079.1.jpg" xlink:href="039/01/079/1.jpg"></figure><lb></lb>contactus, <emph type="italics"></emph>PO<emph.end type="italics"></emph.end><lb></lb>ordinatim ap<lb></lb>plicata ad dia<lb></lb>metrum prin<lb></lb>cipalem, <emph type="italics"></emph>PM<emph.end type="italics"></emph.end><lb></lb>tangens dia<lb></lb>metro princi<lb></lb>pali occurrens <lb></lb>in <emph type="italics"></emph>M,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SN,<emph.end type="italics"></emph.end><lb></lb>linea perpen<lb></lb>dicularis ab umbilico in tangentem. </s> <s>Jungatur <emph type="italics"></emph>AN,<emph.end type="italics"></emph.end>& ob æquales <lb></lb><emph type="italics"></emph>MS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SP, MN<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NP, MA<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AO,<emph.end type="italics"></emph.end>parallelæ erunt rectæ <lb></lb><emph type="italics"></emph>AN<emph.end type="italics"></emph.end>& <emph type="italics"></emph>OP,<emph.end type="italics"></emph.end>& inde triangulum <emph type="italics"></emph>SAN<emph.end type="italics"></emph.end>rectangulum erit ad <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <lb></lb>ſimile triangulis æqualibus <emph type="italics"></emph>SNM, SNP:<emph.end type="italics"></emph.end>Ergo <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>SN,<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SA. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. <emph type="italics"></emph><expan abbr="PSq.">PSque</expan><emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph><expan abbr="SNq.">SNque</expan><emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SA.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et ob datam <emph type="italics"></emph>SA,<emph.end type="italics"></emph.end>eſt <emph type="italics"></emph><expan abbr="SNq.">SNque</expan><emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PS.<emph.end type="italics"></emph.end><pb xlink:href="039/01/080.jpg" pagenum="52"></pb><arrow.to.target n="note28"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note28"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et concurſus tangentis cujuſvis <emph type="italics"></emph>PM<emph.end type="italics"></emph.end>cum recta <emph type="italics"></emph>SN,<emph.end type="italics"></emph.end><lb></lb>quæ ab umbilico in ipſam perpendicularis eſt, incidit in rectam <emph type="italics"></emph>AN,<emph.end type="italics"></emph.end><lb></lb>quæ Parabolam tangit in vertice principali. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO. XIII. PROBLEMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Moveatur corpus in perimetro Parabolæ: requiritur Lex vis centri<lb></lb>petæ tendentis ad umbilicum hujus figuræ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Maneat conſtructio Lemmatis, ſitque <emph type="italics"></emph>P<emph.end type="italics"></emph.end>corpus in perimetro Pa<lb></lb>rabolæ, & a loco <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>in quem corpus proxime movetur, age ipſi <emph type="italics"></emph>SP<emph.end type="italics"></emph.end><lb></lb>parallelam <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>& perpendicularem <emph type="italics"></emph>QT,<emph.end type="italics"></emph.end>necnon <emph type="italics"></emph>Qv<emph.end type="italics"></emph.end>tangenti pa<lb></lb>rallelam & occurrentem tum diametro <emph type="italics"></emph>YPG<emph.end type="italics"></emph.end>in <emph type="italics"></emph>v,<emph.end type="italics"></emph.end>tum diſtantiæ <lb></lb><emph type="italics"></emph>SP<emph.end type="italics"></emph.end>in <emph type="italics"></emph>x.<emph.end type="italics"></emph.end>Jam ob ſimilia triangula <emph type="italics"></emph>Pxv, SPM<emph.end type="italics"></emph.end>& æqualia unius <lb></lb>latera <emph type="italics"></emph>SM, SP,<emph.end type="italics"></emph.end>æqualia ſunt alterius latera <emph type="italics"></emph>Px<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Pv.<emph.end type="italics"></emph.end><lb></lb>Sed, ex Conicis, quadratum ordinatæ <emph type="italics"></emph>Qv<emph.end type="italics"></emph.end>æquale eſt rectangulo ſub <lb></lb>latere recto & ſegmento diametri <emph type="italics"></emph>Pv,<emph.end type="italics"></emph.end>id eſt (per Lem. </s> <s>XIII.) rectangu<lb></lb>lo 4 <emph type="italics"></emph>PSXPv,<emph.end type="italics"></emph.end>ſeu 4 <emph type="italics"></emph>PSXQR<emph.end type="italics"></emph.end>; & punctis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>coeuntibus, ra<lb></lb>tio <emph type="italics"></emph>Qv<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Qx<emph.end type="italics"></emph.end>per (per Corol. </s> <s>2 Lem. </s> <s>VII.) fit ratio æqualitatis. </s> <s>Er<lb></lb>go <emph type="italics"></emph>Qxquad.<emph.end type="italics"></emph.end>eo <lb></lb><figure id="id.039.01.080.1.jpg" xlink:href="039/01/080/1.jpg"></figure><lb></lb>in caſu, æquale <lb></lb>eſt rectangu<lb></lb>lo 4 <emph type="italics"></emph>PSXQR.<emph.end type="italics"></emph.end><lb></lb>Eſt autem (ob <lb></lb>ſimilia trian<lb></lb>gula <emph type="italics"></emph>QxT, <lb></lb>SPN) <expan abbr="Qxq.">Qxque</expan><emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph><expan abbr="QTq.">QTque</expan><emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph><expan abbr="PSq.">PSque</expan><emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="SNq.">SNque</expan><emph.end type="italics"></emph.end><lb></lb>hoc eſt (per <lb></lb>Corol. </s> <s>1. Lem. </s> <s>XIV.) ut <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SA,<emph.end type="italics"></emph.end>id eſt ut 4 <emph type="italics"></emph>PSXQR<emph.end type="italics"></emph.end><lb></lb>ad 4<emph type="italics"></emph>SAXQR,<emph.end type="italics"></emph.end>& inde (per Prop. </s> <s>IX. Lib. </s> <s>v. </s> <s>Elem.) <emph type="italics"></emph><expan abbr="QTq.">QTque</expan><emph.end type="italics"></emph.end>& <lb></lb>4<emph type="italics"></emph>SAXQR<emph.end type="italics"></emph.end>æquantur. </s> <s>Ducantur hæc æqualia in (<emph type="italics"></emph>SPq./QR<emph.end type="italics"></emph.end>), & fiet <lb></lb>(<emph type="italics"></emph>SPq.XQTq./QR<emph.end type="italics"></emph.end>) æquale <emph type="italics"></emph>SPq.X4SA:<emph.end type="italics"></emph.end>& propterea (per Corol. </s> <s>1 & 5 <lb></lb>Prop. </s> <s>VI.) vis centripeta eſt reciproce ut <emph type="italics"></emph>SPq.X4SA,<emph.end type="italics"></emph.end>id eſt, ob da<lb></lb>tam 4<emph type="italics"></emph>SA,<emph.end type="italics"></emph.end>reciproce in duplicata ratione diſtantiæ <emph type="italics"></emph>SP. Q.E.I.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/081.jpg" pagenum="53"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Ex tribus noviſſimis Propoſitionibus conſequens eſt, quod </s></p> <p type="main"> <s><arrow.to.target n="note29"></arrow.to.target>ſi corpus quodvis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſecundum lineam quamvis rectam <emph type="italics"></emph>PR,<emph.end type="italics"></emph.end>qua<lb></lb>cunque cum velocitate exeat de loco <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& vi centripeta quæ ſit re<lb></lb>ciproce proportionalis quadrato diſtantiæ loeorum a centro, ſimul <lb></lb>agitetur; movebitur hoc corpus in aliqua ſectionum Conicarum <lb></lb>umbilicum habente in centro virium; & contra. </s> <s>Nam datis umbi<lb></lb>lico & puncto contactus & poſitione tangentis, deſcribi poteſt ſectio <lb></lb>Conica quæ curvaturam datam ad punctum illud habebit. </s> <s>Datur <lb></lb>autem curvatura ex data vi centripeta: & Orbes duo ſe mutuo tan<lb></lb>gentes, eadem vi centripeta deſcribi non poſſunt. </s></p> <p type="margin"> <s><margin.target id="note29"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si velocitas, quacum corpus exit de loco ſuo <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ea <lb></lb>ſit, qua lineola <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>in minima aliqua temporis particula deſcribi <lb></lb>poſſit, & vis centripeta potis ſit eodem tempore corpus idem mo<lb></lb>vere per ſpatium <emph type="italics"></emph>QR:<emph.end type="italics"></emph.end>movebitur hoc corpus in Conica aliqua ſe<lb></lb>ctione, cujus latus rectum principale eſt quantitas illa (<emph type="italics"></emph>QTq./QR<emph.end type="italics"></emph.end>) quæ <lb></lb>ultimo fit ubi lineolæ <emph type="italics"></emph>PR, QR<emph.end type="italics"></emph.end>in infinitum diminuuntur. </s> <s>Circu<lb></lb>lum in his Corollariis refero ad Ellipſin, & caſum excipio ubi cor<lb></lb>pus recta deſcendit ad centrum. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XIV. THEOREMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpora plura revolvantur circa centrum commune, & vis centri<lb></lb>peta ſit reciproce in duplicata ratione diſtantiæ loeorum a centro; <lb></lb>dico quod Orbium Latera recta principalia ſunt in duplicata ratio<lb></lb>one arearum quas corpora, radiis ad centrum ductis, eodem tempore <lb></lb>deſcribunt.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam, per Corol. </s> <s>2. Prop. </s> <s>XIII, Latus rectum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>æquale eſt quan<lb></lb>titati (<emph type="italics"></emph>QTq./QR<emph.end type="italics"></emph.end>) quæ ultimo fit ubi coeunt puncta <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end>Sed linea <lb></lb>minima <emph type="italics"></emph>QR,<emph.end type="italics"></emph.end>dato tempore, eſt ut vis centripeta generans, hoc <lb></lb>eſt (per Hypotheſin) reciproce ut <emph type="italics"></emph><expan abbr="SPq.">SPque</expan><emph.end type="italics"></emph.end>Ergo (<emph type="italics"></emph>QTq./QR<emph.end type="italics"></emph.end>) eſt ut <lb></lb><emph type="italics"></emph><expan abbr="QTq.XSPq.">QTq.XSPque</expan><emph.end type="italics"></emph.end>hoc eſt, latus rectum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>in duplicata ratione areæ <lb></lb><emph type="italics"></emph>QTXSP. Q.E.D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/082.jpg" pagenum="54"></pb><arrow.to.target n="note30"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note30"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc Ellipſeos area tota, eique proportionale rectangu<lb></lb>lum ſub axibus, eſt in ratione compoſita ex ſubduplicata ratione <lb></lb>lateris recti & ratione temporis periodici. </s> <s>Namque area tota eſt <lb></lb>ut area <emph type="italics"></emph>QTXSP,<emph.end type="italics"></emph.end>quæ dato tempore deſcribitur, ducta in &c. </s> <s>ducta in tempus periodicum. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XV. THEOREMA VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Iiſdem poſitis, dico quod Tempora periodica in Ellipſibus ſunt in ratione <lb></lb>ſeſquiplicata majorum axium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Namque axis minor eſt medius proportionalis inter axem majo<lb></lb>rem & latus rectum, atque adeo rectangulum ſub axibus eſt in ra<lb></lb>tione compoſita ex ſubduplicata ratione lateris recti & ſeſquiplicata <lb></lb>ratione axis majoris. </s> <s>Sed hoc rectangulum, per Corollarium Prop. </s> <s><lb></lb>XIV. eſt in ratione compoſita ex ſubduplicata ratione lateris recti <lb></lb>& ratione periodici temporis. </s> <s>Dematur utrobique ſubduplicata <lb></lb>ratio lateris recti, & manebit ſeſquiplicata ratio majoris axis æqua<lb></lb>lis rationi periodici temporis. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Sunt igitur tempora periodica in Ellipſibus eadem ac in <lb></lb>Circulis, quorum diametri æquantur majoribus axibus Ellipſeon. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XVI. THEOREMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, & actis ad corpora lineis rectis, quæ ibidem tangant Or<lb></lb>bitas, demiſſiſque ab umbilico communi ad has tangentes perpendi<lb></lb>cularibus: dico quod Velocitates corporum ſunt in ratione compoſi<lb></lb>ta ex ratione perpendiculorum inverſe & ſubduplicata ratione la<lb></lb>terum rectorum principalium directe.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Ab umbilico <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad tangentem <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>demitte perpendiculum <emph type="italics"></emph>SY<emph.end type="italics"></emph.end><lb></lb>& velocitas corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erit reciproce in ſubduplicata ratione quan<lb></lb>titatis (<emph type="italics"></emph>SYq/L<emph.end type="italics"></emph.end>). Nam velocitas illa eſt ut arcus quam minimus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end><lb></lb>in data temporis particula deſcriptus, hoc eſt (per Lem. </s> <s>VII.) ut <lb></lb>tangens <emph type="italics"></emph>PR,<emph.end type="italics"></emph.end>id eſt (ob proportionales <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>QT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SY<emph.end type="italics"></emph.end>) ut <lb></lb>(<emph type="italics"></emph>SPXQT/SY<emph.end type="italics"></emph.end>), ſive ut <emph type="italics"></emph>SY<emph.end type="italics"></emph.end>reciproce & <emph type="italics"></emph>SPXQT<emph.end type="italics"></emph.end>directe; eſtque <pb xlink:href="039/01/083.jpg" pagenum="55"></pb><emph type="italics"></emph>SPXQT<emph.end type="italics"></emph.end>ut area dato tempore deſcripta, id eſt, per Prop. </s> <s>XIV. </s></p> <p type="main"> <s><arrow.to.target n="note31"></arrow.to.target>in ſubduplicata ratione lateris recti. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note31"></margin.target>LIBER <lb></lb>PRIMUS.</s></p><figure id="id.039.01.083.1.jpg" xlink:href="039/01/083/1.jpg"></figure> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Latera recta principalia ſunt in ratione compoſita ex <lb></lb>duplicata ratione perpendiculorum & duplicata ratione veloci<lb></lb>tatum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Velocitates corporum in maximis & minimis ab umbi<lb></lb>lico communi diſtantiis, ſunt in ratione compoſita ex ratione di<lb></lb>ſtantiarum inverſe & ſubduplicata ratione laterum rectorum princi<lb></lb>palium directe. </s> <s>Nam perpendicula jam ſunt ipſæ diſtantiæ. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Ideoque velocitas in Conica ſectione, in maxima vel <lb></lb>minima ab umbilico diſtantia, eſt ad velocitatem in Circulo in ea<lb></lb>dem à centro diſtantia, in ſubduplicata ratione lateris recti princi<lb></lb>palis ad duplam illam diſtantiam. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Corporum in Ellipſibus gyrantium velocitates in medi<lb></lb>ocribus diſtantiis ab umbilico communi ſunt eædem quæ corporum <lb></lb>gyrantium in Circulis ad eaſdem diſtantias; hoc eſt (per Corol 6. <lb></lb>Prop. </s> <s>IV.) reciproce in ſubduplicata ratione diſtantiarum. </s> <s>Nam <lb></lb>perpendicula jam ſunt ſemi-axes minores; & hi ſunt ut mediæ <lb></lb>proportionales inter diſtantias & latera recta. </s> <s>Componatur hæc <lb></lb>ratio inverſe cum ſubduplicata ratione laterum rectorum directe, & <lb></lb>fiet ratio ſubduplicata diſtantiarum inverſe. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. In eadem figura, vel etiam in figuris diverſis, quarum <pb xlink:href="039/01/084.jpg" pagenum="56"></pb><arrow.to.target n="note32"></arrow.to.target>latera recta principalia ſunt æqualia, velocitas corporis eſt reciproce <lb></lb>ut perpendiculum demiſſum ab umbilico ad tangentem. </s></p> <p type="margin"> <s><margin.target id="note32"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. In Parabola, velocitas eſt reciproce in ſubduplicata ra<lb></lb>tione diſtantiæ corporis ab umbilico figuræ; in Ellipſi magis varia<lb></lb>tur, in Hyperbola minus, quam in hac ratione. </s> <s>Nam (per Corol. </s> <s><lb></lb>2. Lem. </s> <s>XIV.) perpendiculum demiſſum ab umbilico ad tangentem <lb></lb>Parabolæ eſt in ſubduplicata ratione diſtantiæ. </s> <s>In Hyperbola per<lb></lb>pendiculum minus variatur, in Ellipſi magis. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. In Parabola, velocitas corporis ad quamvis ab umbili<lb></lb>co diſtantiam, eſt ad velocitatem corporis revolventis in Circulo <lb></lb>ad eandem a centro diſtantiam, in ſubduplicata ratione numeri bi<lb></lb>narii ad unitatem; in Ellipſi minor eſt, in Hyperbola major quam <lb></lb>in hac ratione. </s> <s>Nam per hujus Corollarium ſecundum, velocitas <lb></lb>in vertice Parabolæ eſt in hac ratione, & per Corollaria ſexta hu<lb></lb>jus & Propoſitionis quartæ, ſervatur eadem proportio in omnibus <lb></lb>diſtantiis. </s> <s>Hinc etiam in Parabola velocitas ubique æqualis eſt ve<lb></lb>locitati corporis revolventis in Circulo ad dimidiam diſtantiam, in <lb></lb>Ellipſi minor eſt, in Hyperbola major. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Velocitas gyrantis in Sectione quavis Conica eſt ad ve<lb></lb>locitatem gyrantis in Circulo in diſtantia dimidii lateris recti princi<lb></lb>palis Sectionis, ut diſtantia illa ad perpendiculum ab umbilico in <lb></lb>tangentem Sectionis demiſſum. </s> <s>Patet per Corollarium quintum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>9. Unde cum (per Corol. </s> <s>6. Prop. </s> <s>IV.) velocitas gyrantis <lb></lb>in hoc Circulo ſit ad velocitatem gyrantis in Circulo quovis alio, <lb></lb>reciproce in ſubduplicata ratione diſtantiarum; fiet ex æquo velo<lb></lb>citas gyrantis in Conica ſectione ad velocitatem gyrantis in Circulo <lb></lb>in eadem diſtantia, ut media proportionalis inter diſtantiam illam <lb></lb>communem & ſemiſſem principalis lateris recti ſectionis, ad per<lb></lb>pendiculum ab umbilico communi in tangentem ſectionis de<lb></lb>miſſum. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XVII. PROBLEMA. IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſito quod vis centripeta ſit reciproce proportionalis quadrato diſtan<lb></lb>ſtantiæ loeorum a centro, & quod vis illius quantitas abſoluta ſit <lb></lb>cognita; requiritur Linea quam corpus deſcribit, de loco dato, cum <lb></lb>data velocitate, ſecundum datam rectam egrediens.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Vis centripeta tendens ad punctum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ea ſit qua corpus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>in or<lb></lb>bita quavis data <emph type="italics"></emph>pq<emph.end type="italics"></emph.end>gyretur, & cognoſcatur hujus velocitas in loco <emph type="italics"></emph>p.<emph.end type="italics"></emph.end><pb xlink:href="039/01/085.jpg" pagenum="57"></pb>De loco <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſecundum lineam <emph type="italics"></emph>PR,<emph.end type="italics"></emph.end>exeat corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>cum data velo<lb></lb><arrow.to.target n="note33"></arrow.to.target>citate, & mox inde, cogente vi centripeta, deflectat illud in CoNI<lb></lb>ſectionem <emph type="italics"></emph><expan abbr="Pq.">Pque</expan><emph.end type="italics"></emph.end>Hanc igitur recta <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>tanget in <emph type="italics"></emph>P.<emph.end type="italics"></emph.end>Tangat itidem <lb></lb>recta aliqua <emph type="italics"></emph>pr<emph.end type="italics"></emph.end>Orbitam <emph type="italics"></emph>pq<emph.end type="italics"></emph.end>in <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>& ſi ab <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad eas tangentes demitti <lb></lb>intelligantur perpendicula, erit (per Corol. </s> <s>1. Prop. </s> <s>XVI.) latus re<lb></lb>ctum principale Coniſectionis ad latus rectum principale Orbitæ, in <lb></lb>ratione compoſita ex duplicata ratione perpendiculorum & dupli<lb></lb>cata ratione velocitatum, atque adeo datur. </s> <s>Sit iſtud <emph type="italics"></emph>L.<emph.end type="italics"></emph.end>Da<lb></lb>tur præterea Coniſe<lb></lb><figure id="id.039.01.085.1.jpg" xlink:href="039/01/085/1.jpg"></figure><lb></lb>ctionis umbilicus <emph type="italics"></emph>S.<emph.end type="italics"></emph.end><lb></lb>Anguli <emph type="italics"></emph>RPS<emph.end type="italics"></emph.end>com<lb></lb>plementum ad du<lb></lb>os rectos fiat angu<lb></lb>lus <emph type="italics"></emph>RPH,<emph.end type="italics"></emph.end>& dabi<lb></lb>tur poſitione linea <lb></lb><emph type="italics"></emph>PH,<emph.end type="italics"></emph.end>in qua umbilicus <lb></lb>alter <emph type="italics"></emph>H<emph.end type="italics"></emph.end>locatur. </s> <s>De<lb></lb>miſſo ad <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>perpen<lb></lb>diculo <emph type="italics"></emph>SK,<emph.end type="italics"></emph.end>erigi intelligatur ſemiaxis conjugatus <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>& erit <lb></lb><emph type="italics"></emph>SPq.-2KPH+PHq.=SHq.=4CHq.=4BHq-4BCq.= <lb></lb>—SP+PH: quad. -LX—SP+PH=SPq.+2SPH+PHq. <lb></lb>-LX—SP+PH.<emph.end type="italics"></emph.end>Addantur utrobique 2<emph type="italics"></emph>KPH-SPq-PHq <lb></lb>+LX—SP+PH,<emph.end type="italics"></emph.end>& fiet <emph type="italics"></emph>LX—SP+PH=2SPH+2KPH,<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>SP+PH,<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PH,<emph.end type="italics"></emph.end>ut 2<emph type="italics"></emph>SP+2KP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>L.<emph.end type="italics"></emph.end>Unde datur <emph type="italics"></emph>PH<emph.end type="italics"></emph.end><lb></lb>tam longitudine quam poſitione. </s> <s>Nimirum ſi ea fit corporis &c. </s> <s>in <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>velocitas, ut latus rectum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>minus fuerit quam 2 <emph type="italics"></emph>SP+2KP,<emph.end type="italics"></emph.end><lb></lb>jacebit <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>ad eandem partem tangentis <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>cum linea <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end><lb></lb>adeoque figura erit Ellipſis, & ex datis umbilicis <emph type="italics"></emph>S, H,<emph.end type="italics"></emph.end>& axe <lb></lb>principali <emph type="italics"></emph>SP+PH,<emph.end type="italics"></emph.end>dabitur: Sin tanta ſit corporis velocitas ut <lb></lb>latus rectum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>æquale fuerit 2 <emph type="italics"></emph>SP+2KP,<emph.end type="italics"></emph.end>longitudo <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>infi<lb></lb>nita erit, & propterea figura erit Parabola axem habens <emph type="italics"></emph>SH<emph.end type="italics"></emph.end>paral<lb></lb>lelum lineæ <emph type="italics"></emph>PK,<emph.end type="italics"></emph.end>& inde dabitur. </s> <s>Quod ſi corpus majori adhuc <lb></lb>cum velocitate de loco ſuo <emph type="italics"></emph>P<emph.end type="italics"></emph.end>exeat, capienda erit longitudo <emph type="italics"></emph>PH<emph.end type="italics"></emph.end><lb></lb>ad alteram partem tangentis, adeoque tangente inter umbilicos per<lb></lb>gente, figura erit Hyperbola axem habens principalem æqualem dif<lb></lb>ferentiæ linearum <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PH,<emph.end type="italics"></emph.end>& inde dabitur. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note33"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc in omni Coniſectione ex dato vertice principali <emph type="italics"></emph>D,<emph.end type="italics"></emph.end><lb></lb>latere recto <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>& umbilico <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>datur umbilicus alter <emph type="italics"></emph>H<emph.end type="italics"></emph.end>capiendo <emph type="italics"></emph>DH,<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>DS<emph.end type="italics"></emph.end>ut eſt latus rectum ad differentiam inter latus rectum & <lb></lb>4 <emph type="italics"></emph>DS.<emph.end type="italics"></emph.end>Nam proportio <emph type="italics"></emph>SP+PH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>ut 2 <emph type="italics"></emph>SP+2KP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>L,<emph.end type="italics"></emph.end><pb xlink:href="039/01/086.jpg" pagenum="58"></pb><arrow.to.target n="note34"></arrow.to.target>in caſu hujus Corollarii, ſit <emph type="italics"></emph>DS+DH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DH<emph.end type="italics"></emph.end>ut 4 <emph type="italics"></emph>DS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>& <lb></lb>diviſim <emph type="italics"></emph>DS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DH<emph.end type="italics"></emph.end>ut 4 <emph type="italics"></emph>DS-L<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>L.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note34"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Unde ſi datur corporis velocitas in vertice principali <emph type="italics"></emph>D,<emph.end type="italics"></emph.end><lb></lb>invenietur Orbita expedite, capiendo ſcilicet latus rectum ejus, ad <lb></lb>duplam diſtantiam <emph type="italics"></emph>DS,<emph.end type="italics"></emph.end>in duplicata ratione velocitatis hujus datæ <lb></lb>ad velocitatem corporis in Circulo, ad diſtantiam <emph type="italics"></emph>DS,<emph.end type="italics"></emph.end>gyrantis (per <lb></lb>Corol. </s> <s>3. Prop. </s> <s>XVI.) dein <emph type="italics"></emph>DH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DS<emph.end type="italics"></emph.end>ut latus rectum ad differen<lb></lb>tiam inter latus rectum & 4 <emph type="italics"></emph>DS.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Hinc etiam ſi corpus moveatur in Sectione quacunque <lb></lb>Conica, & ex Orbe ſuo impulſu quocunque exturbetur; cognoſci <lb></lb>poteſt Orbis in quo poſtea curſum ſuum peraget. </s> <s>Nam componen<lb></lb>do proprium corporis motum cum motu illo quem impulſus ſolus <lb></lb>generaret, habebitur motus quocum corpus de dato impulſus loco, <lb></lb>ſecundum rectam poſitione datam, exibit. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Et ſi corpus illud vi aliqua extrinſecus impreſſa conti<lb></lb>nuo perturbetur, innoteſcet curſus quam proxime, colligendo mu<lb></lb>tationes quas vis illa in punctis quibuſdam inducit, & ex ſeriei ana<lb></lb>logia mutationes continuas in locis intermediis æſtimando. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>vi centripeta ad <lb></lb><figure id="id.039.01.086.1.jpg" xlink:href="039/01/086/1.jpg"></figure><lb></lb>punctum quodcunQ.E.D.tum <emph type="italics"></emph>R<emph.end type="italics"></emph.end><lb></lb>tendente moveatur in perimetro <lb></lb>datæ cujuſcunque Sectionis co<lb></lb>nicæ cujus centrum ſit <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>& re<lb></lb>quiratur Lex vis centripetæ: du<lb></lb>catur <emph type="italics"></emph>CG<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>RP<emph.end type="italics"></emph.end>paralle<lb></lb>la, & Orbis tangenti <emph type="italics"></emph>PG<emph.end type="italics"></emph.end>oc<lb></lb>currens in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>; & vis illa (per <lb></lb>Corol. </s> <s>1 & Schol. </s> <s>Prop. </s> <s>X, & Corol. </s> <s>3 Prop. </s> <s>VII.) erit ut <lb></lb>(<emph type="italics"></emph>CG cub./RP quad.<emph.end type="italics"></emph.end>) <pb xlink:href="039/01/087.jpg" pagenum="59"></pb><arrow.to.target n="note35"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note35"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Inventione Orbium Elliptieorum, Parabolieorum & Hyperbolico<lb></lb>rum ex umbilico dato.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ab Ellipſeos vel Hyperbolæ cujuſvis umbilicis duobus<emph.end type="italics"></emph.end>S, H, <emph type="italics"></emph>ad <lb></lb>punctum quodvis tertium<emph.end type="italics"></emph.end>V <emph type="italics"></emph>inflectantur rectæ duæ<emph.end type="italics"></emph.end>SV, HV, <lb></lb><emph type="italics"></emph>quarum una<emph.end type="italics"></emph.end>HV <emph type="italics"></emph>æqualis ſit axi principali figuræ, altera<emph.end type="italics"></emph.end>SV <emph type="italics"></emph>a <lb></lb>perpendiculo<emph.end type="italics"></emph.end>TR <emph type="italics"></emph>in ſe demiſſo bi-<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.087.1.jpg" xlink:href="039/01/087/1.jpg"></figure><lb></lb><emph type="italics"></emph>ſecetur in<emph.end type="italics"></emph.end>T; <emph type="italics"></emph>perpendiculum illud<emph.end type="italics"></emph.end><lb></lb>TR <emph type="italics"></emph>ſectionem Conicam alicubi tan<lb></lb>get: & contra, ſi tangit, erit<emph.end type="italics"></emph.end>HV <lb></lb><emph type="italics"></emph>æqualis axi principali figuræ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Secet enim perpendiculum <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>re<lb></lb>ctam <emph type="italics"></emph>HV<emph.end type="italics"></emph.end>productam, ſi opus fuerit, <lb></lb>in <emph type="italics"></emph>R<emph.end type="italics"></emph.end>; & jungatur <emph type="italics"></emph>SR.<emph.end type="italics"></emph.end>Ob æquales <lb></lb><emph type="italics"></emph>TS, TV,<emph.end type="italics"></emph.end>æquales erunt & rectæ <emph type="italics"></emph>SR, VR<emph.end type="italics"></emph.end>& anguli <emph type="italics"></emph>TRS, TRV.<emph.end type="italics"></emph.end><lb></lb>Unde punctum <emph type="italics"></emph>R<emph.end type="italics"></emph.end>erit ad Sectionem Conicam, & perpendiculum <lb></lb><emph type="italics"></emph>TR<emph.end type="italics"></emph.end>tanget eandem: & contra. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XVIII. PROBLEMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Datis umbilico & axibus principalibus deſcribere Trajectorias Ellipti<lb></lb>cas & Hyperbolicas, quæ tranſibunt per puncta data, & rectas po<lb></lb>ſitione datas contingent.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>S<emph.end type="italics"></emph.end>communis umbilicus figurarum; <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>longitudo axis prin<lb></lb>cipalis Trajectoriæ cujuſvis; <emph type="italics"></emph>P<emph.end type="italics"></emph.end>punctum per quod Trajectoria de<lb></lb>bet tranſire; & <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>recta quam debet tangere. </s> <s>Centro <emph type="italics"></emph>P<emph.end type="italics"></emph.end>inter<lb></lb>vallo <emph type="italics"></emph>AB-SP,<emph.end type="italics"></emph.end>ſi orbita ſit Ellipſis, vel <emph type="italics"></emph>AB+SP,<emph.end type="italics"></emph.end>ſi ea ſit Hy<lb></lb>perbola, deſcribatur circulus <emph type="italics"></emph>HG.<emph.end type="italics"></emph.end>Ad tangentem <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>demittatur <lb></lb>perpendiculum <emph type="italics"></emph>ST,<emph.end type="italics"></emph.end>& producatur idem ad <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>TV<emph.end type="italics"></emph.end>æqualis <lb></lb><emph type="italics"></emph>ST<emph.end type="italics"></emph.end>; centroque <emph type="italics"></emph>V<emph.end type="italics"></emph.end>& intervallo <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>deſcribatur circulus <emph type="italics"></emph>FH.<emph.end type="italics"></emph.end>Hac <pb xlink:href="039/01/088.jpg" pagenum="60"></pb><arrow.to.target n="note36"></arrow.to.target>methodo ſive dentur duo puncta <emph type="italics"></emph>P, p,<emph.end type="italics"></emph.end>ſive duæ tangentes <emph type="italics"></emph>TR, <lb></lb>tr,<emph.end type="italics"></emph.end>ſive punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& tangens <lb></lb><figure id="id.039.01.088.1.jpg" xlink:href="039/01/088/1.jpg"></figure><lb></lb><emph type="italics"></emph>TR,<emph.end type="italics"></emph.end>deſcribendi ſunt circuli duo. </s> <s><lb></lb>Sit <emph type="italics"></emph>H<emph.end type="italics"></emph.end>eorum interſectio com<lb></lb>munis, & umbilicis <emph type="italics"></emph>S, H,<emph.end type="italics"></emph.end>axe illo <lb></lb>dato deſcribatur Trajectoria. </s> <s><lb></lb>Dico factum. </s> <s>Nam Trajecto<lb></lb>ctoria deſcripta (eo quod <emph type="italics"></emph>PH <lb></lb>+SP<emph.end type="italics"></emph.end>in Ellipſi, & <emph type="italics"></emph>PH-SP<emph.end type="italics"></emph.end><lb></lb>in Hyperbola æquatur axi) <lb></lb>tranſibit per punctum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& <lb></lb>(per Lemma ſuperius) tanget <lb></lb>rectam <emph type="italics"></emph>TR.<emph.end type="italics"></emph.end>Et eodem argu<lb></lb>mento vel tranſibit eadem per <lb></lb>puncta duo <emph type="italics"></emph>P, p,<emph.end type="italics"></emph.end>vel tanget re<lb></lb>ctas duas <emph type="italics"></emph>TR, tr. </s> <s>q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note36"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XIX. PROBLEMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Circa datum umbilicum Trajectoriam Parabolicam deſcribere, quæ <lb></lb>tranſibit per puncta data, & rectas poſitione datas continget.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>S<emph.end type="italics"></emph.end>umbilicus, <emph type="italics"></emph>P<emph.end type="italics"></emph.end>punctum & <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>tangens Trajectoriæ deſcri<lb></lb>bendæ. </s> <s>Centro <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>deſcribe cir<lb></lb><figure id="id.039.01.088.2.jpg" xlink:href="039/01/088/2.jpg"></figure><lb></lb>culum <emph type="italics"></emph>FG.<emph.end type="italics"></emph.end>Ab umbilico ad tangentem demit<lb></lb>te perpendicularem <emph type="italics"></emph>ST,<emph.end type="italics"></emph.end>& produc eam ad <emph type="italics"></emph>V,<emph.end type="italics"></emph.end><lb></lb>ut ſit <emph type="italics"></emph>TV<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>ST.<emph.end type="italics"></emph.end>Eodem modo deſcri<lb></lb>bendus eſt alter circulus <emph type="italics"></emph>fg,<emph.end type="italics"></emph.end>ſi datur alterum <lb></lb>punctum <emph type="italics"></emph>p<emph.end type="italics"></emph.end>; vel inveniendum alterum punctum <lb></lb><emph type="italics"></emph>v,<emph.end type="italics"></emph.end>ſi datur altera tangens <emph type="italics"></emph>tr<emph.end type="italics"></emph.end>; dein ducenda re<lb></lb>cta <emph type="italics"></emph>IF<emph.end type="italics"></emph.end>quæ tangat duos circulos <emph type="italics"></emph>FG, fg<emph.end type="italics"></emph.end>ſi <lb></lb>dantur duo puncta <emph type="italics"></emph>P, p,<emph.end type="italics"></emph.end>vel tranſeat per duo <lb></lb>puncta <emph type="italics"></emph>V, v,<emph.end type="italics"></emph.end>ſi dantur duæ tangentes <emph type="italics"></emph>TR, tr,<emph.end type="italics"></emph.end>vel <lb></lb>tangat circulum <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>& tranſeat per punctum <emph type="italics"></emph>V,<emph.end type="italics"></emph.end><lb></lb>ſi datur punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& tangens <emph type="italics"></emph>TR.<emph.end type="italics"></emph.end>Ad <emph type="italics"></emph>FI<emph.end type="italics"></emph.end>demitte perpendicula<lb></lb>rem <emph type="italics"></emph>SI,<emph.end type="italics"></emph.end>eamque biſeca in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>; & axe <emph type="italics"></emph>SK,<emph.end type="italics"></emph.end>vertice principali <emph type="italics"></emph>K<emph.end type="italics"></emph.end>de<lb></lb>ſcribatur Parabola. </s> <s>Dico factum. </s> <s>Nam Parabola, ob æquales <lb></lb><emph type="italics"></emph>SK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IK, SP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FP,<emph.end type="italics"></emph.end>tranſibit per punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>; & (per Lem<lb></lb>matis XIV. Corol. </s> <s>3.) ob æquales <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>& <emph type="italics"></emph>TV<emph.end type="italics"></emph.end>& angulum rectum <lb></lb><emph type="italics"></emph>STR,<emph.end type="italics"></emph.end>tanget rectam <emph type="italics"></emph>TR. q.E.F.<emph.end type="italics"></emph.end><pb xlink:href="039/01/089.jpg" pagenum="61"></pb><arrow.to.target n="note37"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note37"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XX. PROBLEMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Circa datum umbilicum Trajectoriam quamvis ſpecie datam deſcribe<lb></lb>re, quæ per data puncta tranſibit & rectas tanget pofitione datas.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Dato umbilico <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>deſcribenda ſit Trajectoria <emph type="italics"></emph>ABC<emph.end type="italics"></emph.end>per <lb></lb>puncta duo <emph type="italics"></emph>B, C.<emph.end type="italics"></emph.end>Quoniam Trajectoria datur ſpecie, dabitur ra<lb></lb>tio axis principalis ad diſtantiam <lb></lb><figure id="id.039.01.089.1.jpg" xlink:href="039/01/089/1.jpg"></figure><lb></lb>umbilieorum. </s> <s>In ea ratione cape <lb></lb><emph type="italics"></emph>KB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BS,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>LC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CS.<emph.end type="italics"></emph.end>Cen<lb></lb>tris <emph type="italics"></emph>B, C,<emph.end type="italics"></emph.end>intervallis <emph type="italics"></emph>BK, CL,<emph.end type="italics"></emph.end>de<lb></lb>ſcribe circulos duos, & ad rectam <lb></lb><emph type="italics"></emph>KL,<emph.end type="italics"></emph.end>quæ tangat eoſdem in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>L,<emph.end type="italics"></emph.end>demitte perpendiculum <emph type="italics"></emph>SG,<emph.end type="italics"></emph.end>idemque ſeca in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>a,<emph.end type="italics"></emph.end>ita ut ſit <lb></lb><emph type="italics"></emph>GA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Ga<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>aS,<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>KB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BS,<emph.end type="italics"></emph.end>& axe &c. <emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end>verticibus <lb></lb><emph type="italics"></emph>A, a,<emph.end type="italics"></emph.end>deſcribatur Trajectoria. </s> <s>Dico factum. </s> <s>Sit enim <emph type="italics"></emph>H<emph.end type="italics"></emph.end>umbilicus <lb></lb>alter Figuræ deſcriptæ, & cum ſit <emph type="italics"></emph>GA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Ga<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>aS,<emph.end type="italics"></emph.end>erit di<lb></lb>viſim <emph type="italics"></emph>Ga-GA<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>aS-AS<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>SH<emph.end type="italics"></emph.end>in eadem &c. </s> <s>ratione, <lb></lb>adeoQ.E.I. ratione quam habet axis principalis Figuræ deſcribendæ <lb></lb>ad diſtantiam umbilieorum ejus; & propterea Figura deſcripta eſt <lb></lb>ejuſdem ſpeciei cum deſcribenda. </s> <s>Cumque ſint <emph type="italics"></emph>KB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>LC<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>CS<emph.end type="italics"></emph.end>in eadem ratione, tranſibit hæc Figura per puncta <emph type="italics"></emph>B, C,<emph.end type="italics"></emph.end>ut <lb></lb>ex Conicis manifeſtum eſt. </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Dato umbilico <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>deſcribenda ſit Trajectoria quæ rectas <lb></lb>duas <emph type="italics"></emph>TR, tr<emph.end type="italics"></emph.end>alicubi contingat. </s> <s>Ab umbilico in tangentes demitte <lb></lb>perpendicula <emph type="italics"></emph>ST, St<emph.end type="italics"></emph.end>& produc ea<lb></lb><figure id="id.039.01.089.2.jpg" xlink:href="039/01/089/2.jpg"></figure><lb></lb>dem ad <emph type="italics"></emph>V, v,<emph.end type="italics"></emph.end>ut ſint <emph type="italics"></emph>TV, tv<emph.end type="italics"></emph.end>æ<lb></lb>quales <emph type="italics"></emph>TS, tS.<emph.end type="italics"></emph.end>Biſeca <emph type="italics"></emph>Vv<emph.end type="italics"></emph.end>in <emph type="italics"></emph>O,<emph.end type="italics"></emph.end><lb></lb>& erige perpendiculum infinitum <lb></lb><emph type="italics"></emph>OH,<emph.end type="italics"></emph.end>rectamque <emph type="italics"></emph>VS<emph.end type="italics"></emph.end>infinite pro<lb></lb>ductam ſeca in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>k<emph.end type="italics"></emph.end>ita, ut ſit <lb></lb><emph type="italics"></emph>VK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Vk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>kS<emph.end type="italics"></emph.end>ut eſt <lb></lb>Trajectoriæ deſcribendæ axis prin<lb></lb>cipalis ad umbilieorum diſtantiam. </s> <s><lb></lb>Super diametro <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>deſcribatur <lb></lb>circulus ſecans <emph type="italics"></emph>OH<emph.end type="italics"></emph.end>in <emph type="italics"></emph>H<emph.end type="italics"></emph.end>; & umbilicis <emph type="italics"></emph>S, H,<emph.end type="italics"></emph.end>axe principali ipſam <lb></lb><emph type="italics"></emph>VH<emph.end type="italics"></emph.end>æquante, deſcribatur Trajectoria. </s> <s>Dico factum. </s> <s>Nam biſeca <lb></lb><emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>in <emph type="italics"></emph>X,<emph.end type="italics"></emph.end>& junge <emph type="italics"></emph>HX, HS, HV, Hv.<emph.end type="italics"></emph.end>Quoniam eſt <emph type="italics"></emph>VK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KS<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>Vk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>kS<emph.end type="italics"></emph.end>; & compofite ut <emph type="italics"></emph>VK+Vk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KS+kS<emph.end type="italics"></emph.end>; diviſimque <pb xlink:href="039/01/090.jpg" pagenum="62"></pb><arrow.to.target n="note38"></arrow.to.target>ut <emph type="italics"></emph>Vk-VK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>kS-KS,<emph.end type="italics"></emph.end>id eſt ut 2 <emph type="italics"></emph>VX<emph.end type="italics"></emph.end>ad 2 <emph type="italics"></emph>KX<emph.end type="italics"></emph.end>& 2 <emph type="italics"></emph>KX<emph.end type="italics"></emph.end>ad <lb></lb>2 <emph type="italics"></emph>SX,<emph.end type="italics"></emph.end>adeoque ut <emph type="italics"></emph>VX<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HX<emph.end type="italics"></emph.end>& <emph type="italics"></emph>HX<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SX,<emph.end type="italics"></emph.end>ſimilia erunt tri<lb></lb>angula <emph type="italics"></emph>VXH, HXS,<emph.end type="italics"></emph.end>& propterea <emph type="italics"></emph>VH<emph.end type="italics"></emph.end>erit ad <emph type="italics"></emph>SH<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>VX<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>XH,<emph.end type="italics"></emph.end><lb></lb>adeoque ut <emph type="italics"></emph>VK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KS.<emph.end type="italics"></emph.end>Habet igitur Trajectoriæ deſcriptæ axis <lb></lb>principalis <emph type="italics"></emph>VH<emph.end type="italics"></emph.end>eam rationem ad ipſius umbilieorum diſtantiam <emph type="italics"></emph>SH,<emph.end type="italics"></emph.end><lb></lb>quam habet Trajectoriæ deſcribendæ axis principalis ad ipſius um<lb></lb>bilieorum diſtantiam, & propterea ejuſdem eſt ſpeciei. </s> <s>Inſuper cum <lb></lb><emph type="italics"></emph>VH, vH<emph.end type="italics"></emph.end>æquentur axi principali, & <emph type="italics"></emph>VS, vS<emph.end type="italics"></emph.end>a rectis <emph type="italics"></emph>TR, tr<emph.end type="italics"></emph.end><lb></lb>perpendiculariter biſecentur, liquet, ex Lemmate XV, rectas illas <lb></lb>Trajectoriam deſcriptam tangere. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note38"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Dato umbilico <emph type="italics"></emph>S<emph.end type="italics"></emph.end>deſcribenda ſit Trajectoria quæ rect<lb></lb>am <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>tanget in puncto dato <emph type="italics"></emph>R.<emph.end type="italics"></emph.end>In rectam <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>demitte perpen<lb></lb>dicularem <emph type="italics"></emph>ST,<emph.end type="italics"></emph.end>& produc eandem ad <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>TV<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>ST.<emph.end type="italics"></emph.end>Junge <lb></lb><emph type="italics"></emph>VR,<emph.end type="italics"></emph.end>& rectam <emph type="italics"></emph>VS<emph.end type="italics"></emph.end>infinite productam ſeca in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>k,<emph.end type="italics"></emph.end>ita ut ſit <lb></lb><emph type="italics"></emph>VK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Vk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Sk<emph.end type="italics"></emph.end>ut Ellipſeos deſcribendæ axis principalis <lb></lb>ad diſtantiam umbilieorum; circuloque ſuper diametro <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>de<lb></lb>ſcripto, ſecetur producta recta <emph type="italics"></emph>VR<emph.end type="italics"></emph.end>in <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>& umbilicis <emph type="italics"></emph>S, H,<emph.end type="italics"></emph.end>axe <lb></lb>principali rectam <emph type="italics"></emph>VH<emph.end type="italics"></emph.end>æquante, deſcribatur Trajectoria. </s> <s>Dico fa<lb></lb>ctum. </s> <s>Namque <emph type="italics"></emph>VH<emph.end type="italics"></emph.end>eſſe ad <lb></lb><figure id="id.039.01.090.1.jpg" xlink:href="039/01/090/1.jpg"></figure><lb></lb><emph type="italics"></emph>SH<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>VK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SK,<emph.end type="italics"></emph.end>atque adeo <lb></lb>ut axis principalis Trajectoriæ <lb></lb>deſcribendæ ad diſtantiam um<lb></lb>bilieorum ejus, patet ex demon<lb></lb>ſtratis in Caſu ſecundo, & prop<lb></lb>terea Trajectoriam deſcriptam <lb></lb>ejuſdem eſſe ſpeciei cum deſcri<lb></lb>benda; rectam vero <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>qua an<lb></lb>gulus <emph type="italics"></emph>VRS<emph.end type="italics"></emph.end>biſecatur, tangere Trajectoriam in puncto <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>patet ex <lb></lb>Conicis. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>4. Circa umbilicum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>deſcribenda jam ſit Trajectoria <emph type="italics"></emph>APB,<emph.end type="italics"></emph.end><lb></lb>quæ tangat rectam <emph type="italics"></emph>TR,<emph.end type="italics"></emph.end>tranſeatque per punctum quodvis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>extra <lb></lb>tangentem datum, quæque ſimilis ſit Figuræ <emph type="italics"></emph>apb,<emph.end type="italics"></emph.end>axe principali <lb></lb><emph type="italics"></emph>ab<emph.end type="italics"></emph.end>& umbilicis <emph type="italics"></emph>s, h<emph.end type="italics"></emph.end>deſcriptæ. </s> <s>In tangentem <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>demitte per<lb></lb>pendiculum <emph type="italics"></emph>ST,<emph.end type="italics"></emph.end>& produc idem ad <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>TV<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>ST.<emph.end type="italics"></emph.end>An<lb></lb>gulis autem <emph type="italics"></emph>VSP, SVP<emph.end type="italics"></emph.end>fac angulos <emph type="italics"></emph>hsq, shq<emph.end type="italics"></emph.end>æquales; cen<lb></lb>troque <emph type="italics"></emph>q<emph.end type="italics"></emph.end>& intervallo quod ſit ad <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>VS<emph.end type="italics"></emph.end>deſcribe circu<lb></lb>lum ſecantem Figuram <emph type="italics"></emph>apb<emph.end type="italics"></emph.end>in <emph type="italics"></emph>p.<emph.end type="italics"></emph.end>Junge <emph type="italics"></emph>sp<emph.end type="italics"></emph.end>& age <emph type="italics"></emph>SH<emph.end type="italics"></emph.end>quæ ſit ad <lb></lb><emph type="italics"></emph>sh<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>sp,<emph.end type="italics"></emph.end>quæque angulum <emph type="italics"></emph>PSH<emph.end type="italics"></emph.end>angulo <emph type="italics"></emph>psh<emph.end type="italics"></emph.end>& angulum <lb></lb><emph type="italics"></emph>VSH<emph.end type="italics"></emph.end>angulo <emph type="italics"></emph>psq<emph.end type="italics"></emph.end>æquales conſtituat. </s> <s>Denique umbilicis <emph type="italics"></emph>S, H,<emph.end type="italics"></emph.end><lb></lb>& axe principali <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>diſtantiam <emph type="italics"></emph>VH<emph.end type="italics"></emph.end>æquante, deſcribatur ſectio <lb></lb>Conica. </s> <s>Dico factum. </s> <s>Nam ſi agatur <emph type="italics"></emph>sv<emph.end type="italics"></emph.end>quæ ſit ad <emph type="italics"></emph>sp<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>sh<emph.end type="italics"></emph.end><pb xlink:href="039/01/091.jpg" pagenum="63"></pb>ad <emph type="italics"></emph>sq,<emph.end type="italics"></emph.end>quæque conſtituat angulum <emph type="italics"></emph>vsp<emph.end type="italics"></emph.end>angulo <emph type="italics"></emph>hsq<emph.end type="italics"></emph.end>& angulum <lb></lb><arrow.to.target n="note39"></arrow.to.target><emph type="italics"></emph>vsh<emph.end type="italics"></emph.end>angulo <emph type="italics"></emph>psq<emph.end type="italics"></emph.end>æquales, triangula <emph type="italics"></emph>svh, spq<emph.end type="italics"></emph.end>erunt ſimilia, & prop<lb></lb>terea <emph type="italics"></emph>vh<emph.end type="italics"></emph.end>erit ad <emph type="italics"></emph>pq<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>sh<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>sq,<emph.end type="italics"></emph.end>id eſt (ob ſimilia triangula <lb></lb><figure id="id.039.01.091.1.jpg" xlink:href="039/01/091/1.jpg"></figure><lb></lb><emph type="italics"></emph>VSP, hsq<emph.end type="italics"></emph.end>) ut eſt <emph type="italics"></emph>VS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="pq.">pque</expan><emph.end type="italics"></emph.end>Æquantur ergo <lb></lb><emph type="italics"></emph>vh<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ab.<emph.end type="italics"></emph.end>Porro ob ſimilia triangula <emph type="italics"></emph>VSH. vsh,<emph.end type="italics"></emph.end>eſt <emph type="italics"></emph>VH<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>SH<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>vh<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>sh,<emph.end type="italics"></emph.end>id eſt, axis Conicæ ſectionis jam deſcriptæ ad <lb></lb>illius umbilieorum intervallum, ut axis <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>ad umbilieorum inter<lb></lb>vallum <emph type="italics"></emph>sh<emph.end type="italics"></emph.end>; & propterea Figura jam deſeripta ſimilis eſt Figuræ <lb></lb><emph type="italics"></emph>apb.<emph.end type="italics"></emph.end>Tranſit autem hæc Figura per punctum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>eo quod trian<lb></lb>gulum <emph type="italics"></emph>PSH<emph.end type="italics"></emph.end>ſimile ſit triangulo <emph type="italics"></emph>psh<emph.end type="italics"></emph.end>; & quia <emph type="italics"></emph>VH<emph.end type="italics"></emph.end>æquatur ipſius <lb></lb>axi & <emph type="italics"></emph>VS<emph.end type="italics"></emph.end>biſecatur perpendiculariter a recta <emph type="italics"></emph>TR,<emph.end type="italics"></emph.end>tangit eadem <lb></lb>rectam <emph type="italics"></emph>TR. q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note39"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>A datis tribus punctis ad quartum non datum inflectere tres rectas <lb></lb>quarum differentiæ vel dantur vel nullæ ſunt.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Sunto puncta illa data <emph type="italics"></emph>A, B, C<emph.end type="italics"></emph.end>& punctum quartum <emph type="italics"></emph>Z,<emph.end type="italics"></emph.end><lb></lb>quod invenire oportet; Ob datam differentiam linearum <emph type="italics"></emph>AZ, BZ,<emph.end type="italics"></emph.end><lb></lb>locabitur punctum <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>in Hyperbola cujus umbilici ſunt <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& <lb></lb>principalis axis differentia illa data. </s> <s>Sit axis ille <emph type="italics"></emph>MN.<emph.end type="italics"></emph.end>Cape <emph type="italics"></emph>PM.<emph.end type="italics"></emph.end><pb xlink:href="039/01/092.jpg" pagenum="64"></pb><arrow.to.target n="note40"></arrow.to.target>ad <emph type="italics"></emph>MA<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>& erecta <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>perpendiculari ad <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end><lb></lb>demiſſaque <emph type="italics"></emph>ZR<emph.end type="italics"></emph.end>perpendiculari ad <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>; erit, ex natura hujus Hy<lb></lb>perbolæ, <emph type="italics"></emph>ZR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Simili diſcurſu punctum <lb></lb><emph type="italics"></emph>Z<emph.end type="italics"></emph.end>locabitur in alia Hyperbola, cujus umbilici ſunt <emph type="italics"></emph>A, C<emph.end type="italics"></emph.end>& princi<lb></lb>palis axis differentia inter <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CZ,<emph.end type="italics"></emph.end>ducique poteſt <emph type="italics"></emph>QS<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>AC<emph.end type="italics"></emph.end><lb></lb>perpendicularis, ad quam ſi ab Hyperbolæ hujus puncto quovis <emph type="italics"></emph>Z<emph.end type="italics"></emph.end><lb></lb>demittatur normalis <emph type="italics"></emph>ZS,<emph.end type="italics"></emph.end>hæc fuerit ad <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>ut eſt differentia inter <lb></lb><emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CZ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC.<emph.end type="italics"></emph.end>Dantur ergo rationes ipſarum <emph type="italics"></emph>ZR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ZS<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>AZ,<emph.end type="italics"></emph.end>& idcirco datur earun<lb></lb><figure id="id.039.01.092.1.jpg" xlink:href="039/01/092/1.jpg"></figure><lb></lb>dem <emph type="italics"></emph>ZR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ZS<emph.end type="italics"></emph.end>ratio ad invicem; <lb></lb>ideoque ſi rectæ <emph type="italics"></emph>RP, SQ<emph.end type="italics"></emph.end>concur<lb></lb>rant in <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>& agatur <emph type="italics"></emph>TZ,<emph.end type="italics"></emph.end>figura <lb></lb><emph type="italics"></emph>TRZS,<emph.end type="italics"></emph.end>dabitur ſpecie, & recta <lb></lb><emph type="italics"></emph>TZ<emph.end type="italics"></emph.end>in qua punctum <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>alicubi lo<lb></lb>catur, dabitur poſitione. </s> <s>Eadem <lb></lb>methodo per Hyperbolam ter<lb></lb>tiam, cujus umbilici ſunt <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>C<emph.end type="italics"></emph.end><lb></lb>& axis principalis differentia re<lb></lb>ctarum <emph type="italics"></emph>BZ, CZ,<emph.end type="italics"></emph.end>inveniri poteſt <lb></lb>alia recta in qua <expan abbr="pũctum">punctum</expan> <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>locatur. </s> <s><lb></lb>Habitis autem duobus Locis recti<lb></lb>lineis, habetur punctum quæſitum <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>in eorum interſectione. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note40"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Si duæ ex tribus lineis, puta <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BZ<emph.end type="italics"></emph.end>æquantur, pun<lb></lb>ctum <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>locabitur in perpendiculo biſecante diſtantiam <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>& lo<lb></lb>cus alius rectilineus invenietur ut ſupra. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Si omnes tres æquantur, locabitur punctum <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>in centro <lb></lb>Circuli per puncta <emph type="italics"></emph>A, B, C<emph.end type="italics"></emph.end>tranſeuntis. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Solvitur etiam hoc Lemma problematicum per Librum Tactio<lb></lb>num <emph type="italics"></emph>Apollonii<emph.end type="italics"></emph.end>a <emph type="italics"></emph>Vieta<emph.end type="italics"></emph.end>reſtitutum. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXI. PROBLEMA XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam circa datum umbilicum deſcribere, quæ tranſibit per <lb></lb>puncta data & rectas poſitione datas continget.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Detur umbilicus <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>punctum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& tangens <emph type="italics"></emph>TR,<emph.end type="italics"></emph.end>& invenien<lb></lb>dus ſit umbilicus alter <emph type="italics"></emph>H.<emph.end type="italics"></emph.end>Ad tangentem demitte perpendiculum <lb></lb><emph type="italics"></emph>ST,<emph.end type="italics"></emph.end>& produc idem ad <emph type="italics"></emph>Y,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>TY<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>ST,<emph.end type="italics"></emph.end>& erit <emph type="italics"></emph>YH<emph.end type="italics"></emph.end>æ<lb></lb>qualis axi principali. </s> <s>Junge <emph type="italics"></emph>SP, HP,<emph.end type="italics"></emph.end>& erit <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>differentia inter <lb></lb><emph type="italics"></emph>HP<emph.end type="italics"></emph.end>& axem principalem. </s> <s>Hoc modo ſi dentur plures tangen-<pb xlink:href="039/01/093.jpg" pagenum="65"></pb>tes <emph type="italics"></emph>TR,<emph.end type="italics"></emph.end>vel plura puncta <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>devenietur ſemper ad lineas totidem <lb></lb><arrow.to.target n="note41"></arrow.to.target><emph type="italics"></emph>YH,<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>PH,<emph.end type="italics"></emph.end>a dictis punctis <emph type="italics"></emph>Y<emph.end type="italics"></emph.end>vel <lb></lb><figure id="id.039.01.093.1.jpg" xlink:href="039/01/093/1.jpg"></figure><lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>ad umbilicum <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ductas, quæ vel <lb></lb>æquantur axibus, vel datis longitu<lb></lb>dinibus <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>differunt ab iiſdem, at<lb></lb>que adeo quæ vel æquantur ſibi invi<lb></lb>cem, vel datas habent differentias; & <lb></lb>inde, per Lemma ſuperius, datur umbi<lb></lb>licus ille alter <emph type="italics"></emph>H.<emph.end type="italics"></emph.end>Habitis autem um<lb></lb>bilicis una cum axis longitudine (quæ <lb></lb>vel eſt <emph type="italics"></emph>YH<emph.end type="italics"></emph.end>; vel, ſi Trajectoria Ellipſis eſt, <emph type="italics"></emph>PH+SP<emph.end type="italics"></emph.end>; ſin Hy<lb></lb>perbola, <emph type="italics"></emph>PH-SP<emph.end type="italics"></emph.end>) habetur Trajectoria. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note41"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Caſus ubi dantur tria puncta ſic ſolvitur expeditius. </s> <s>Dentur <lb></lb>puncta <emph type="italics"></emph>B, C, D.<emph.end type="italics"></emph.end>Junctas <emph type="italics"></emph>BC, CD<emph.end type="italics"></emph.end>produc ad <emph type="italics"></emph>E, F,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>EB<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>EC<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SC,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SD.<emph.end type="italics"></emph.end>Ad <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ductam <lb></lb>& productam demitte normales <emph type="italics"></emph>SG, BH,<emph.end type="italics"></emph.end>inque <emph type="italics"></emph>GS<emph.end type="italics"></emph.end>infinite <lb></lb>producta cape <emph type="italics"></emph>GA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Ga<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>aS<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>HB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BS<emph.end type="italics"></emph.end>; & erit <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>vertex, & <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>axis principalis Trajectoriæ: quæ, perinde ut <emph type="italics"></emph>GA<emph.end type="italics"></emph.end><lb></lb>major, æqualis, vel minor fuerit quam <emph type="italics"></emph>AS,<emph.end type="italics"></emph.end>erit Ellipſis, Parabola <lb></lb>vel Hyperbola; pun<lb></lb><figure id="id.039.01.093.2.jpg" xlink:href="039/01/093/2.jpg"></figure><lb></lb>cto <emph type="italics"></emph>a<emph.end type="italics"></emph.end>in primo caſu <lb></lb>cadente ad eandem <lb></lb>partem lineæ <emph type="italics"></emph>GF<emph.end type="italics"></emph.end><lb></lb>cum puncto <emph type="italics"></emph>A<emph.end type="italics"></emph.end>; in <lb></lb>ſecundo caſu abeunte <lb></lb>in infinitum; in tertio <lb></lb>cadente ad contrari<lb></lb>am partem lineæ <emph type="italics"></emph>GF.<emph.end type="italics"></emph.end><lb></lb>Nam ſi demittantur <lb></lb>ad <emph type="italics"></emph>GF<emph.end type="italics"></emph.end>perpendicula <lb></lb><emph type="italics"></emph>CI, DK<emph.end type="italics"></emph.end>; erit <emph type="italics"></emph>IC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HB<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>EC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EB,<emph.end type="italics"></emph.end>hoc eſt, ut <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SB<emph.end type="italics"></emph.end>; & vi<lb></lb>ciſſim <emph type="italics"></emph>IC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>HB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SB<emph.end type="italics"></emph.end>ſive ut <emph type="italics"></emph>GA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SA.<emph.end type="italics"></emph.end>Et ſimili argumento <lb></lb>probabitur eſſe <emph type="italics"></emph>KD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SD<emph.end type="italics"></emph.end>in eadem ratione. </s> <s>Jacent ergo puncta <emph type="italics"></emph>B, <lb></lb>C, D<emph.end type="italics"></emph.end>in Coniſectione circa umbilicum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ita deſcripta, ut rectæ omnes <lb></lb>ab umbilico <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad ſingula Sectionis puncta ductæ, ſint ad perpendicula <lb></lb>a punctis iiſdem ad rectam <emph type="italics"></emph>GF<emph.end type="italics"></emph.end>demiſſa in data illa ratione. </s></p> <p type="main"> <s>Methodo haud multum diſſimili hujus problematis ſolutionem <lb></lb>tradit Clariſſimus Geometra <emph type="italics"></emph>de la Hire,<emph.end type="italics"></emph.end>Conieorum ſuorum Lib. </s> <s><lb></lb>VIII. Prop. XXV. <pb xlink:href="039/01/094.jpg" pagenum="66"></pb><arrow.to.target n="note42"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note42"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Inventio Orbium ubi umbilicus neuter datur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si a datæ Conicæ Sectionis puncto quovis<emph.end type="italics"></emph.end>P, <emph type="italics"></emph>ad Trapezii alicujus<emph.end type="italics"></emph.end><lb></lb>ABDC, <emph type="italics"></emph>in Conica illa ſectione inſcripti, latera quatuor infinite <lb></lb>producta<emph.end type="italics"></emph.end>AB, CD, AC, DB, <emph type="italics"></emph>totidem rectæ<emph.end type="italics"></emph.end>PQ, PR, PS, PT <lb></lb><emph type="italics"></emph>in datis angulis ducantur, ſingulæ ad ſingula: rectangulum duc<lb></lb>tarum ad oppoſita duo latera<emph.end type="italics"></emph.end>PQXPR, <emph type="italics"></emph>erit ad rectangulum duc<lb></lb>tarum ad alia duo latera oppoſita<emph.end type="italics"></emph.end>PSXPT <emph type="italics"></emph>in data ratione.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Ponamus primo lineas ad <lb></lb><figure id="id.039.01.094.1.jpg" xlink:href="039/01/094/1.jpg"></figure><lb></lb>oppoſita latera ductas parallelas eſ<lb></lb>ſe alterutri reliquorum laterum, <lb></lb>puta <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>lateri <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PS<emph.end type="italics"></emph.end><lb></lb>ac <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>lateri <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>SintQ.E.I.ſuper <lb></lb>latera duo ex oppoſitis, puta <emph type="italics"></emph>AC<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>BD,<emph.end type="italics"></emph.end>ſibi invicem paralle<lb></lb>la. </s> <s>Et recta quæ biſecat paralle<lb></lb>la illa latera erit una ex diametris <lb></lb>Conicæ ſectionis, & biſecabit eti<lb></lb>am <emph type="italics"></emph><expan abbr="Rq.">Rque</expan><emph.end type="italics"></emph.end>Sit <emph type="italics"></emph>O<emph.end type="italics"></emph.end>punctum in quo <lb></lb><emph type="italics"></emph>RQ<emph.end type="italics"></emph.end>biſecatur, & erit <emph type="italics"></emph>PO<emph.end type="italics"></emph.end>ordinatim applicata ad diametrum illam. </s> <s><lb></lb>Produc <emph type="italics"></emph>PO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>K<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>OK<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>PO,<emph.end type="italics"></emph.end>& erit <emph type="italics"></emph>OK<emph.end type="italics"></emph.end>ordinatim <lb></lb>applicata ad contrarias partes diametri. </s> <s>Cum igitur puncta <emph type="italics"></emph>A, B, <lb></lb>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end>ſint ad Conicam ſectionem, & <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>ſecet <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in dato an<lb></lb>gulo, erit (per Prop.17 & 18 Lib. </s> <s>III Conieorum <emph type="italics"></emph>Apollonii<emph.end type="italics"></emph.end>) rectangu<lb></lb>lum <emph type="italics"></emph>PQK<emph.end type="italics"></emph.end>ad rectangulum <emph type="italics"></emph>AQB<emph.end type="italics"></emph.end>in data ratione. </s> <s>Sed <emph type="italics"></emph>QK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PR<emph.end type="italics"></emph.end><lb></lb>æquales ſunt, utpote æqualium <emph type="italics"></emph>OK, OP,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>OQ, OR<emph.end type="italics"></emph.end>differentiæ, <lb></lb>& inde etiam rectangula <emph type="italics"></emph>PQK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end>æqualia ſunt; at<lb></lb>que adeo rectangulum <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end>eſt ad rectangulum <emph type="italics"></emph>AQB,<emph.end type="italics"></emph.end>hoc <lb></lb>eſt ad rectangulum <emph type="italics"></emph>PSXPT<emph.end type="italics"></emph.end>in data ratione. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/095.jpg" pagenum="67"></pb> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Ponamus jam Trapezii latera oppoſita <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>non <lb></lb><arrow.to.target n="note43"></arrow.to.target>eſſe parallela. </s> <s>Age <emph type="italics"></emph>Bd<emph.end type="italics"></emph.end>parallelam <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>& occurrentem tum rectæ <lb></lb><emph type="italics"></emph>ST<emph.end type="italics"></emph.end>in <emph type="italics"></emph>t,<emph.end type="italics"></emph.end>tum Conicæ ſectioni in <emph type="italics"></emph>d.<emph.end type="italics"></emph.end>Junge <emph type="italics"></emph>Cd<emph.end type="italics"></emph.end>ſecantem <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>r,<emph.end type="italics"></emph.end><lb></lb>& ipſi <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>parallelam age <emph type="italics"></emph>DM<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.095.1.jpg" xlink:href="039/01/095/1.jpg"></figure><lb></lb>ſecantem <emph type="italics"></emph>Cd<emph.end type="italics"></emph.end>in <emph type="italics"></emph>M<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in <emph type="italics"></emph>N.<emph.end type="italics"></emph.end><lb></lb>Jam ob ſimilia triangula <emph type="italics"></emph>BTt, <lb></lb>DBN<emph.end type="italics"></emph.end>; eſt <emph type="italics"></emph>Bt<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Tt<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>NB.<emph.end type="italics"></emph.end>Sic & <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>eſt ad <lb></lb><emph type="italics"></emph>AQ<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DM<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AN.<emph.end type="italics"></emph.end><lb></lb>Ergo, ducendo antecedentes in <lb></lb>antecedentes & conſequentes in <lb></lb>conſequentes, ut rectangulum <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>eſt ad rectangulum <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>in <lb></lb><emph type="italics"></emph>Tt,<emph.end type="italics"></emph.end>ita rectangulum <emph type="italics"></emph>NDM<emph.end type="italics"></emph.end>eſt <lb></lb>ad rectangulum <emph type="italics"></emph>ANB,<emph.end type="italics"></emph.end>& (per Caſ.1) ita rectangulum <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Pr<emph.end type="italics"></emph.end>eſt <lb></lb>ad rectangulum <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Pt,<emph.end type="italics"></emph.end>ac diviſim ita rectangulum <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end><lb></lb>eſt ad rectangulum <emph type="italics"></emph>PSXPT. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note43"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Ponamus denique lineas <lb></lb><figure id="id.039.01.095.2.jpg" xlink:href="039/01/095/2.jpg"></figure><lb></lb>quatuor <emph type="italics"></emph>PQ, PR, PS, PT<emph.end type="italics"></emph.end>non <lb></lb>eſſe parallelas lateribus <emph type="italics"></emph>AC, AB,<emph.end type="italics"></emph.end><lb></lb>ſed ad ea utcunQ.E.I.clinatas. </s> <s>Ea<lb></lb>rum vice age <emph type="italics"></emph>Pq, Pr<emph.end type="italics"></emph.end>parallelas <lb></lb>ipſi <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>Ps, Pt<emph.end type="italics"></emph.end>parallelas <lb></lb>ipſi <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; & propter datos angu<lb></lb>los triangulorum <emph type="italics"></emph>PQq, PRr, <lb></lb>PSs, PTt,<emph.end type="italics"></emph.end>dabuntur rationes <lb></lb><emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Pq, PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Pr, PS<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>Ps,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Pt<emph.end type="italics"></emph.end>; atque adeo rationes compoſitæ <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>PqXPr,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PSXPT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PsXPt.<emph.end type="italics"></emph.end>Sed, per ſuperius de<lb></lb>monſtrata, ratio <emph type="italics"></emph>PqXPr<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PsXPt<emph.end type="italics"></emph.end>data eſt: Ergo & ratio <lb></lb><emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PSXPT. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, ſi rectangulum ductarum ad oppoſita duo latera Tra<lb></lb>pezii<emph.end type="italics"></emph.end>PQXPR <emph type="italics"></emph>ſit ad rectangulum ductarum ad reliqua duo late<lb></lb>ra<emph.end type="italics"></emph.end>PSXPT <emph type="italics"></emph>in data ratione; punctum<emph.end type="italics"></emph.end>P, <emph type="italics"></emph>a quo lineæ ducuntur, <lb></lb>tanget Conicam ſectionem circa Trapezium deſcriptam.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/096.jpg" pagenum="68"></pb> <p type="main"> <s>Per puncta <emph type="italics"></emph>A, B, C, D<emph.end type="italics"></emph.end>& aliquod infinitorum punctorum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>pu<lb></lb><arrow.to.target n="note44"></arrow.to.target>ta <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>concipe Conicam ſectionem deſcribi: dico punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>hanc <lb></lb>ſemper tangere. </s> <s>Si negas, <lb></lb><figure id="id.039.01.096.1.jpg" xlink:href="039/01/096/1.jpg"></figure><lb></lb>junge <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ſecantem hanc <lb></lb>Conicam ſectionem alibi <lb></lb>quam in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſi fieri poteſt, <lb></lb>puta in <emph type="italics"></emph>b.<emph.end type="italics"></emph.end>Ergo ſi ab his <lb></lb>punctis <emph type="italics"></emph>p<emph.end type="italics"></emph.end>& <emph type="italics"></emph>b<emph.end type="italics"></emph.end>ducantur in <lb></lb>datis angulis ad latera Tra<lb></lb>pezii rectæ <emph type="italics"></emph>pq, pr, ps, pt<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>bk, br, bſ, bd<emph.end type="italics"></emph.end>; erit <lb></lb>ut <emph type="italics"></emph>bkXb<emph.end type="italics"></emph.end>r ad <emph type="italics"></emph>bſXbd<emph.end type="italics"></emph.end>ita <lb></lb>(per Lem. </s> <s>XVII) <emph type="italics"></emph>pqXpr<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>psXpt,<emph.end type="italics"></emph.end>& ita (per <lb></lb>Hypoth.) <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>PSXPT.<emph.end type="italics"></emph.end>Eſt & prop<lb></lb>ter ſimilitudinem Trapeziorum <emph type="italics"></emph>bkAſ, PQAS,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>bk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>bſ<emph.end type="italics"></emph.end>ita <lb></lb><emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PS.<emph.end type="italics"></emph.end>Quare, applicando terminos prioris proportionis ad <lb></lb>terminos correſpondentes hujus, erit <emph type="italics"></emph>b<emph.end type="italics"></emph.end>r ad <emph type="italics"></emph>bd<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PT.<emph.end type="italics"></emph.end>Er<lb></lb>go Trapezia æquiangula <emph type="italics"></emph>Dr bd, DRPT<emph.end type="italics"></emph.end>ſimilia ſunt, & eorum <lb></lb>diagonales <emph type="italics"></emph>Db, DP<emph.end type="italics"></emph.end>propterea coincidunt. </s> <s>Incidit itaque <emph type="italics"></emph>b<emph.end type="italics"></emph.end>in <lb></lb>interſectionem rectarum <emph type="italics"></emph>AP, DP<emph.end type="italics"></emph.end>adeoque coincidit cum puncto <lb></lb><emph type="italics"></emph>P.<emph.end type="italics"></emph.end>Quare punctum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ubicunque ſumatur, incidit in aſſignatam <lb></lb>Conicam ſectionem. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note44"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc ſi rectæ tres <emph type="italics"></emph>PQ, PR, PS<emph.end type="italics"></emph.end>a puncto communi <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>ad alias totidem poſitione datas rectas <emph type="italics"></emph>AB, CD, AC,<emph.end type="italics"></emph.end>ſingulæ ad <lb></lb>ſingulas, in datis angulis ducantur, ſitque rectangulum ſub duabus <lb></lb>ductis <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end>ad quadratum tertiæ <emph type="italics"></emph>PS quad.<emph.end type="italics"></emph.end>in data ratione: <lb></lb>punctum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>a quibus rectæ ducuntur, locabitur in ſectione Conica <lb></lb>quæ tangit lineas <emph type="italics"></emph>AB, CD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>C<emph.end type="italics"></emph.end>; & contra. </s> <s>Nam coeat linea <lb></lb><emph type="italics"></emph>BD<emph.end type="italics"></emph.end>cum linea <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>manente poſitione trium <emph type="italics"></emph>AB, CD, AC<emph.end type="italics"></emph.end>; de<lb></lb>in coeat etiam linea <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>cum linea <emph type="italics"></emph>PS:<emph.end type="italics"></emph.end>& rectangulum <emph type="italics"></emph>PSXPT<emph.end type="italics"></emph.end><lb></lb>evadet <emph type="italics"></emph>PS quad.<emph.end type="italics"></emph.end>rectæque <emph type="italics"></emph>AB, CD<emph.end type="italics"></emph.end>quæ curvam in punctis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B, <lb></lb>C<emph.end type="italics"></emph.end>& <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ſecabant, jam Curvam in punctis illis coeuntibus non am<lb></lb>plius ſecare poſſunt ſed tantum tangent. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Nomen Conicæ ſectionis in hoc Lemmate late ſumitur, ita ut <lb></lb>ſectio tam Rectilinea per verticem Coni tranſiens, quam Circularis <lb></lb>baſi parallela includatur. </s> <s>Nam ſi punctum <emph type="italics"></emph>p<emph.end type="italics"></emph.end>incidit in rectam, qua <lb></lb>quævis ex punctis quatuor <emph type="italics"></emph>A, B, C, D<emph.end type="italics"></emph.end>junguntur, Conica ſectio <pb xlink:href="039/01/097.jpg" pagenum="69"></pb>vertetur in geminas Rectas, quarum una eſt recta illa in quam pun<lb></lb><arrow.to.target n="note45"></arrow.to.target>ctum <emph type="italics"></emph>p<emph.end type="italics"></emph.end>incidit, & altera eſt recta qua alia duo ex punctis quatuor jun<lb></lb>guntur. </s> <s>Si Trapezii anguli duo oppoſiti ſimul ſumpti æquentur <lb></lb>duobus rectis, & lineæ quatuor <emph type="italics"></emph>PQ, PR, PS, PT<emph.end type="italics"></emph.end>ducantur ad <lb></lb>latera ejus vel perpendiculariter vel in angulis quibuſvis æqualibus, <lb></lb>ſitque rectangulum ſub duabus ductis <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end>æquale rectangu<lb></lb>lo ſub duabus aliis <emph type="italics"></emph>PSXPT,<emph.end type="italics"></emph.end>Sectio conica evadet Circulus. </s> <s>Idem <lb></lb>fiet ſi lineæ quatuor ducantur in angulis quibuſvis & rectangulum <lb></lb>ſub duabus ductis <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end>ſit ad rectangulum ſub aliis duabus <lb></lb><emph type="italics"></emph>PSXPT<emph.end type="italics"></emph.end>ut rectangulum ſub ſinubus angulorum <emph type="italics"></emph>S, T,<emph.end type="italics"></emph.end>in quibus <lb></lb>duæ ultimæ <emph type="italics"></emph>PS, PT<emph.end type="italics"></emph.end>ducuntur, ad rectangulum ſub ſinubus angu<lb></lb>lorum <emph type="italics"></emph>Q, R,<emph.end type="italics"></emph.end>in quibus duæ primæ <emph type="italics"></emph>PQ, PR<emph.end type="italics"></emph.end>ducuntur. </s> <s>Cæteris <lb></lb>in caſibus Locus puncti <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erit aliqua trium figurarum quæ vulgo <lb></lb>nominantur Sectiones Conicæ. </s> <s>Vice autem Trapezii <emph type="italics"></emph>ABCD<emph.end type="italics"></emph.end>ſub<lb></lb>ſtitui poteſt Quadrilaterum cujus latera duo oppoſita ſe mutuo in<lb></lb>ſtar diagonalium decuſſant. </s> <s>Sed & e punctis quatuor <emph type="italics"></emph>A, B, C, D<emph.end type="italics"></emph.end><lb></lb>poſſunt unum vel duo abire ad infinitum, eoque pacto latera fi<lb></lb>guræ quæ ad puncta illa convergunt, evadere parallela: quo in <lb></lb>caſu Sectio Conica tranſibit per cætera puncta, & in plagas paralle<lb></lb>larum abibit in infinitum. </s></p> <p type="margin"> <s><margin.target id="note45"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Invenire <expan abbr="punctũ">punctum</expan><emph.end type="italics"></emph.end>P, <emph type="italics"></emph>a quo ſi rectæ<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.097.1.jpg" xlink:href="039/01/097/1.jpg"></figure><lb></lb><emph type="italics"></emph>quatuor<emph.end type="italics"></emph.end>PQ, PR, PS, PT, <lb></lb><emph type="italics"></emph>ad alias totidem poſitione da<lb></lb>tas rectas<emph.end type="italics"></emph.end>AB, CD, AC, BD, <lb></lb><emph type="italics"></emph>ſingulæ ad ſingulas in datis <lb></lb>angulis ducantur, <expan abbr="rectangulũ">rectangulum</expan> <lb></lb>ſub duabus ductis,<emph.end type="italics"></emph.end>PQXPR, <lb></lb><emph type="italics"></emph>ſit ad rectangulum ſub aliis <lb></lb>duabus,<emph.end type="italics"></emph.end>PSXPT, <emph type="italics"></emph>in data ra<lb></lb>tione.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Lineæ <emph type="italics"></emph>AB, CD,<emph.end type="italics"></emph.end>ad quas rectæ duæ <emph type="italics"></emph>PQ, PR,<emph.end type="italics"></emph.end>unum rectan<lb></lb>gulorum continentes ducuntur, conveniant cum aliis duabus poſi<lb></lb>tione datis lineis in punctis <emph type="italics"></emph>A, B, C, D.<emph.end type="italics"></emph.end>Ab eorum aliquo <emph type="italics"></emph>A<emph.end type="italics"></emph.end>age <lb></lb>rectam quamlibet <emph type="italics"></emph>AH,<emph.end type="italics"></emph.end>in qua velis punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>reperiri. </s> <s>Secet ea <lb></lb>lineas oppoſitas <emph type="italics"></emph>BD, CD,<emph.end type="italics"></emph.end>nimirum <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>H<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>& ob <lb></lb>datos omnes angulos figuræ, dabuntur rationes <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PA<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PA<emph.end type="italics"></emph.end><pb xlink:href="039/01/098.jpg" pagenum="70"></pb><arrow.to.target n="note46"></arrow.to.target>ad <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end>adeoque ratio <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <lb></lb><figure id="id.039.01.098.1.jpg" xlink:href="039/01/098/1.jpg"></figure><lb></lb><emph type="italics"></emph>PS.<emph.end type="italics"></emph.end>Auferendo hanca data ra<lb></lb>tione <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PSXPT,<emph.end type="italics"></emph.end><lb></lb>dabitur ratio <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>& <lb></lb>addendo datas rationes <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>PR,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>dabitur <lb></lb>ratio <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>atque adeo <lb></lb>punctum <emph type="italics"></emph>P. Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note46"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc etiam ad Loci <lb></lb>punctorum infinitorum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>pun<lb></lb>ctum quodvis <emph type="italics"></emph>D<emph.end type="italics"></emph.end>tangens duci <lb></lb>poteſt. </s> <s>Nam chorda <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ubi <lb></lb>puncta <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>D<emph.end type="italics"></emph.end>conveniunt, hoc <lb></lb>eſt, ubi <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>ducitur per punctum <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>tangens evadit. </s> <s>Quo in caſu, <lb></lb>ultima ratio evaneſcentium <emph type="italics"></emph>IP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>invenietur ut ſupra. </s> <s>Ipſi <lb></lb>igitur <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>due parallelam <emph type="italics"></emph>CF,<emph.end type="italics"></emph.end>occurrentem <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>& in ea ul<lb></lb>tima ratione ſectam in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>tangens erit, propterea quod <emph type="italics"></emph>CF<emph.end type="italics"></emph.end><lb></lb>& evaneſcens <emph type="italics"></emph>IH<emph.end type="italics"></emph.end>parallelæ ſunt, & in <emph type="italics"></emph>E<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>fimiliter ſectæ. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Hinc etiam Locus punctorum omnium <emph type="italics"></emph>P<emph.end type="italics"></emph.end>definiri poteſt. </s> <s><lb></lb>Per quodvis punctorum <emph type="italics"></emph>A, B, C, D,<emph.end type="italics"></emph.end>puta <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>duc Loci tangentem <lb></lb><emph type="italics"></emph>AE<emph.end type="italics"></emph.end>& per aliud quodvis punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>duc tangenti parallelam <emph type="italics"></emph>BF<emph.end type="italics"></emph.end><lb></lb>occurrentem Loco in <emph type="italics"></emph>F.<emph.end type="italics"></emph.end>Invenie<lb></lb><figure id="id.039.01.098.2.jpg" xlink:href="039/01/098/2.jpg"></figure><lb></lb>tur autem punctum <emph type="italics"></emph>F<emph.end type="italics"></emph.end>per Lem. </s> <s>XIX. </s> <s><lb></lb>Biſeca <emph type="italics"></emph>BF<emph.end type="italics"></emph.end>in <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>& acta indefinita <lb></lb><emph type="italics"></emph>AG<emph.end type="italics"></emph.end>erit poſitio diametri ad quam <lb></lb><emph type="italics"></emph>BG<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>ordinatim applicantur. </s> <s><lb></lb>Hæc <emph type="italics"></emph>AG<emph.end type="italics"></emph.end>occurrat Loco in <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>& <lb></lb>erit <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>diameter ſive latus tranſ<lb></lb>verſum, ad quod latus rectum erit <lb></lb>ut <emph type="italics"></emph><expan abbr="BGq.">BGque</expan><emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AGH.<emph.end type="italics"></emph.end>Si <emph type="italics"></emph>AG<emph.end type="italics"></emph.end>nullibi <lb></lb>occurrit Loco, linea <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>exiſtente <lb></lb>infinita, Locus erit Parabola & la<lb></lb>rum rectum ejus ad diametrum <emph type="italics"></emph>AG<emph.end type="italics"></emph.end><lb></lb>pertinens erit (<emph type="italics"></emph>BGq./AG<emph.end type="italics"></emph.end>) Sin ea alicubi occurrit, Locus Hyperbola erit <lb></lb>ubi puncta <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ſita ſunt ad eaſdem partes ipſius <emph type="italics"></emph>G:<emph.end type="italics"></emph.end>& Ellipſis, <lb></lb>ubi <emph type="italics"></emph>G<emph.end type="italics"></emph.end>intermedium eſt, niſi forte angulus <emph type="italics"></emph>AGB<emph.end type="italics"></emph.end>rectus ſit & inſuper <lb></lb><emph type="italics"></emph>BG quad.<emph.end type="italics"></emph.end>æquale rectangulo <emph type="italics"></emph>AGH,<emph.end type="italics"></emph.end>quo in caſu Circulus habebitur. </s></p> <p type="main"> <s>AtQ.E.I.a Problematis Veterum de quatuor lineis ab <emph type="italics"></emph>Euclide<emph.end type="italics"></emph.end>incæp<lb></lb>ti & ab <emph type="italics"></emph>Apollonio<emph.end type="italics"></emph.end>continuati non calculus, ſed compoſitio Geometri<lb></lb>ca, qualem Veteres quærebant, in hoc Corollario exhibetur. <pb xlink:href="039/01/099.jpg" pagenum="71"></pb><arrow.to.target n="note47"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note47"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Parallelogrammum quodvis<emph.end type="italics"></emph.end>ASPQ <emph type="italics"></emph>angulis duobus oppoſitis<emph.end type="italics"></emph.end>A <emph type="italics"></emph>&<emph.end type="italics"></emph.end><lb></lb>P <emph type="italics"></emph>tangit ſectionem quamvis Conicam in punctis<emph.end type="italics"></emph.end>A <emph type="italics"></emph>&<emph.end type="italics"></emph.end>P; <emph type="italics"></emph>&, lateri<lb></lb>bus unius angulorum illorum infinite productis<emph.end type="italics"></emph.end>AQ, AS, <emph type="italics"></emph>occurrit <lb></lb>eidem ſectioni Conicæ in<emph.end type="italics"></emph.end>B <emph type="italics"></emph>&<emph.end type="italics"></emph.end>C; <emph type="italics"></emph>a punctis autem occurſuum<emph.end type="italics"></emph.end>B <emph type="italics"></emph>&<emph.end type="italics"></emph.end><lb></lb>C <emph type="italics"></emph>ad quintum quodvis ſectionis Conicæ punctum<emph.end type="italics"></emph.end>D <emph type="italics"></emph>agantur rec<lb></lb>tæ duæ<emph.end type="italics"></emph.end>BD, CD <emph type="italics"></emph>occurrentes alteris duobus infinite productis pa<lb></lb>rallelogrammi lateribus<emph.end type="italics"></emph.end>PS, PQ <emph type="italics"></emph>in<emph.end type="italics"></emph.end>T <emph type="italics"></emph>&<emph.end type="italics"></emph.end>R: <emph type="italics"></emph>erunt ſemper abſciſſæ <lb></lb>laterum partes<emph.end type="italics"></emph.end>PR <emph type="italics"></emph>&<emph.end type="italics"></emph.end>PT <emph type="italics"></emph>adinvicem in data ratione. </s> <s>Et contra, ſi <lb></lb>partes illæ abſciſſæ ſunt ad invicem in data ratione, punctum<emph.end type="italics"></emph.end>D <emph type="italics"></emph>tan<lb></lb>get Sectionem Conicam per puncta quatuor<emph.end type="italics"></emph.end>A, B, C, P <emph type="italics"></emph>tranſeuntem.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Jungantur <emph type="italics"></emph>BP, CP<emph.end type="italics"></emph.end>& a puncto <emph type="italics"></emph>D<emph.end type="italics"></emph.end>agantur rectæ duæ <lb></lb><emph type="italics"></emph>DG, DE,<emph.end type="italics"></emph.end>quarum prior <lb></lb><figure id="id.039.01.099.1.jpg" xlink:href="039/01/099/1.jpg"></figure><lb></lb><emph type="italics"></emph>DG<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>parallela ſit & <lb></lb>occurrat <emph type="italics"></emph>PB, PQ, CA<emph.end type="italics"></emph.end>in <lb></lb><emph type="italics"></emph>H, I, G<emph.end type="italics"></emph.end>; altera <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>paral<lb></lb>lela ſit ipfi <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>& occurrat <lb></lb><emph type="italics"></emph>PC, PS, AB<emph.end type="italics"></emph.end>in <emph type="italics"></emph>F, K, E:<emph.end type="italics"></emph.end><lb></lb>& erit (per Lemma XVII.) re<lb></lb>ctangulum <emph type="italics"></emph>DEXDF<emph.end type="italics"></emph.end>ad re<lb></lb>ctangulum <emph type="italics"></emph>DGXDH<emph.end type="italics"></emph.end>in ra<lb></lb>tione data. </s> <s>Sed eſt <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>DE<emph.end type="italics"></emph.end>(ſeu <emph type="italics"></emph>IQ<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>PB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HB,<emph.end type="italics"></emph.end><lb></lb>adeoque ut <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DH<emph.end type="italics"></emph.end>; & <lb></lb>viciſſim <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DH.<emph.end type="italics"></emph.end>Eſt & <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>RC<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>DC,<emph.end type="italics"></emph.end>adeoque ut (<emph type="italics"></emph>IG<emph.end type="italics"></emph.end>vel) <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DG,<emph.end type="italics"></emph.end>& viciſſim <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PS<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>; & conjunctis rationibus fit rectangulum <emph type="italics"></emph>PQXPR<emph.end type="italics"></emph.end><lb></lb>ad rectangulum <emph type="italics"></emph>PSXPT<emph.end type="italics"></emph.end>ut rectangulum <emph type="italics"></emph>DEXDF<emph.end type="italics"></emph.end>ad rectan<lb></lb>gulum <emph type="italics"></emph>DGXDH,<emph.end type="italics"></emph.end>atque adeo in data ratione. </s> <s>Sed dantur <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>& propterea ratio <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>datur. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Quod ſi <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ponantur in data ratione ad invi<lb></lb>cem, tum ſimili ratiocinio regrediendo, ſequetur eſſe rectangulum <lb></lb><emph type="italics"></emph>DEXDF<emph.end type="italics"></emph.end>ad rectangulum <emph type="italics"></emph>DGXDH<emph.end type="italics"></emph.end>in ratione data, adeoque <lb></lb>punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>(per Lemma XVIII.) contingere Conicam ſectionem <lb></lb>tranſeuntem per puncta <emph type="italics"></emph>A, B, C, P. Q.E.D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/100.jpg" pagenum="72"></pb><arrow.to.target n="note48"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note48"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi agatur <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>ſecans <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>r,<emph.end type="italics"></emph.end>& in <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>capiatur <lb></lb><emph type="italics"></emph>Pt<emph.end type="italics"></emph.end>in ratione ad <emph type="italics"></emph>Pr<emph.end type="italics"></emph.end>quam habet <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PR:<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>Bt<emph.end type="italics"></emph.end>tangens <lb></lb>Conicæ ſectionis ad punctum <emph type="italics"></emph>B.<emph.end type="italics"></emph.end>Nam concipe punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>coire <lb></lb>cum puncto <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ita ut, chorda <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>evaneſcente, <emph type="italics"></emph>BT<emph.end type="italics"></emph.end>tangens eva<lb></lb>dat; & <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>BT<emph.end type="italics"></emph.end>coincident cum <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Bt.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et vice verſa ſi <lb></lb><figure id="id.039.01.100.1.jpg" xlink:href="039/01/100/1.jpg"></figure><lb></lb><emph type="italics"></emph>Bt<emph.end type="italics"></emph.end>fit tangens, & ad quod<lb></lb>vis Conicæ ſectionis punc<lb></lb>tum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>conveniant <emph type="italics"></emph>BD, <lb></lb>CD<emph.end type="italics"></emph.end>; erit <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ut <lb></lb>ut <emph type="italics"></emph>Pr<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Pt.<emph.end type="italics"></emph.end>Et contra, <lb></lb>ſi ſit <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Pr<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>Pt:<emph.end type="italics"></emph.end>convenient <emph type="italics"></emph>BD, CD<emph.end type="italics"></emph.end><lb></lb>ad Conicæ Sectionis punc<lb></lb>um aliquod <emph type="italics"></emph>D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Conica ſectio <lb></lb>non ſecat Conicam ſectio<lb></lb>nem in punctis pluribus quam quatuor. </s> <s>Nam, ſi fieri poteſt, tranſ<lb></lb>eant duæ Conicæ ſectiones per quinque puncta <emph type="italics"></emph>A, B, C, P, O<emph.end type="italics"></emph.end>; eaſ<lb></lb>que ſecet recta <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>in punctis <emph type="italics"></emph>D, d,<emph.end type="italics"></emph.end>& ipſam <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ſecet recta <emph type="italics"></emph>Cd<emph.end type="italics"></emph.end><lb></lb>in r. </s> <s>Ergo <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>P<emph.end type="italics"></emph.end>r ad <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>; unde <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>r ſibi <lb></lb>invicem æquantur, contra Hypotheſin. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si rectæ duæ mobiles & infinitæ<emph.end type="italics"></emph.end>BM, CM <emph type="italics"></emph>per data puncta<emph.end type="italics"></emph.end>B, C, <emph type="italics"></emph>ceu <lb></lb>polos ductæ, concurſu ſuo<emph.end type="italics"></emph.end>M <emph type="italics"></emph>deſcribant tertiam poſitione da<lb></lb>tam rectam<emph.end type="italics"></emph.end>MN; <emph type="italics"></emph>& aliæ duæ infinitæ rectæ<emph.end type="italics"></emph.end>BD, CD <emph type="italics"></emph>cum <lb></lb>prioribus duabus ad puncta illa data<emph.end type="italics"></emph.end>B, C <emph type="italics"></emph>datos angulos<emph.end type="italics"></emph.end><lb></lb>MBD, MCD <emph type="italics"></emph>efficientes ducantur; dico quod hæ duæ<emph.end type="italics"></emph.end>BD, <lb></lb>CD <emph type="italics"></emph>concurſu ſuo<emph.end type="italics"></emph.end>D <emph type="italics"></emph>deſcribent ſectionem Conicam per puncta<emph.end type="italics"></emph.end><lb></lb>B, C <emph type="italics"></emph>tranſeuntem. </s> <s>Et vice verſa, ſi rectæ<emph.end type="italics"></emph.end>BD, CD <emph type="italics"></emph>concurſu <lb></lb>ſuo<emph.end type="italics"></emph.end>D <emph type="italics"></emph>deſcribant Sectionem Conicam per data puncta<emph.end type="italics"></emph.end>B, C, A <lb></lb><emph type="italics"></emph>tranſeuntem, & ſit angulus<emph.end type="italics"></emph.end>DBM <emph type="italics"></emph>ſemper æqualis angulo dato<emph.end type="italics"></emph.end><lb></lb>ABC, <emph type="italics"></emph>anguluſque<emph.end type="italics"></emph.end>DCM <emph type="italics"></emph>ſemper æqualis angulo dato<emph.end type="italics"></emph.end>ACB: <lb></lb><emph type="italics"></emph>punctum<emph.end type="italics"></emph.end>M <emph type="italics"></emph>continget rectam poſitione datam.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/101.jpg" pagenum="73"></pb> <p type="main"> <s><arrow.to.target n="note49"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note49"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Nam in recta <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>detur punctum <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>& ubi punctum mobile <lb></lb><emph type="italics"></emph>M<emph.end type="italics"></emph.end>incidit in immotum <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>incidat punctum mobile <emph type="italics"></emph>D<emph.end type="italics"></emph.end>in immo<lb></lb>tum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>Junge <emph type="italics"></emph>CN, BN,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.101.1.jpg" xlink:href="039/01/101/1.jpg"></figure><lb></lb><emph type="italics"></emph>CP, BP,<emph.end type="italics"></emph.end>& a puncto <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>age rectas <emph type="italics"></emph>PT, PR<emph.end type="italics"></emph.end><lb></lb>occurrentes ipſis <emph type="italics"></emph>BD, <lb></lb>CD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>& fa<lb></lb>cientes angulum <emph type="italics"></emph>BPT<emph.end type="italics"></emph.end><lb></lb>æqualem angulo dato <lb></lb><emph type="italics"></emph>BNM,<emph.end type="italics"></emph.end>& angulum <lb></lb><emph type="italics"></emph>CPR<emph.end type="italics"></emph.end>æqualem angu<lb></lb>gulo dato <emph type="italics"></emph>CNM.<emph.end type="italics"></emph.end>Cum <lb></lb>ergo (ex Hypotheſi) <lb></lb>æquales ſint anguli <lb></lb><emph type="italics"></emph>MBD, NBP,<emph.end type="italics"></emph.end>ut & <lb></lb>anguli <emph type="italics"></emph>MCD, NCP<emph.end type="italics"></emph.end>; <lb></lb>aufer communes <emph type="italics"></emph>NBD<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>NCD,<emph.end type="italics"></emph.end>& reſtabunt <lb></lb>æquales <emph type="italics"></emph>NBM<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PBT, <lb></lb>NCM<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PCR:<emph.end type="italics"></emph.end>adeoque triangula <emph type="italics"></emph>NBM, PBT<emph.end type="italics"></emph.end>ſimilia ſunt, ut <lb></lb>& triangula <emph type="italics"></emph>NCM, PCR.<emph.end type="italics"></emph.end>Quare <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>NM<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PB<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>NB,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>NM<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>NC.<emph.end type="italics"></emph.end>Sunt autem puncta <emph type="italics"></emph>B, C, N, P<emph.end type="italics"></emph.end><lb></lb>immobilia. </s> <s>Ergo <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>datam habent rationem ad <emph type="italics"></emph>NM,<emph.end type="italics"></emph.end>pro<lb></lb>indeQ.E.D.tam rationem inter ſe; atque adeo, per Lemma xx, <lb></lb>punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>(perpetuus rectarum mobilium <emph type="italics"></emph>BT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CR<emph.end type="italics"></emph.end>concurſus) <lb></lb>contingit ſectionem Conicam, per puncta <emph type="italics"></emph>B, C, P<emph.end type="italics"></emph.end>tranſeuntem. <lb></lb><emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Et contra, ſi punctum mobile <emph type="italics"></emph>D<emph.end type="italics"></emph.end>contingat ſectionem Conicam <lb></lb>tranſeuntem per data puncta <emph type="italics"></emph>B, C, A,<emph.end type="italics"></emph.end>& ſit angulus <emph type="italics"></emph>DBM<emph.end type="italics"></emph.end>ſemper <lb></lb>æqualis angulo dato <emph type="italics"></emph>ABC,<emph.end type="italics"></emph.end>& angulus <emph type="italics"></emph>DCM<emph.end type="italics"></emph.end>ſemper æqualis angu<lb></lb>lo dato <emph type="italics"></emph>ACB,<emph.end type="italics"></emph.end>& ubi punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>incidit ſucceſſive in duo quævis ſe<lb></lb>ctionis puncta immobilia <emph type="italics"></emph>p, P,<emph.end type="italics"></emph.end>punctum mobile <emph type="italics"></emph>M<emph.end type="italics"></emph.end>incidat ſucceſſive <lb></lb>in puncta duo immobilia <emph type="italics"></emph>n, N:<emph.end type="italics"></emph.end>per eadem <emph type="italics"></emph>n, N<emph.end type="italics"></emph.end>agatur Recta <emph type="italics"></emph>n N,<emph.end type="italics"></emph.end><lb></lb>& hæc erit Locus perpetuus puncti illius mobilis <emph type="italics"></emph>M.<emph.end type="italics"></emph.end>Nam, ſi fieri <lb></lb>poteſt, verſetur punctum <emph type="italics"></emph>M<emph.end type="italics"></emph.end>in linea aliqua Curva. </s> <s>Tanget ergo <lb></lb>punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ſectionem Conicam per puncta quinque <emph type="italics"></emph>B, CA, p, P,<emph.end type="italics"></emph.end><lb></lb>tranſeuntem, ubi punctum <emph type="italics"></emph>M<emph.end type="italics"></emph.end>perpetuo tangit lineam Curvam. </s> <s>Sed <lb></lb>& ex jam demonſtratis tanget etiam punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ſectionem CoNI<lb></lb>cam per eadem quinque puncta <emph type="italics"></emph>B, C, A, p, P<emph.end type="italics"></emph.end>tranſeuntem, ubi pun-</s></p><pb xlink:href="039/01/102.jpg" pagenum="74"></pb> <p type="main"> <s><arrow.to.target n="note50"></arrow.to.target>ctum <emph type="italics"></emph>M<emph.end type="italics"></emph.end>perpetuo tangit lineam Rectam. </s> <s>Ergo duæ ſectiones Co<lb></lb>nicæ tranſibunt per eadem quinque puncta, contra Corol. </s> <s>3. Lem. </s> <s><lb></lb>xx. </s> <s>Igitur punctum <emph type="italics"></emph>M<emph.end type="italics"></emph.end>verſari in linea Curva abſurdum eſt. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note50"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXII. PROBLEMA. XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam per data quinque puncta deſcribere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Dentur puncta quinque <emph type="italics"></emph>A, B, C, P, D.<emph.end type="italics"></emph.end>Ab eorum aliquo <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <lb></lb>alia duo quævis <emph type="italics"></emph>B, C,<emph.end type="italics"></emph.end>quæ poli nominentur, age rectas <emph type="italics"></emph>AB, AC,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.102.1.jpg" xlink:href="039/01/102/1.jpg"></figure><lb></lb>hiſque parallelas <emph type="italics"></emph>TPS, PRQ<emph.end type="italics"></emph.end>per punctum quartum <emph type="italics"></emph>P.<emph.end type="italics"></emph.end>De<lb></lb>inde a polis duobus <emph type="italics"></emph>B, C<emph.end type="italics"></emph.end>age per punctum quintum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>infiNI<lb></lb>tas duas <emph type="italics"></emph>BDT, CRD,<emph.end type="italics"></emph.end>noviſſime ductis <emph type="italics"></emph>TPS, PRQ<emph.end type="italics"></emph.end>(prio<lb></lb>rem priori & poſteriorem poſteriori) occurrentes in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>R.<emph.end type="italics"></emph.end>De<lb></lb>niQ.E.D. rectis <emph type="italics"></emph>PT, PR,<emph.end type="italics"></emph.end>acta recta <emph type="italics"></emph>tr<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>parallela, ab<lb></lb>ſcinde quaſvis <emph type="italics"></emph>Pt, Pr<emph.end type="italics"></emph.end>ipſis <emph type="italics"></emph>PT, PR<emph.end type="italics"></emph.end>proportionales; & ſi per <lb></lb>earum terminos <emph type="italics"></emph>t, r<emph.end type="italics"></emph.end>& polos <emph type="italics"></emph>B, C<emph.end type="italics"></emph.end>actæ <emph type="italics"></emph>Bt, Cr<emph.end type="italics"></emph.end>concurrant in <lb></lb><emph type="italics"></emph>d,<emph.end type="italics"></emph.end>locabitur punctum illud <emph type="italics"></emph>d<emph.end type="italics"></emph.end>in Trajectoria quæſita. </s> <s>Nam punc<lb></lb>tum illud <emph type="italics"></emph>d<emph.end type="italics"></emph.end>(per Lemma xx) verſatur in Conica Sectione per <lb></lb>puncta quatuor <emph type="italics"></emph>A, B, C, P<emph.end type="italics"></emph.end>tranſeunte; &, lineis <emph type="italics"></emph>Rr, Tt<emph.end type="italics"></emph.end>evane<lb></lb>ſcentibus, coit punctum <emph type="italics"></emph>d<emph.end type="italics"></emph.end>cum puncto <emph type="italics"></emph>D.<emph.end type="italics"></emph.end>Tranſit ergo ſectio Co<lb></lb>nica per puncta quinque <emph type="italics"></emph>A, B, C, P, D. Q.E.D.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/103.jpg" pagenum="75"></pb> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Idem aliter.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note51"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note51"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>E punctis datis junge tria quævis <emph type="italics"></emph>A, B, C<emph.end type="italics"></emph.end>; &, circum duo eorum <lb></lb><emph type="italics"></emph>B, C<emph.end type="italics"></emph.end>ceu polos, rotando angulos magnitudine datos <emph type="italics"></emph>ABC, <lb></lb>ACB,<emph.end type="italics"></emph.end>applicentur cru<lb></lb><figure id="id.039.01.103.1.jpg" xlink:href="039/01/103/1.jpg"></figure><lb></lb>ra <emph type="italics"></emph>BA, CA<emph.end type="italics"></emph.end>primo ad <lb></lb>punctum <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>deinde <lb></lb>ad punctum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& no<lb></lb>tentur puncta <emph type="italics"></emph>M, N<emph.end type="italics"></emph.end>in <lb></lb>quibus altera crura <lb></lb><emph type="italics"></emph>BL, CL<emph.end type="italics"></emph.end>caſu utroque <lb></lb>ſe decuſſant. </s> <s>Agatur <lb></lb>recta infinita <emph type="italics"></emph>MN,<emph.end type="italics"></emph.end>& <lb></lb>rotentur anguli illi mo<lb></lb>biles circum polos ſuos <lb></lb><emph type="italics"></emph>B, C,<emph.end type="italics"></emph.end>ea lege ut cru<lb></lb>rum <emph type="italics"></emph>BL, CL<emph.end type="italics"></emph.end>vel <lb></lb><emph type="italics"></emph>BM, CM<emph.end type="italics"></emph.end>interſectio <lb></lb>quæ jam ſit <emph type="italics"></emph>m<emph.end type="italics"></emph.end>incidat <lb></lb>ſemper in rectam illam <lb></lb>infinitam <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>& cru<lb></lb>rum <emph type="italics"></emph>BA, CA,<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>BD, CD<emph.end type="italics"></emph.end>interſectio, quæ jam ſit <emph type="italics"></emph>d,<emph.end type="italics"></emph.end>Trajecto<lb></lb>riam quæſitam <emph type="italics"></emph>PAD dB<emph.end type="italics"></emph.end>delineabit. </s> <s>Nam punctum <emph type="italics"></emph>d,<emph.end type="italics"></emph.end>per Lem. </s> <s><lb></lb>XXI, continget ſectionem Conicam per puncta <emph type="italics"></emph>B, C<emph.end type="italics"></emph.end>tranſeuntem; & <lb></lb>ubi punctum <emph type="italics"></emph>m<emph.end type="italics"></emph.end>accedit ad puncta <emph type="italics"></emph>L, M, N,<emph.end type="italics"></emph.end>punctum <emph type="italics"></emph>d<emph.end type="italics"></emph.end>(per con<lb></lb>ſtructionem) accedet ad puncta <emph type="italics"></emph>A, D, P.<emph.end type="italics"></emph.end>Deſcribetur itaque ſec<lb></lb>tio Conica tranſiens per puncta quinque <emph type="italics"></emph>A, B, C, P, D. q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc recta expedite duci poteſt quæ Trajectoriam quæ<lb></lb>ſitam, in puncto quovis dato <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>continget. </s> <s>Accedat punctum <emph type="italics"></emph>d<emph.end type="italics"></emph.end>ad <lb></lb>punctum <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& recta <emph type="italics"></emph>Bd<emph.end type="italics"></emph.end>evadet tangens quæſita. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Unde etiam Trajectoriarum Centra, Diametri & Latera <lb></lb>recta inveniri poſſunt, ut in Corollario ſecundo Lemmatis XIX. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Conſtructio prior evadet paulo ſimplicior jungendo <emph type="italics"></emph>BP,<emph.end type="italics"></emph.end>& in ea, <lb></lb>ſi opus eſt, producta capiendo <emph type="italics"></emph>Bp<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BP<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>; & <lb></lb>per <emph type="italics"></emph>p<emph.end type="italics"></emph.end>agendo rectam infinitam <emph type="italics"></emph>p<emph.end type="italics"></emph.end>d ipſi <emph type="italics"></emph>SPT<emph.end type="italics"></emph.end>parallelam, inque ea <lb></lb>capiendo ſemper <emph type="italics"></emph>p<emph.end type="italics"></emph.end>d æqualem <emph type="italics"></emph>Pr<emph.end type="italics"></emph.end>; & agendo rectas <emph type="italics"></emph>Bd, Cr<emph.end type="italics"></emph.end>con<lb></lb>currentes in <emph type="italics"></emph>d.<emph.end type="italics"></emph.end>Nam cum ſint <emph type="italics"></emph>Pr<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Pt, PR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PT, pB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PB, <lb></lb>p<emph.end type="italics"></emph.end>d ad <emph type="italics"></emph>Pt<emph.end type="italics"></emph.end>in eadem ratione; erunt <emph type="italics"></emph>p<emph.end type="italics"></emph.end>d & <emph type="italics"></emph>Pr<emph.end type="italics"></emph.end>ſemper æqua-<pb xlink:href="039/01/104.jpg" pagenum="76"></pb>les. </s> <s>Hac methodo puncta Trajectoriæ inveniuntur expeditiſſime, </s></p> <p type="main"> <s><arrow.to.target n="note52"></arrow.to.target>niſi mavis Curvam, ut in conſtructione ſecunda, deſeribere Me<lb></lb>chanice. </s></p> <p type="margin"> <s><margin.target id="note52"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIII. PROBLEMA XV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam deſcribere quæ per data quatuor puncta tranſibit, & rec<lb></lb>tam continget poſitione datam.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Dentur tangens <emph type="italics"></emph>HB,<emph.end type="italics"></emph.end>punctum contactus <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& alia tria <lb></lb>puncta <emph type="italics"></emph>C, D, P.<emph.end type="italics"></emph.end>Junge <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>& agendo <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>parallelam <emph type="italics"></emph>BH,<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>parallelam <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>comple parallelogrammum <emph type="italics"></emph><expan abbr="BSPq.">BSPque</expan><emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.104.1.jpg" xlink:href="039/01/104/1.jpg"></figure><lb></lb>Age <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ſecantem <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ſecantem <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>R.<emph.end type="italics"></emph.end>De<lb></lb>nique, agendo quamvis <emph type="italics"></emph>tr<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>parallelam, de <emph type="italics"></emph>PQ, PS<emph.end type="italics"></emph.end><lb></lb>abſcinde <emph type="italics"></emph>Pr, Pt<emph.end type="italics"></emph.end>ipſis <emph type="italics"></emph>PR, PT<emph.end type="italics"></emph.end>proportionales reſpective; & <lb></lb>actarum <emph type="italics"></emph>Cr, Bt<emph.end type="italics"></emph.end>concurſus <emph type="italics"></emph>d<emph.end type="italics"></emph.end>(per Lem. </s> <s>xx) incidet ſemper in <lb></lb>Trajectoriam deſcribendam. <pb xlink:href="039/01/105.jpg" pagenum="77"></pb><arrow.to.target n="note53"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note53"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Idem aliter.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Revolvatur tum angulus magnitudine datus <emph type="italics"></emph>CBH<emph.end type="italics"></emph.end>circa polum <lb></lb><emph type="italics"></emph>B,<emph.end type="italics"></emph.end>tum radius quilibet rectilineus & utrinque productus <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>cir<lb></lb>ca polum <emph type="italics"></emph>C.<emph.end type="italics"></emph.end>Notentur puncta <emph type="italics"></emph>M, N<emph.end type="italics"></emph.end>in quibus anguli crus <emph type="italics"></emph>BC<emph.end type="italics"></emph.end><lb></lb>ſecat radium illum ubi crus alterum <emph type="italics"></emph>BH<emph.end type="italics"></emph.end>concurrit cum eodem ra<lb></lb>dio in punctis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>D.<emph.end type="italics"></emph.end>Deinde ad actam infinitam <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>con<lb></lb><figure id="id.039.01.105.1.jpg" xlink:href="039/01/105/1.jpg"></figure><lb></lb>currant perpetuo radius ille <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>& anguli crus <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>& <lb></lb>cruris alterius <emph type="italics"></emph>BH<emph.end type="italics"></emph.end>concurſus cum radio delineabit Trajectoriam <lb></lb>quæſitam. </s></p> <p type="main"> <s>Nam ſi in conſtructionibus Problematis ſuperioris accedat punc<lb></lb>tum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad punctum <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>lineæ <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>coincident, & linea <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in <lb></lb>ultimo ſuo ſitu fiet tangens <emph type="italics"></emph>BH,<emph.end type="italics"></emph.end>atque adeo conſtructiones ibi po<lb></lb>ſitæ evadent eædem cum conſtructionibus hic deſcriptis. </s> <s>Delinea<lb></lb>bit igitur cruris <emph type="italics"></emph>BH<emph.end type="italics"></emph.end>concurſus cum radio ſectionem Conicam per <lb></lb>puncta <emph type="italics"></emph>C, D, P<emph.end type="italics"></emph.end>tranſeuntem, & rectam <emph type="italics"></emph>BH<emph.end type="italics"></emph.end>tangentem in puncto <lb></lb><emph type="italics"></emph>B. q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Dentur puncta quatuor <emph type="italics"></emph>B, C, D, P<emph.end type="italics"></emph.end>extra tangentem <lb></lb><emph type="italics"></emph>HI<emph.end type="italics"></emph.end>ſita. </s> <s>Junge bina lineis <emph type="italics"></emph>BD, CP<emph.end type="italics"></emph.end>concurrentibus in <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>tangen-<pb xlink:href="039/01/106.jpg" pagenum="78"></pb><arrow.to.target n="note54"></arrow.to.target>tique occurrentibus in <emph type="italics"></emph>H<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I.<emph.end type="italics"></emph.end>Secetur tangens in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ita ut ſit <lb></lb><emph type="italics"></emph>HA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AI,<emph.end type="italics"></emph.end>ut eſt rectan<lb></lb><figure id="id.039.01.106.1.jpg" xlink:href="039/01/106/1.jpg"></figure><lb></lb>gulum ſub media proportio<lb></lb>nali inter <emph type="italics"></emph>CG<emph.end type="italics"></emph.end>& <emph type="italics"></emph>GP<emph.end type="italics"></emph.end>& me<lb></lb>dia proportionali inter <emph type="italics"></emph>BH<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>HD,<emph.end type="italics"></emph.end>ad rectangulum ſub me<lb></lb>dia proportionali inter <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>GB<emph.end type="italics"></emph.end>& media proportionali in<lb></lb>ter <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IC<emph.end type="italics"></emph.end>; & erit <emph type="italics"></emph>A<emph.end type="italics"></emph.end>punc<lb></lb>tum contactus. </s> <s>Nam ſi rectæ <lb></lb><emph type="italics"></emph>PI<emph.end type="italics"></emph.end>parallela <emph type="italics"></emph>HX<emph.end type="italics"></emph.end>Trajecto<lb></lb>riam ſecet in punctis quibuſ<lb></lb>vis <emph type="italics"></emph>X<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Y:<emph.end type="italics"></emph.end>erit (ex Conicis) <lb></lb>punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ita locandum, ut fuerit <emph type="italics"></emph>HA quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AI quad.<emph.end type="italics"></emph.end>in ra<lb></lb>tione compoſita ex ratione rectanguli <emph type="italics"></emph>XHY<emph.end type="italics"></emph.end>ad rectangulum <emph type="italics"></emph>BHD<emph.end type="italics"></emph.end><lb></lb>ſeu rectanguli <emph type="italics"></emph>CGP<emph.end type="italics"></emph.end>ad rectangulum <emph type="italics"></emph>DGB<emph.end type="italics"></emph.end>& ex ratione rectan<lb></lb>guli <emph type="italics"></emph>BHD<emph.end type="italics"></emph.end>ad rectangulum <emph type="italics"></emph>PIC.<emph.end type="italics"></emph.end>Invento autem contactus <lb></lb>puncto <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>deſcribetur Trajectoria ut in caſu primo. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end><lb></lb>Capi autem poteſt punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>vel inter puncta <emph type="italics"></emph>H<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>vel extra; <lb></lb>& perinde Trajectoria dupliciter deſcribi. </s></p> <p type="margin"> <s><margin.target id="note54"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIV. PROBLEMA XVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam deſcribere quæ tranſibit per data tria puncta & rectas <lb></lb>duas poſitione datas continget.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Dentur tangentes <emph type="italics"></emph>HI, KL<emph.end type="italics"></emph.end>& <lb></lb><figure id="id.039.01.106.2.jpg" xlink:href="039/01/106/2.jpg"></figure><lb></lb>puncta <emph type="italics"></emph>B, C, D.<emph.end type="italics"></emph.end>Per punctorum <lb></lb>duo quævis <emph type="italics"></emph>B, D<emph.end type="italics"></emph.end>age rectam in<lb></lb>finitam <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>tangentibus occur<lb></lb>rentem in punctis <emph type="italics"></emph>H, K.<emph.end type="italics"></emph.end>Deinde <lb></lb>etiam per alia duo quævis <emph type="italics"></emph>C, D<emph.end type="italics"></emph.end><lb></lb>age infinitam <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>tangentibus oc<lb></lb>currentem in punctis <emph type="italics"></emph>I, L.<emph.end type="italics"></emph.end>Actas <lb></lb>ita ſeca in <emph type="italics"></emph>R<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>HR<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>KR<emph.end type="italics"></emph.end>ut eſt media proportionalis <lb></lb>inter <emph type="italics"></emph>BH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>HD<emph.end type="italics"></emph.end>ad mediam <lb></lb>proportionalem inter <emph type="italics"></emph>BK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>KD<emph.end type="italics"></emph.end>; <lb></lb>& <emph type="italics"></emph>IS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>LS<emph.end type="italics"></emph.end>ut eſt media pro<lb></lb>portionalis inter <emph type="italics"></emph>CI<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ID<emph.end type="italics"></emph.end>ad me<lb></lb>diam proportionalem inter <emph type="italics"></emph>CL<emph.end type="italics"></emph.end><pb xlink:href="039/01/107.jpg" pagenum="79"></pb>& <emph type="italics"></emph>LD.<emph.end type="italics"></emph.end>Seca autem pro lubitu vel inter puncta <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>H,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note55"></arrow.to.target><emph type="italics"></emph>I<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>vel extra eadem: dein age <emph type="italics"></emph>RS<emph.end type="italics"></emph.end>ſecantem tangentes in <emph type="italics"></emph>A<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& erunt <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>puncta contactuum. </s> <s>Nam ſi <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>ſupponantur eſſe puncta contactuum alicubi in tangentibus ſi<lb></lb>ta; & per punctorum <emph type="italics"></emph>H, I, K, L<emph.end type="italics"></emph.end>quodvis <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>in tangente al<lb></lb>terutra <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>ſitum, agatur recta <emph type="italics"></emph>IY<emph.end type="italics"></emph.end>tangenti alteri <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>paral<lb></lb>lela, quæ occurrat curvæ in <emph type="italics"></emph>X<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Y,<emph.end type="italics"></emph.end>& in ea ſumatur <emph type="italics"></emph>IZ<emph.end type="italics"></emph.end>me<lb></lb>dia proportionalis inter <emph type="italics"></emph>IX<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IY:<emph.end type="italics"></emph.end>erit, ex Conicis, rectangulum <lb></lb><emph type="italics"></emph>XIY<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>IZ quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>LP quad.<emph.end type="italics"></emph.end>ut rectangulum <emph type="italics"></emph>CID<emph.end type="italics"></emph.end>ad rectan<lb></lb>gulum <emph type="italics"></emph>CLD,<emph.end type="italics"></emph.end>id eſt (per conſtructionem) ut <emph type="italics"></emph>SI quad.<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>SL quad:<emph.end type="italics"></emph.end>atque adeo <emph type="italics"></emph>IZ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>LP<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SL.<emph.end type="italics"></emph.end>Jacent ergo punc<lb></lb>ta <emph type="italics"></emph>S, P, Z<emph.end type="italics"></emph.end>in una recta. </s> <s>Porro tangentibus concurrentibus in <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>e<lb></lb>rit (ex Conicis) rectangulum <emph type="italics"></emph>XIY<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>IZ quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>IA quad.<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>GP quad<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GA quad:<emph.end type="italics"></emph.end>adeoque <emph type="italics"></emph>IZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IA<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>GP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GA.<emph.end type="italics"></emph.end>Jacent <lb></lb>ergo puncta <emph type="italics"></emph>P, Z<emph.end type="italics"></emph.end>& <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in una recta, adeoque puncta <emph type="italics"></emph>S, P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>A<emph.end type="italics"></emph.end><lb></lb>ſunt in una recta. </s> <s>Et eodem argumento probabitur quod puncta <lb></lb><emph type="italics"></emph>R, P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſunt in una recta. </s> <s>Jacent igitur puncta contactuum <emph type="italics"></emph>A<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in recta <emph type="italics"></emph>RS.<emph.end type="italics"></emph.end>Hiſce autem inventis, Trajectoria deſeribetur <lb></lb>ut in caſu primo Problematis ſuperioris. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note55"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Figuras in alias ejuſdem generis figur as mutare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Tranſmutanda ſit figura quævis <emph type="italics"></emph>HGI.<emph.end type="italics"></emph.end>Ducantur pro lubitu <lb></lb>rectæ duæ parallelæ <emph type="italics"></emph>AO, BL<emph.end type="italics"></emph.end>tertiam quamvis poſitione datam <lb></lb><emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ſecantes in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.107.1.jpg" xlink:href="039/01/107/1.jpg"></figure><lb></lb>& a figuræ puncto quo<lb></lb>vis <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>ad rectam <emph type="italics"></emph>AB<emph.end type="italics"></emph.end><lb></lb>ducatur quævis <emph type="italics"></emph>GD,<emph.end type="italics"></emph.end><lb></lb>ipſi <emph type="italics"></emph>OA<emph.end type="italics"></emph.end>parallela. </s> <s>De<lb></lb>inde a puncto aliquo <emph type="italics"></emph>O,<emph.end type="italics"></emph.end><lb></lb>in linea <emph type="italics"></emph>OA<emph.end type="italics"></emph.end>dato, ad <lb></lb>punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ducatur <lb></lb>recta <emph type="italics"></emph>OD,<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>BL<emph.end type="italics"></emph.end>oc<lb></lb>currens in <emph type="italics"></emph>d,<emph.end type="italics"></emph.end>& a puncto <lb></lb>occurſus erigatur recta <lb></lb><emph type="italics"></emph>dg<emph.end type="italics"></emph.end>datum quemvis angulum cum recta <emph type="italics"></emph>BL<emph.end type="italics"></emph.end>continens, atque eam <lb></lb>habens rationem ad <emph type="italics"></emph>Od<emph.end type="italics"></emph.end>quam habet <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OD<emph.end type="italics"></emph.end>; & erit <emph type="italics"></emph>g<emph.end type="italics"></emph.end>punc<lb></lb>tum in figura nova <emph type="italics"></emph>hgi<emph.end type="italics"></emph.end>puncto <emph type="italics"></emph>G<emph.end type="italics"></emph.end>reſpondens. </s> <s>Eadem ratione <lb></lb>puncta ſingula figuræ primæ dabunt puncta totidem figura novæ. <pb xlink:href="039/01/108.jpg" pagenum="80"></pb><arrow.to.target n="note56"></arrow.to.target>Concipe igitur punctum <emph type="italics"></emph>G<emph.end type="italics"></emph.end>motu continuo percurrere puncta om<lb></lb>nia figuræ primæ, & punctum <emph type="italics"></emph>g<emph.end type="italics"></emph.end>motu itidem continuo percurret <lb></lb>puncta omnia figuræ novæ & eandem deſcribet. </s> <s>Diſtinctionis gra<lb></lb>tia nominemus <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>ordinatam primam, <emph type="italics"></emph>dg<emph.end type="italics"></emph.end>ordinatam novam; <lb></lb><emph type="italics"></emph>AD<emph.end type="italics"></emph.end>abſciſſam primam, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end>abſciſſam novam; <emph type="italics"></emph>O<emph.end type="italics"></emph.end>polum, <emph type="italics"></emph>OD<emph.end type="italics"></emph.end>ra<lb></lb>dium abſcidentem, <emph type="italics"></emph>OA<emph.end type="italics"></emph.end>radium ordinatum primum, & <emph type="italics"></emph>Oa<emph.end type="italics"></emph.end>(qno <lb></lb>parallelogrammum <emph type="italics"></emph>OABa<emph.end type="italics"></emph.end>completur) radium ordinatum novum. </s></p> <p type="margin"> <s><margin.target id="note56"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Dico jam quod, ſi punctum <emph type="italics"></emph>G<emph.end type="italics"></emph.end>tangit rectam Lineam poſitione da<lb></lb>tam, punctum <emph type="italics"></emph>g<emph.end type="italics"></emph.end>tanget etiam Lineam rectam poſitione datam. </s> <s>Si <lb></lb>punctum <emph type="italics"></emph>G<emph.end type="italics"></emph.end>tangit Conicam ſectionem, punctum <emph type="italics"></emph>g<emph.end type="italics"></emph.end>tanget etiam <lb></lb>Conicam ſectionem. </s> <s>Conicis ſectionibus hic Circulum annumero. </s> <s><lb></lb>Porro ſi punctum <emph type="italics"></emph>G<emph.end type="italics"></emph.end>tan<lb></lb><figure id="id.039.01.108.1.jpg" xlink:href="039/01/108/1.jpg"></figure><lb></lb>git Lineam tertii ordinis <lb></lb>Analytici, punctum <emph type="italics"></emph>g<emph.end type="italics"></emph.end><lb></lb>tanget Lineam tertii iti<lb></lb>dem ordinis; & ſic de <lb></lb>curvis lineis ſuperiorum <lb></lb>ordinum. </s> <s>Lineæ duæ e<lb></lb>runt ejuſdem ſemper or<lb></lb>dinis Analytici quas pun<lb></lb>cta <emph type="italics"></emph>G, g<emph.end type="italics"></emph.end>tangunt. </s> <s>Et<lb></lb>enim ut eſt <emph type="italics"></emph>ad<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OA<emph.end type="italics"></emph.end><lb></lb>ita ſunt <emph type="italics"></emph>Od<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OD, dg<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DG,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>; adeoque <emph type="italics"></emph>AD<emph.end type="italics"></emph.end><lb></lb>æqualis eſt (<emph type="italics"></emph>OAXAB/ad<emph.end type="italics"></emph.end>), & <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>æqualis eſt (<emph type="italics"></emph>OAXdg/ad<emph.end type="italics"></emph.end>). Jam ſi punc<lb></lb>tum <emph type="italics"></emph>G<emph.end type="italics"></emph.end>tangit rectam Lineam, atque adeo in æquatione quavis, <lb></lb>qua relatio inter abſciſſam <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>& ordinatam <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>habetur, in<lb></lb>determinatæ illæ <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>ad unicam tantum dimenſionem <lb></lb>aſcendunt, ſcribendo in hac æquatione (<emph type="italics"></emph>OAXAB/ad<emph.end type="italics"></emph.end>) pro <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>& <lb></lb>(<emph type="italics"></emph>OAXdg/ad<emph.end type="italics"></emph.end>) pro <emph type="italics"></emph>DG,<emph.end type="italics"></emph.end>producetur æquatio nova, in qua abſciſſa no<lb></lb>va <emph type="italics"></emph>ad<emph.end type="italics"></emph.end>& ordinata nova <emph type="italics"></emph>dg<emph.end type="italics"></emph.end>ad unicam tantum dimenſionem aſcen<lb></lb>dent, atque adeo quæ deſignat Lineam rectam. </s> <s>Sin <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DG<emph.end type="italics"></emph.end><lb></lb>(vel earum alterutra) aſcendebant ad duas dimenſiones in æquati<lb></lb>one prima, aſcendent itidem <emph type="italics"></emph>ad<emph.end type="italics"></emph.end>& <emph type="italics"></emph>dg<emph.end type="italics"></emph.end>ad duas in æquatione ſecun<lb></lb>da. </s> <s>Et ſic de tribus vel pluribus dimenſionibus. </s> <s>Indeterminatæ <lb></lb><emph type="italics"></emph>ad, dg<emph.end type="italics"></emph.end>in æquatione ſecunda & <emph type="italics"></emph>AD, DG<emph.end type="italics"></emph.end>in prima aſcendent ſem<lb></lb>per ad eundem dimenſionum numerum, & propterea Lineæ, quas <lb></lb>puncta <emph type="italics"></emph>G, g<emph.end type="italics"></emph.end>tangunt, ſunt ejuſdem ordinis Analytici. </s></p><pb xlink:href="039/01/109.jpg" pagenum="81"></pb> <p type="main"> <s>Dico præterea quod ſi recta aliqua tangat lineam curvam in fi<lb></lb><arrow.to.target n="note57"></arrow.to.target>gura prima; hæc recta eodem modo cum curva in figuram novam <lb></lb>tranſlata tanget lineam illam curvam in figura nova: & contra. </s> <s>Nam <lb></lb>ſi Curvæ puncta quævis duo accedunt ad invicem & coeunt in fi<lb></lb>gura prima, puncta eadem tranſlata accedent ad invicem & coibunt <lb></lb>in figura nova, atque adeo rectæ, quibus hæc puncta junguntur, ſi<lb></lb>mul evadent curvarum tangentes in figura utraque. </s> <s>Componi poſ<lb></lb>ſent harum aſſertionum Demonſtrationes more magis Geometrico. </s> <s><lb></lb>Sed brevitati conſulo. </s></p> <p type="margin"> <s><margin.target id="note57"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Igitur ſi figura rectilinea in aliam tranſmutanda eſt, ſufficit rec<lb></lb>tarum a quibus conflatur interſectiones transferre, & per eaſdem <lb></lb>in figura nova lineas rectas ducere. </s> <s>Sin curvilineam tranſmutare <lb></lb>oportet, transferenda ſunt puncta, tangentes & aliæ rectæ quarum <lb></lb>ope curva linea definitur. </s> <s>Inſervit autem hoc Lemma ſolutioni <lb></lb>difficiliorum Problematum, tranſmutando figuras propoſitas in ſim<lb></lb>pliciores. </s> <s>Nam rectæ quævis convergentes tranſmutantur in pa<lb></lb>rallelas, adhibendo pro radio ordinato primo, lineam quam<lb></lb>vis rectam quæ per concurſum convergentium tranſit: id adeo quia <lb></lb>concurſus ille hoc pacto abit in infinitum, lineæ autem parallelæ <lb></lb>ſunt quæ ad punctum infinite diſtans tendunt. </s> <s>Poſtquam autem <lb></lb>Problema ſolvitur in figura nova, ſi per inverſas operationes tranſ<lb></lb>mutetur hæc figura in figuram primam, habebitur ſolutio quæſita. </s></p> <p type="main"> <s>Utile eſt etiam hoc Lemma in ſolutione Solidorum Problema<lb></lb>tum. </s> <s>Nam quoties duæ ſectiones Conicæ obvenerint, quarum in<lb></lb>terſectione Problema ſolvi poteſt, tranſmutare licet earum alter<lb></lb>utram, ſi Hyperbola ſit vel Parabola, in Ellipſin: deinde Ellipſis <lb></lb>facile mutatur in Circulum. </s> <s>Recta item & ſectio Conica, in con<lb></lb>ſtructione Planorum Problematum, vertuntur in Rectam & Cir<lb></lb>culum. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXV. PROBLEMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam deſcribere qua per data duo puncta tranſibit & rectas <lb></lb>tres continget poſitione datas.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Per concurſum tangentium quarumvis duarum cum ſe invicem, & <lb></lb>concurſum tangentis tertiæ cum recta illa, quæ per puncta duo data <lb></lb>tranſit, age rectam infinitam; eaque adhibita pro radio ordinato pri<lb></lb>mo, tranſmutetur figura, per Lemma ſuperius, in figuram novam. </s> <s>In <pb xlink:href="039/01/110.jpg" pagenum="82"></pb><arrow.to.target n="note58"></arrow.to.target>hac figura tangentes illæ duæ evadent ſibi invicem parallelæ, & tan<lb></lb>gens tertia fiet parallela rectæ per <lb></lb><figure id="id.039.01.110.1.jpg" xlink:href="039/01/110/1.jpg"></figure><lb></lb>puncta duo data tranſeunti. </s> <s>Sunto <lb></lb><emph type="italics"></emph>hi, kl<emph.end type="italics"></emph.end>tangentes illæ duæ parallelæ, <lb></lb><emph type="italics"></emph>ik<emph.end type="italics"></emph.end>tangens tertia, & <emph type="italics"></emph>hl<emph.end type="italics"></emph.end>recta huic <lb></lb>parallela tranſiens per puncta illa <lb></lb><emph type="italics"></emph>a, b,<emph.end type="italics"></emph.end>per quæ Conica ſectio in hac <lb></lb>figura nova tranſire debet, & pa<lb></lb>rallelogrammum <emph type="italics"></emph>hikl<emph.end type="italics"></emph.end>complens. </s> <s><lb></lb>Secentur rectæ <emph type="italics"></emph>hi, ik, kl<emph.end type="italics"></emph.end>in <emph type="italics"></emph>c, d, e,<emph.end type="italics"></emph.end><lb></lb>ita ut ſit <emph type="italics"></emph>hc<emph.end type="italics"></emph.end>ad latus quadratum <lb></lb>rectanguli <emph type="italics"></emph>ahb, ic<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>id,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ke<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>kd<emph.end type="italics"></emph.end>ut eſt ſumma rectarum <emph type="italics"></emph>hi<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>kl<emph.end type="italics"></emph.end>ad ſummam trium linea<lb></lb>rum quarum prima eſt recta <emph type="italics"></emph>ik,<emph.end type="italics"></emph.end>& alteræ duæ ſunt latera quadrata <lb></lb>rectangulorum <emph type="italics"></emph>ahb<emph.end type="italics"></emph.end>& <emph type="italics"></emph>alb<emph.end type="italics"></emph.end>& erunt <emph type="italics"></emph>c, d, e<emph.end type="italics"></emph.end>puncta contactuum. </s> <s>Et<lb></lb>enim, ex Conicis, ſunt <emph type="italics"></emph>hc<emph.end type="italics"></emph.end>quadratum ad rectangulum <emph type="italics"></emph>ahb,<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>ic<emph.end type="italics"></emph.end>quadratum ad <emph type="italics"></emph>id<emph.end type="italics"></emph.end>quadratum, & <emph type="italics"></emph>ke<emph.end type="italics"></emph.end>quadratum ad <emph type="italics"></emph>kd<emph.end type="italics"></emph.end>quadratum, <lb></lb>& <emph type="italics"></emph>el<emph.end type="italics"></emph.end>quadratum ad rectangulum <emph type="italics"></emph>alb<emph.end type="italics"></emph.end>in eadem ratione; & propter<lb></lb>ea <emph type="italics"></emph>hc<emph.end type="italics"></emph.end>ad latus quadratum ipſius <emph type="italics"></emph>ahb, ic<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>id, ke<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>kd,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>el<emph.end type="italics"></emph.end>ad <lb></lb>latus quadratum ipſius <emph type="italics"></emph>alb<emph.end type="italics"></emph.end>ſunt in ſubduplicata illa ratione, & <lb></lb>compoſite, in data ratione omnium antecedentium <emph type="italics"></emph>hi<emph.end type="italics"></emph.end>& <emph type="italics"></emph>kl<emph.end type="italics"></emph.end>ad <lb></lb>omnes conſequentes, quæ ſunt latus quadratum rectanguli <emph type="italics"></emph>ahb<emph.end type="italics"></emph.end>& <lb></lb>recta <emph type="italics"></emph>ik<emph.end type="italics"></emph.end>& latus quadratum rectanguli <emph type="italics"></emph>alb.<emph.end type="italics"></emph.end>Habentur igitur ex <lb></lb>data illa ratione puncta contactuum <emph type="italics"></emph>c, d, e,<emph.end type="italics"></emph.end>in figura nova. </s> <s>Per <lb></lb>inverſas operationes Lemmatis noviſſimi transferantur hæc pun<lb></lb>cta in figuram primam & ibi, per Probl. </s> <s>XIV, deſcribetur <lb></lb>Trajectoria. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end>Ceterum perinde ut puncta <emph type="italics"></emph>a, b<emph.end type="italics"></emph.end>ja<lb></lb>cent vel inter puncta <emph type="italics"></emph>h, l,<emph.end type="italics"></emph.end>vel extra, debent puncta <emph type="italics"></emph>c, d, e<emph.end type="italics"></emph.end>vel <lb></lb>inter puncta <emph type="italics"></emph>h, i, k, l<emph.end type="italics"></emph.end>capi, vel extra. </s> <s>Si punctorum <emph type="italics"></emph>a, b<emph.end type="italics"></emph.end>al<lb></lb>terutrum cadit inter puncta <emph type="italics"></emph>h, l,<emph.end type="italics"></emph.end>& alterum extra, Problema im<lb></lb>poſſibile eſt. </s></p> <p type="margin"> <s><margin.target id="note58"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVI. PROBLEMA XVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam deſcribere quæ tranſibit per punctum datum & rectas <lb></lb>quatuor poſitione datas continget.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ab interſectione communi duarum quarumlibet tangentium ad <lb></lb>interſectionem communem reliquarum duarum agatur recta infini-<pb xlink:href="039/01/111.jpg" pagenum="83"></pb>ta, & eadem pro radio ordinato primo adhibita, tranſmutetur fi<lb></lb><arrow.to.target n="note59"></arrow.to.target>gura (per Lem. </s> <s>XXII) in figuram novam, & tangentes binæ, quæ ad <lb></lb>radium ordinatum primum concurrebant, jam evadent parallelæ. </s> <s>Sun<lb></lb>to illæ <emph type="italics"></emph>hi<emph.end type="italics"></emph.end>& <emph type="italics"></emph>kl, ik<emph.end type="italics"></emph.end>& <emph type="italics"></emph>hl<emph.end type="italics"></emph.end>continentes parallelogrammum <emph type="italics"></emph>hikl.<emph.end type="italics"></emph.end>Sit<lb></lb>que <emph type="italics"></emph>p<emph.end type="italics"></emph.end>punctum in hac nova figura, puncto in figura prima dato <lb></lb>reſpondens. </s> <s>Per figuræ centrum <emph type="italics"></emph>O<emph.end type="italics"></emph.end>agatur <emph type="italics"></emph>pq,<emph.end type="italics"></emph.end>& exiſtente <emph type="italics"></emph>Oq<emph.end type="italics"></emph.end>æ<lb></lb>quali <emph type="italics"></emph>Op,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>q<emph.end type="italics"></emph.end>punctum alterum per quod ſectio Conica in hac <lb></lb>figura nova tranſire debet. </s> <s>Per Lemmatis XXII operationem in<lb></lb>verſam transferatur hoc punctum in figuram primam, & ibi habe<lb></lb>buntur puncta duo per quæ Trajectoria deſcribenda eſt. </s> <s>Per ea<lb></lb>dem vero deſcribi poteſt Trajectoria illa per Prob. </s> <s>XVII. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note59"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si rectæ duæ poſitione datæ<emph.end type="italics"></emph.end>AC, BD <emph type="italics"></emph>ad data puncta<emph.end type="italics"></emph.end>A, B, <emph type="italics"></emph>ter<lb></lb>minentur, datamque habeant rationem ad invicem, & recta<emph.end type="italics"></emph.end><lb></lb>CD, <emph type="italics"></emph>qua puncta indeterminata<emph.end type="italics"></emph.end>C, D <emph type="italics"></emph>junguntur, ſecetur in ra<lb></lb>tione data in<emph.end type="italics"></emph.end>K: <emph type="italics"></emph>dico quod punctum<emph.end type="italics"></emph.end>K <emph type="italics"></emph>locabitur in recta poſi<lb></lb>tione data.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Concurrant enim rectæ <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.111.1.jpg" xlink:href="039/01/111/1.jpg"></figure><lb></lb><emph type="italics"></emph>BD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>& in <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>BG<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>ſit<lb></lb>que <emph type="italics"></emph>FD<emph.end type="italics"></emph.end>ſemper æqualis datæ <lb></lb><emph type="italics"></emph>EG<emph.end type="italics"></emph.end>; & erit ex conſtructione <lb></lb><emph type="italics"></emph>EC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GD,<emph.end type="italics"></emph.end>hoc eſt, ad <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BD,<emph.end type="italics"></emph.end>adeoQ.E.I. ratione <lb></lb>data, & propterea dabitur ſpecie <lb></lb>triangulum <emph type="italics"></emph>EFC.<emph.end type="italics"></emph.end>Secetur <emph type="italics"></emph>CF<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>CL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CF<emph.end type="italics"></emph.end>in ratio<lb></lb>ne <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>; &, ob datam il<lb></lb>lam rationem, dabitur etiam ſpecie triangulum <emph type="italics"></emph>EFL<emph.end type="italics"></emph.end>; proindeque <lb></lb>punctum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>locabitur in recta <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>poſitione data. </s> <s>Junge <emph type="italics"></emph>LK,<emph.end type="italics"></emph.end>& <lb></lb>ſimilia erunt triangula <emph type="italics"></emph>CLK, CFD<emph.end type="italics"></emph.end>; &, ob datam <emph type="italics"></emph>FD<emph.end type="italics"></emph.end>& datam <lb></lb>rationem <emph type="italics"></emph>LK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FD,<emph.end type="italics"></emph.end>dabitur <emph type="italics"></emph>LK.<emph.end type="italics"></emph.end>Huic æqualis capiatur <emph type="italics"></emph>EH,<emph.end type="italics"></emph.end><lb></lb>& erit ſemper <emph type="italics"></emph>ELKH<emph.end type="italics"></emph.end>parallelogrammum. </s> <s>Locatur igitur punc<lb></lb>tum <emph type="italics"></emph>K<emph.end type="italics"></emph.end>in parallelogrammi illius latere poſitione dato <emph type="italics"></emph>HK. Q.E.D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/112.jpg" pagenum="84"></pb><arrow.to.target n="note60"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note60"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si rectæ tres tangant quamcunque Coniſectionem, quarum duæ pa<lb></lb>rallelæ ſint ac dentur poſitione; dico quod Sectionis ſemidia<lb></lb>meter hiſce duabus parallela, ſit media proportionalis inter ha<lb></lb>rum ſegmenta, punctis contactuum & tangenti tertiæ inter<lb></lb>jecta.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sunto <emph type="italics"></emph>AF, GB<emph.end type="italics"></emph.end>pa<lb></lb><figure id="id.039.01.112.1.jpg" xlink:href="039/01/112/1.jpg"></figure><lb></lb>rallelæ duæ Coniſec<lb></lb>tionem <emph type="italics"></emph>ADB<emph.end type="italics"></emph.end>tan<lb></lb>gentes in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B; EF<emph.end type="italics"></emph.end><lb></lb>recta tertia Coniſec<lb></lb>tionem tangens in <emph type="italics"></emph>I,<emph.end type="italics"></emph.end><lb></lb>& occurrens prioribus <lb></lb>tangentibus in <emph type="italics"></emph>F<emph.end type="italics"></emph.end>& <emph type="italics"></emph>G<emph.end type="italics"></emph.end>; <lb></lb>ſitque <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ſemidiame<lb></lb>ter Figuræ tangenti<lb></lb>bus parallela: Dico <lb></lb>quod <emph type="italics"></emph>AF, CD, BG<emph.end type="italics"></emph.end><lb></lb>ſunt continue proportionales. </s></p> <p type="main"> <s>Nam ſi diametri conjugatæ <emph type="italics"></emph>AB, DM<emph.end type="italics"></emph.end>tangenti <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>occurrant <lb></lb>in <emph type="italics"></emph>E<emph.end type="italics"></emph.end>& <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>ſeque mutuo ſecent in <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>& compleatur parallelogram<lb></lb>mum <emph type="italics"></emph>IKCL<emph.end type="italics"></emph.end>; erit, ex natura Sectionum Conicarum, ut <emph type="italics"></emph>EC<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>CA<emph.end type="italics"></emph.end>ita <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CL,<emph.end type="italics"></emph.end>& ita diviſim <emph type="italics"></emph>EC-CA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CA-CL,<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>EA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AL,<emph.end type="italics"></emph.end>& compoſite <emph type="italics"></emph>EA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EA+AL<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>EC<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>EC+CA<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>EB<emph.end type="italics"></emph.end>; adeoque (ob ſimilitudinem triangulorum <emph type="italics"></emph>EAF, <lb></lb>ELI, ECH, EBG) AF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>LI<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BG.<emph.end type="italics"></emph.end>Eſt itidem, <lb></lb>ex natura Sectionum Conicarum, <emph type="italics"></emph>LI<emph.end type="italics"></emph.end>(ſeu <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>) ad <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>CH<emph.end type="italics"></emph.end>; atque, adeo ex æquo perturbate, <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BG. <lb></lb>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi tangentes duæ <emph type="italics"></emph>FG, PQ<emph.end type="italics"></emph.end>tangentibus parallelis <lb></lb><emph type="italics"></emph>AF, BG<emph.end type="italics"></emph.end>occurrant in <emph type="italics"></emph>F<emph.end type="italics"></emph.end>& <emph type="italics"></emph>G, P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end>ſeque mutuo ſecent in <emph type="italics"></emph>O<emph.end type="italics"></emph.end>; <lb></lb>erit (ex æquo perturbate) <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BG,<emph.end type="italics"></emph.end>& diviſim <lb></lb>ut <emph type="italics"></emph>FP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GQ,<emph.end type="italics"></emph.end>atque adeo ut <emph type="italics"></emph>FO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OG.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Unde etiam rectæ duæ <emph type="italics"></emph>PG, FQ<emph.end type="italics"></emph.end>per puncta <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>G, <lb></lb>F<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>ductæ, concurrent ad rectam <emph type="italics"></emph>ACB<emph.end type="italics"></emph.end>per centrum Figuræ & <lb></lb>puncta contactuum <emph type="italics"></emph>A, B<emph.end type="italics"></emph.end>tranſeuntem. <pb xlink:href="039/01/113.jpg" pagenum="85"></pb><arrow.to.target n="note61"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note61"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si parallelogrammi latera quatuor infinite producta tangant Sectio<lb></lb>nem quamcunque Conicam, & abſcindantur ad tangentem quamvis <lb></lb>quintam; ſumantur autem laterum quorumvis duorum contermi<lb></lb>norum abſciſſæ terminatæ ad angulos oppoſitos parallelogrammi: <lb></lb>dico quod abſciſſa alterutra ſit ad latus illud a quo est abſciſſa, ut <lb></lb>pars lateris alterius contermini inter punctum contactus & latus <lb></lb>tertium, est ad abſciſſarum alteram.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Tangant parallelogrammi <emph type="italics"></emph>MLIK<emph.end type="italics"></emph.end>latera quatuor <emph type="italics"></emph>ML, IK, KL, <lb></lb>MI<emph.end type="italics"></emph.end>ſectionem Conicam in <emph type="italics"></emph>A, B, C, D,<emph.end type="italics"></emph.end>& ſecet tangens quinta <emph type="italics"></emph>FQ<emph.end type="italics"></emph.end><lb></lb>hæc latera in <emph type="italics"></emph>F, Q, H<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.113.1.jpg" xlink:href="039/01/113/1.jpg"></figure><lb></lb>& <emph type="italics"></emph>E<emph.end type="italics"></emph.end>; ſumantur autem <lb></lb>laterum <emph type="italics"></emph>MI, KI<emph.end type="italics"></emph.end>ab<lb></lb>ſciſſæ <emph type="italics"></emph>ME, KQ,<emph.end type="italics"></emph.end>vel <lb></lb>laterum <emph type="italics"></emph>KL, ML<emph.end type="italics"></emph.end>ab<lb></lb>ſciſſæ <emph type="italics"></emph>KH, MF:<emph.end type="italics"></emph.end>di<lb></lb>co quod ſit <emph type="italics"></emph>ME<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>MI<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>BK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KQ<emph.end type="italics"></emph.end>; <lb></lb>& <emph type="italics"></emph>KH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>AM<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MF.<emph.end type="italics"></emph.end>Nam <lb></lb>per Corollarium ſe<lb></lb>cundum Lemmatis ſuperioris, eſt <emph type="italics"></emph>ME<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EI<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>AM<emph.end type="italics"></emph.end>ſeu) <emph type="italics"></emph>BK<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>BQ,<emph.end type="italics"></emph.end>& componendo <emph type="italics"></emph>ME<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MI<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>BK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="Kq.">Kque</expan> Q.E.D.<emph.end type="italics"></emph.end><lb></lb>Item <emph type="italics"></emph>KH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HL<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>BK<emph.end type="italics"></emph.end>ſeu) <emph type="italics"></emph>AM<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AF,<emph.end type="italics"></emph.end>& dividendo <emph type="italics"></emph>KH<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>KL<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AM<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MF. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi datur parallelogramum <emph type="italics"></emph>IKLM,<emph.end type="italics"></emph.end>circa datam Sec<lb></lb>tionem Conicam deſeriptum, dabitur rectangulum <emph type="italics"></emph>KQXME,<emph.end type="italics"></emph.end>ut <lb></lb>& huic æquale rectangulum <emph type="italics"></emph>KHXMF.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et ſi ſexta ducatur tangens <emph type="italics"></emph>eq<emph.end type="italics"></emph.end>tangentibus <emph type="italics"></emph>KI, MI<emph.end type="italics"></emph.end><lb></lb>occurrens in <emph type="italics"></emph>q<emph.end type="italics"></emph.end>& <emph type="italics"></emph>e<emph.end type="italics"></emph.end>; rectangulum <emph type="italics"></emph>KQXME<emph.end type="italics"></emph.end>æquabitur rectan<lb></lb>gulo <emph type="italics"></emph>KqXMe<emph.end type="italics"></emph.end>; eritque <emph type="italics"></emph>KQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Me<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Kq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ME,<emph.end type="italics"></emph.end>& diviſim ut <lb></lb><emph type="italics"></emph>Qq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ee.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Unde etiam ſi <emph type="italics"></emph>Eq, eQ<emph.end type="italics"></emph.end>jungantur & biſecentur, & recta <lb></lb>per puncta biſectionum agatur, tranſibit hæc per centrum Sectio<lb></lb>nis Conicæ. </s> <s>Nam cum ſit <emph type="italics"></emph>Qq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ee<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>KQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Me,<emph.end type="italics"></emph.end>tranſibit ea-<pb xlink:href="039/01/114.jpg" pagenum="86"></pb><arrow.to.target n="note62"></arrow.to.target>dem recta per medium omnium <emph type="italics"></emph>Eq, eQ, MK<emph.end type="italics"></emph.end>; (per Lem. </s> <s>XXIII) <lb></lb>& medium rectæ <emph type="italics"></emph>MK<emph.end type="italics"></emph.end>eſt centrum Sectionis. </s></p> <p type="margin"> <s><margin.target id="note62"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVII. PROBLEMA XIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam deſcribere quæ rectas quinque poſitione datas continget.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Dentur pofitione tangentes <emph type="italics"></emph>ABG, BCF, GCD, FDE, EA.<emph.end type="italics"></emph.end><lb></lb>Figuræ quadrilateræ ſub quatuor quibuſvis contentæ <emph type="italics"></emph>ABFE<emph.end type="italics"></emph.end>dia<lb></lb>gonales <emph type="italics"></emph>AF, BE<emph.end type="italics"></emph.end>biſeca, & (per Corol. </s> <s>3. Lem. </s> <s>XXV) recta <emph type="italics"></emph>MN<emph.end type="italics"></emph.end><lb></lb>per puncta biſectionum acta tranſibit per centrum Trajectoriæ. </s> <s><lb></lb>Rurſus Figuræ quadrilateræ <emph type="italics"></emph>BGDF,<emph.end type="italics"></emph.end>ſub aliis quibuſvis quatuor <lb></lb><figure id="id.039.01.114.1.jpg" xlink:href="039/01/114/1.jpg"></figure><lb></lb>tangentibus contentæ, diagonales (ut ita dicam) <emph type="italics"></emph>BD, GF<emph.end type="italics"></emph.end>bi<lb></lb>ſeca in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q:<emph.end type="italics"></emph.end>& recta <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>per puncta biſectionum acta tranſ<lb></lb>ibit per centrum Trajectoriæ. </s> <s>Dabitur ergo centrum in concurſu bi<lb></lb>ſecantium. </s> <s>Sit illud <emph type="italics"></emph>O.<emph.end type="italics"></emph.end>Tangenti cuivis <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>parallelam age <emph type="italics"></emph>KL,<emph.end type="italics"></emph.end><lb></lb>ad eam diſtantiam ut centrum <emph type="italics"></emph>O<emph.end type="italics"></emph.end>in medio inter parallelas locetur, <lb></lb>& acta <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>tanget Trajectoriam deſcribendam. </s> <s>Secet hæc tan-<pb xlink:href="039/01/115.jpg" pagenum="87"></pb>gentes alias quaſvis duas <emph type="italics"></emph>GCD, FDE<emph.end type="italics"></emph.end>in <emph type="italics"></emph>L<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K.<emph.end type="italics"></emph.end>Per harum <lb></lb><arrow.to.target n="note63"></arrow.to.target>tangentium non parallelarum <emph type="italics"></emph>CL, FK<emph.end type="italics"></emph.end>cum parallelis <emph type="italics"></emph>CF, KL<emph.end type="italics"></emph.end><lb></lb>concurſus <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K, F<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end>age <emph type="italics"></emph>CK, FL<emph.end type="italics"></emph.end>concurrentes in <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>& rec<lb></lb>ta <emph type="italics"></emph>OR<emph.end type="italics"></emph.end>ducta & producta ſecabit tangentes parallelas <emph type="italics"></emph>CF, KL<emph.end type="italics"></emph.end>in <lb></lb>punctis contactuum. </s> <s>Patet hoc per Corol. </s> <s>2. Lem. </s> <s>XXIV. </s> <s>Ea<lb></lb>dem methodo invenire licet alia contactuum puncta, & tum de<lb></lb>mum per Probl. </s> <s>XIV. &c. </s> <s>Trajectoriam deſcribere. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note63"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Problemata, ubi dantur Trajectoriarum vel centra vel Aſymp<lb></lb>toti, includuntnr in præcedentibus. </s> <s>Nam datis punctis & tangen<lb></lb>tibus una cum centro, dantur alia totidem puncta aliæque tangen<lb></lb>tes a centro ex altera ejus parte æqualiter diſtantes. </s> <s>Aſymptotos <lb></lb>autem pro tangente habenda eſt, & ejus terminus infinite diſtans <lb></lb>(ſi ita loqui fas ſit) pro puncto contactus. </s> <s>Concipe tangentis cu<lb></lb>juſvis punctum contactus abire in infinitum, & tangens vertetur in <lb></lb>Aſymptoton, atque conſtructiones Problematis XIV & Caſus pri<lb></lb>mi Problematis XV vertentur in conſtructiones Problematum ubi <lb></lb>Aſymptoti dantur. </s></p> <p type="main"> <s>Poſtquam Trajectoria deſcripta eſt, invenire licet axes & umbi<lb></lb>licos ejus hac methodo. </s> <s>In conſtructione & figura Lemmatis XXI, <lb></lb>fac ut angulorum mobi<lb></lb><figure id="id.039.01.115.1.jpg" xlink:href="039/01/115/1.jpg"></figure><lb></lb>lium <emph type="italics"></emph>PBN, PCN<emph.end type="italics"></emph.end>cru<lb></lb>ra <emph type="italics"></emph>BP, CP,<emph.end type="italics"></emph.end>quorum <lb></lb>concurſu Trajectoria de<lb></lb>ſcribebatur, ſint ſibi invi<lb></lb>cem parallela, eumque <lb></lb>ſervantia ſitum revolvan<lb></lb>tur circa polos ſuos <emph type="italics"></emph>B, C<emph.end type="italics"></emph.end><lb></lb>in figura illa. </s> <s>Interea ve<lb></lb>ro deſcribant altera an<lb></lb>gulorum illorum crura <lb></lb><emph type="italics"></emph>CN, BN,<emph.end type="italics"></emph.end>concurſu <lb></lb>ſuo <emph type="italics"></emph>K<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>k,<emph.end type="italics"></emph.end>Circulum <lb></lb><emph type="italics"></emph>IBKGC.<emph.end type="italics"></emph.end>Sit Circuli <lb></lb>hujus centrum <emph type="italics"></emph>O.<emph.end type="italics"></emph.end>Ab <lb></lb>hoc centro ad Regulam <lb></lb><emph type="italics"></emph>MN,<emph.end type="italics"></emph.end>ad quam altera illa crura <emph type="italics"></emph>CN, BN<emph.end type="italics"></emph.end>interea concurrebant <pb xlink:href="039/01/116.jpg" pagenum="88"></pb><arrow.to.target n="note64"></arrow.to.target>dum Trajectoria deſcribebatur, demitte normalem <emph type="italics"></emph>OH<emph.end type="italics"></emph.end>Circulo oc<lb></lb>currentem in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L.<emph.end type="italics"></emph.end>Et ubi crura illa altera <emph type="italics"></emph>CK, BK<emph.end type="italics"></emph.end>concur<lb></lb>runt ad punctum illud <emph type="italics"></emph>K<emph.end type="italics"></emph.end>quod Regulæ propius eſt, crura prima <lb></lb><emph type="italics"></emph>CP, BP<emph.end type="italics"></emph.end>parallela erunt axi majori, & perpendicularia minori; <lb></lb>& contrarium eveniet ſi crura eadem concurrunt ad punctum remo<lb></lb>tius <emph type="italics"></emph>L.<emph.end type="italics"></emph.end>Unde ſi detur Trajectoriæ centrum, dabuntur axes. </s> <s>Hiſce <lb></lb>autem datis, umbilici ſunt in promptu. </s></p> <p type="margin"> <s><margin.target id="note64"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Axium vero quadrata ſunt ad invicem ut <emph type="italics"></emph>KH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>LH,<emph.end type="italics"></emph.end>& inde <lb></lb>facile eſt Trajectoriam <lb></lb><figure id="id.039.01.116.1.jpg" xlink:href="039/01/116/1.jpg"></figure><lb></lb>ſpecie datam per data <lb></lb>quatuor puncta deſcri<lb></lb>bere. </s> <s>Nam ſi duo ex <lb></lb>punctis datis conſtitu<lb></lb>antur poli <emph type="italics"></emph>C, B,<emph.end type="italics"></emph.end>tertium <lb></lb>dabit angulos mobiles <lb></lb><emph type="italics"></emph>PCK, PBK<emph.end type="italics"></emph.end>; his au<lb></lb>tem datis deſcribi poteſt <lb></lb>Circulus <emph type="italics"></emph>IBKGC.<emph.end type="italics"></emph.end><lb></lb>Tum ob datam ſpecie <lb></lb>Trajectoriam, dabitur <lb></lb>ratio <emph type="italics"></emph>OH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OK,<emph.end type="italics"></emph.end>ad<lb></lb>eoQ.E.I.ſa <emph type="italics"></emph>OH.<emph.end type="italics"></emph.end>Cen<lb></lb>tro <emph type="italics"></emph>O<emph.end type="italics"></emph.end>& intervallo <emph type="italics"></emph>OH<emph.end type="italics"></emph.end><lb></lb>deſcribe alium circulum, <lb></lb>& recta quæ tangit hunc circulum, & tranſit per concurſum crurum <lb></lb><emph type="italics"></emph>CK, BK,<emph.end type="italics"></emph.end>ubi crura prima <emph type="italics"></emph>CP, BP<emph.end type="italics"></emph.end>concurrunt ad quartum da<lb></lb>tum punctum erit Regula illa <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>cujus ope Trajectoria deſcri<lb></lb>betur. </s> <s>Unde etiam viciſſim Trapezium ſpecie datum (ſi caſus qui<lb></lb>dam impoſſibiles excipiantur) in data quavis Sectione Conica in<lb></lb>ſcribi poteſt. </s></p> <p type="main"> <s>Sunt & alia Lemmata quorum ope Trajectoriæ ſpecie datæ, <lb></lb>datis punctis & tangentibus, deſcribi poſſunt. </s> <s>Ejus generis <lb></lb>eſt quod, ſi recta linea per punctum quodvis poſitione datum <lb></lb>ducatur, quæ datam Coniſectionem in punctis duobus interſe<lb></lb>cet, & interſectionum intervallum biſecetur, punctum biſectionis <lb></lb>tanget aliam Coniſectionem ejuſdem ſpeciei cum priore, atque <lb></lb>axes habentem prioris axibus parallelos. </s> <s>Sed propero ad magis <lb></lb>utilia. <pb xlink:href="039/01/117.jpg" pagenum="89"></pb><arrow.to.target n="note65"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note65"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Trianguli ſpecie & magnitudine dati tres angulos ad rectas tot<lb></lb>idem poſitione datas, quæ non ſunt omnes parallelæ, ſingulos ad <lb></lb>ſingulas ponere.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Dantur poſitione tres rectæ infinitæ <emph type="italics"></emph>AB, AC, BC,<emph.end type="italics"></emph.end>& opor<lb></lb>tet triangulum <emph type="italics"></emph>DEF<emph.end type="italics"></emph.end>ita locare, ut angulus ejus <emph type="italics"></emph>D<emph.end type="italics"></emph.end>lineam <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end><lb></lb>angulus <emph type="italics"></emph>E<emph.end type="italics"></emph.end>lineam <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.117.1.jpg" xlink:href="039/01/117/1.jpg"></figure><figure id="id.039.01.117.2.jpg" xlink:href="039/01/117/2.jpg"></figure><lb></lb>& angulus <emph type="italics"></emph>F<emph.end type="italics"></emph.end>lineam <lb></lb><emph type="italics"></emph>BC<emph.end type="italics"></emph.end>tangat. </s> <s>Super <emph type="italics"></emph>DE, <lb></lb>DF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>deſcribe <lb></lb>tria circulorum ſeg<lb></lb>menta <emph type="italics"></emph>DRE, DGF, <lb></lb>EMF,<emph.end type="italics"></emph.end>quæ capiant <lb></lb>angulos angulis <emph type="italics"></emph>BAC, <lb></lb>ABC, ACB<emph.end type="italics"></emph.end>æquales <lb></lb>reſpective. </s> <s>Deſcriban<lb></lb>tur autem hæc ſegmen<lb></lb>ta ad eas partes linea<lb></lb>rum <emph type="italics"></emph>DE, DF, EF<emph.end type="italics"></emph.end>ut <lb></lb>literæ <emph type="italics"></emph>DRED<emph.end type="italics"></emph.end>eodem <lb></lb>ordine cum literis <lb></lb><emph type="italics"></emph>BACB,<emph.end type="italics"></emph.end>literæ <emph type="italics"></emph>DGFD<emph.end type="italics"></emph.end><lb></lb>eodem cum literis <lb></lb><emph type="italics"></emph>ABCA,<emph.end type="italics"></emph.end>& literæ <lb></lb><emph type="italics"></emph>EMFE<emph.end type="italics"></emph.end>eodem cum <lb></lb>literis <emph type="italics"></emph>ACBA<emph.end type="italics"></emph.end>in orbem <lb></lb>redeant; deinde com<lb></lb>pleantur hæc ſegmenta <lb></lb>in circulos integros. </s> <s>Se<lb></lb>cent circuli duo prio<lb></lb>res ſe mutuo in <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>ſint<lb></lb>que centra eorum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end>Junctis <emph type="italics"></emph>GP, PQ,<emph.end type="italics"></emph.end><lb></lb>cape <emph type="italics"></emph>Ga<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ut eſt <lb></lb><emph type="italics"></emph>GP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>& cen<lb></lb>tro <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>Ga<emph.end type="italics"></emph.end><lb></lb>deſcribe circulum, qui ſecet circulum primum <emph type="italics"></emph>DGE<emph.end type="italics"></emph.end>in <emph type="italics"></emph>a.<emph.end type="italics"></emph.end>Jungatur <lb></lb>tum <emph type="italics"></emph>aD<emph.end type="italics"></emph.end>ſecans circulum ſecundum <emph type="italics"></emph>DFG<emph.end type="italics"></emph.end>in <emph type="italics"></emph>b,<emph.end type="italics"></emph.end>tum <emph type="italics"></emph>aE<emph.end type="italics"></emph.end>ſecans cir-<pb xlink:href="039/01/118.jpg" pagenum="90"></pb><arrow.to.target n="note66"></arrow.to.target>culum tertium <emph type="italics"></emph>EMF<emph.end type="italics"></emph.end>in <emph type="italics"></emph>c.<emph.end type="italics"></emph.end>Et compleatur Figura <emph type="italics"></emph>ABC def<emph.end type="italics"></emph.end>ſimi<lb></lb>lis & æqualis Figuræ <emph type="italics"></emph>abcDEF.<emph.end type="italics"></emph.end>Dico factum. </s></p> <p type="margin"> <s><margin.target id="note66"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Agatur enim <emph type="italics"></emph>Fc<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>aD<emph.end type="italics"></emph.end>occurrens in <emph type="italics"></emph>n,<emph.end type="italics"></emph.end>& jungantur <emph type="italics"></emph>aG, bG, <lb></lb>QG, QD, PD.<emph.end type="italics"></emph.end>Ex conſtructione eſt angulus <emph type="italics"></emph>EaD<emph.end type="italics"></emph.end>æqualis an<lb></lb>gulo <emph type="italics"></emph>CAB,<emph.end type="italics"></emph.end>& angulus <lb></lb><figure id="id.039.01.118.1.jpg" xlink:href="039/01/118/1.jpg"></figure><figure id="id.039.01.118.2.jpg" xlink:href="039/01/118/2.jpg"></figure><lb></lb><emph type="italics"></emph>acF<emph.end type="italics"></emph.end>æqualis angulo <lb></lb><emph type="italics"></emph>ACB,<emph.end type="italics"></emph.end>adeoque trian<lb></lb>gulum <emph type="italics"></emph>anc<emph.end type="italics"></emph.end>triangulo <lb></lb><emph type="italics"></emph>ABC<emph.end type="italics"></emph.end>æquiangulum. </s> <s><lb></lb>Ergo angulus <emph type="italics"></emph>anc<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>FnD<emph.end type="italics"></emph.end>angulo <emph type="italics"></emph>ABC,<emph.end type="italics"></emph.end><lb></lb>adeoque angulo <emph type="italics"></emph>FbD<emph.end type="italics"></emph.end><lb></lb>æqualis eſt; & propter<lb></lb>ea punctum <emph type="italics"></emph>n<emph.end type="italics"></emph.end>incidit in <lb></lb>punctum <emph type="italics"></emph>b.<emph.end type="italics"></emph.end>Porro an<lb></lb>gulus <emph type="italics"></emph>GPQ,<emph.end type="italics"></emph.end>qui di<lb></lb>midius eſt anguli ad <lb></lb>centrum <emph type="italics"></emph>GPD,<emph.end type="italics"></emph.end>æqua<lb></lb>lis eſt angulo ad cir<lb></lb>cumferentiam <emph type="italics"></emph>GaD<emph.end type="italics"></emph.end>; <lb></lb>& angulus <emph type="italics"></emph>GQP,<emph.end type="italics"></emph.end>qui <lb></lb>dimidius eſt anguli ad <lb></lb>centrum <emph type="italics"></emph>GQD,<emph.end type="italics"></emph.end>æ<lb></lb>qualis eſt complemen<lb></lb>to ad duos rectos an<lb></lb>guli ad circumferenti<lb></lb>am <emph type="italics"></emph>GbD,<emph.end type="italics"></emph.end>adeoque æ<lb></lb>qualis angulo <emph type="italics"></emph>Gba<emph.end type="italics"></emph.end>; <lb></lb>funtQ.E.I.eo triangu<lb></lb>la <emph type="italics"></emph>GPQ, Gab<emph.end type="italics"></emph.end>ſimi<lb></lb>lia; & <emph type="italics"></emph>Ga<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>ab<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>GP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>; id eſt <lb></lb>(ex conſtructione) ut <lb></lb><emph type="italics"></emph>Ga<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Æquan<lb></lb>tur itaque <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; & propterea triangula <emph type="italics"></emph>abc, ABC,<emph.end type="italics"></emph.end>quæ mo<lb></lb>do ſimilia eſſe probavimus, ſunt etiam æqualia. </s> <s>Unde, cum tan<lb></lb>gant inſuper trianguli <emph type="italics"></emph>DEF<emph.end type="italics"></emph.end>anguli <emph type="italics"></emph>D, E, F<emph.end type="italics"></emph.end>trianguli <emph type="italics"></emph>abc<emph.end type="italics"></emph.end>latera <lb></lb><emph type="italics"></emph>ab, ac, bc<emph.end type="italics"></emph.end>reſpective, compleri poteſt Figura <emph type="italics"></emph>ABCdef<emph.end type="italics"></emph.end>Figuræ <lb></lb><emph type="italics"></emph>abc DEF<emph.end type="italics"></emph.end>ſimilis & æqualis, atque eam complendo ſolvetur Pro<lb></lb>blema. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/119.jpg" pagenum="91"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc recta duci poteſt cujus partes longitudine datæ rectis <lb></lb><arrow.to.target n="note67"></arrow.to.target>tribus poſitione datis interjacebunt. </s> <s>Concipe Triangulum <emph type="italics"></emph>DEF,<emph.end type="italics"></emph.end><lb></lb>puncto <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ad latus <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>accedente, & lateribus <emph type="italics"></emph>DE, DF<emph.end type="italics"></emph.end>in di<lb></lb>rectum poſitis, mutari in lineam rectam, cujus pars data <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>rec<lb></lb>tis poſitione datis <emph type="italics"></emph>AB, AC,<emph.end type="italics"></emph.end>& pars data <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>rectis poſitione da<lb></lb>tis <emph type="italics"></emph>AB, BC<emph.end type="italics"></emph.end>interponi debet; & applicando conſtructionem præ<lb></lb>cedentem ad hunc caſum ſolvetur Problema. </s></p> <p type="margin"> <s><margin.target id="note67"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVIII. PROBLEMA XX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam ſpecie & magnitudine datam deſcribere, cujus partes da<lb></lb>tæ rectis tribus poſitione datis interjacebunt.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Deſcribenda ſit Trajectoria quæ ſit ſimilis & æqualis Lineæ cur<lb></lb>væ <emph type="italics"></emph>DEF,<emph.end type="italics"></emph.end>quæque a rectis tribus <emph type="italics"></emph>AB, AC, BC<emph.end type="italics"></emph.end>poſitione datis, in <lb></lb><figure id="id.039.01.119.1.jpg" xlink:href="039/01/119/1.jpg"></figure><lb></lb>partes datis hujus partibus <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ſimiles & æquales ſeca<lb></lb>bitur. </s></p> <p type="main"> <s>Age rectas <emph type="italics"></emph>DE, EF, DF,<emph.end type="italics"></emph.end>& trianguli hujus <emph type="italics"></emph>DEF<emph.end type="italics"></emph.end>pone an<lb></lb>los <emph type="italics"></emph>D, E, F<emph.end type="italics"></emph.end>ad rectas illas poſitione datas (per Lem. </s> <s>XXVI) Dein <lb></lb>circa triangulum deſcribe Trajectoriam Curvæ <emph type="italics"></emph>DEF<emph.end type="italics"></emph.end>ſimilem & <lb></lb>æqualem. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end><pb xlink:href="039/01/120.jpg" pagenum="92"></pb><arrow.to.target n="note68"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note68"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Trapezium ſpecie datum deſcribere cujus anguli ad rectas quatuor po<lb></lb>ſitione datas, quæ neque omnes parallelæ ſunt, neque ad commune <lb></lb>punctum convergunt, ſinguli ad ſingulas conſiſtent.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Dentur poſitione rectæ quatuor <emph type="italics"></emph>ABC, AD, BD, CE,<emph.end type="italics"></emph.end>qua<lb></lb>rum prima ſecet ſecundam in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>tertiam in <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& quartam in <emph type="italics"></emph>C:<emph.end type="italics"></emph.end><lb></lb>& deſcribendum ſit Trapezium <emph type="italics"></emph>fghi<emph.end type="italics"></emph.end>quod ſit Trapezio <emph type="italics"></emph>FGHI<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.120.1.jpg" xlink:href="039/01/120/1.jpg"></figure><lb></lb>ſimile, & cujus angulus <emph type="italics"></emph>f,<emph.end type="italics"></emph.end>angulo dato <emph type="italics"></emph>F<emph.end type="italics"></emph.end>æqualis, tangat rectam <lb></lb><emph type="italics"></emph>ABC,<emph.end type="italics"></emph.end>cæterique anguli <emph type="italics"></emph>g, h, i,<emph.end type="italics"></emph.end>cæteris angulis datis <emph type="italics"></emph>G, H, I<emph.end type="italics"></emph.end>æqua<lb></lb>les, tangant cæteras lineas <emph type="italics"></emph>AD, BD, CE<emph.end type="italics"></emph.end>reſpective. </s> <s>Jungatur <lb></lb><emph type="italics"></emph>FH<emph.end type="italics"></emph.end>& ſuper <emph type="italics"></emph>FG, FH, FI<emph.end type="italics"></emph.end>deſcribantur totidem circulorum ſeg<lb></lb>menta <emph type="italics"></emph>FSG, FTH, FVI<emph.end type="italics"></emph.end>; quorum primum <emph type="italics"></emph>FSG<emph.end type="italics"></emph.end>capiat angu-<pb xlink:href="039/01/121.jpg" pagenum="93"></pb>lum æqualem angulo <emph type="italics"></emph>BAD,<emph.end type="italics"></emph.end>ſecundum <emph type="italics"></emph>FTH<emph.end type="italics"></emph.end>capiat angulum æ<lb></lb><arrow.to.target n="note69"></arrow.to.target>qualem angulo <emph type="italics"></emph>CBD,<emph.end type="italics"></emph.end>ac tertium <emph type="italics"></emph>FVI<emph.end type="italics"></emph.end>capiat angulum æqualem <lb></lb>angulo <emph type="italics"></emph>ACE.<emph.end type="italics"></emph.end>Deſcribi autem debent ſegmenta ad eas partes li<lb></lb>nearum <emph type="italics"></emph>FG, FH, FI,<emph.end type="italics"></emph.end>ut literarum <emph type="italics"></emph>FSGF<emph.end type="italics"></emph.end>idem ſit ordo circula<lb></lb>ris qui literarum <emph type="italics"></emph>BADB,<emph.end type="italics"></emph.end>utque literæ <emph type="italics"></emph>FTHF<emph.end type="italics"></emph.end>eodem ordine cum <lb></lb>literis <emph type="italics"></emph>CBDC,<emph.end type="italics"></emph.end>& literæ <emph type="italics"></emph>FVIF<emph.end type="italics"></emph.end>eodem cum literis <emph type="italics"></emph>ACEA<emph.end type="italics"></emph.end>in or<lb></lb>bem redeant. </s> <s>Compleantur ſegmenta in circulos integros, ſitque <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>centrum circuli primi <emph type="italics"></emph>FSG,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>centrum ſecundi <emph type="italics"></emph>FTH.<emph.end type="italics"></emph.end>Jungatur <lb></lb>& utrinque producatur <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>& in ea capiatur <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>in ea ratione ad <lb></lb><emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>quam habet <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Capiatur autem <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>ad eas partes <lb></lb>puncti <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>ut literarum <emph type="italics"></emph>P, Q, R<emph.end type="italics"></emph.end>idem ſit ordo atque literarum <lb></lb><emph type="italics"></emph>A, B, C:<emph.end type="italics"></emph.end>centroque <emph type="italics"></emph>R<emph.end type="italics"></emph.end>& intervallo <emph type="italics"></emph>RF<emph.end type="italics"></emph.end>deſcribatur circulus quartus <lb></lb><emph type="italics"></emph>FNc<emph.end type="italics"></emph.end>ſecans circulum tertium <emph type="italics"></emph>FVI<emph.end type="italics"></emph.end>in <emph type="italics"></emph>c.<emph.end type="italics"></emph.end>Jungatur <emph type="italics"></emph>Fc<emph.end type="italics"></emph.end>ſecans <lb></lb>circulum primum in <emph type="italics"></emph>a<emph.end type="italics"></emph.end>& ſecundum in <emph type="italics"></emph>b.<emph.end type="italics"></emph.end>Agantur <emph type="italics"></emph>a G, b H, c I,<emph.end type="italics"></emph.end>& <lb></lb>Figuræ <emph type="italics"></emph>abc FGHI<emph.end type="italics"></emph.end>ſimilis conſtituatur Figura <emph type="italics"></emph>ABCfghi:<emph.end type="italics"></emph.end>Eritque <lb></lb>Trapezium <emph type="italics"></emph>fghi<emph.end type="italics"></emph.end>illud ipſum quod conſtituere oportebat. </s></p> <p type="margin"> <s><margin.target id="note69"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Secent enim circuli duo primi <emph type="italics"></emph>FSG, FTH<emph.end type="italics"></emph.end>ſe mutuo in <emph type="italics"></emph>K.<emph.end type="italics"></emph.end><lb></lb>Jungantur <emph type="italics"></emph>PK, QK, RK, a K, b K, c K,<emph.end type="italics"></emph.end>& producatur <emph type="italics"></emph>QP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>L.<emph.end type="italics"></emph.end><lb></lb>Anguli ad circumferentias <emph type="italics"></emph>FaK, FbK, FcK<emph.end type="italics"></emph.end>ſunt ſemiſſes an<lb></lb>gulorum <emph type="italics"></emph>FPK, FQK, FRK<emph.end type="italics"></emph.end>ad centra, adeoque angulorum <lb></lb>illorum dimidiis <emph type="italics"></emph>LPK, LQK, LRK<emph.end type="italics"></emph.end>æquales. </s> <s>Eſt ergo Figura <lb></lb><emph type="italics"></emph>PQRK<emph.end type="italics"></emph.end>Figuræ <emph type="italics"></emph>abcK<emph.end type="italics"></emph.end>æquiangula & ſimilis, & propterea <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>eſt <lb></lb>ad <emph type="italics"></emph>bc<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>QR,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BC.<emph.end type="italics"></emph.end>Angulis inſuper <emph type="italics"></emph>FaG, <lb></lb>FbH, FcI<emph.end type="italics"></emph.end>æquantur <emph type="italics"></emph>fAg, fBh, fCi<emph.end type="italics"></emph.end>per conſtructionem. </s> <s>Er<lb></lb>go Figuræ <emph type="italics"></emph>abcFGHI<emph.end type="italics"></emph.end>Figura ſimilis <emph type="italics"></emph>ABCfghi<emph.end type="italics"></emph.end>compleri poteſt <lb></lb>Quo facto Trapezium <emph type="italics"></emph>fghi<emph.end type="italics"></emph.end>conſtituetur ſimile Trapezio <emph type="italics"></emph>FGHI<emph.end type="italics"></emph.end><lb></lb>& angulis ſuis <emph type="italics"></emph>f, g, h, i<emph.end type="italics"></emph.end>tanget rectas <emph type="italics"></emph>ABC, AD, BD, CE <lb></lb>q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc recta duci poteſt cujus partes, rectis quatuor poſi<lb></lb>tione datis dato ordine interjectæ, datam habebunt proportionem <lb></lb>ad invicem. </s> <s>Augeantur anguli <emph type="italics"></emph>FGH, GHI<emph.end type="italics"></emph.end>uſque eo, ut rectæ <emph type="italics"></emph>FG, <lb></lb>GH, HI<emph.end type="italics"></emph.end>in directum jaceant, & in hoc caſu conſtruendo Proble<lb></lb>ma, ducetur recta <emph type="italics"></emph>fghi<emph.end type="italics"></emph.end>cujus partes <emph type="italics"></emph>fg, gh, hi,<emph.end type="italics"></emph.end>rectis quatuor po<lb></lb>ſitione datis <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AD, AD<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BD, BD<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>interjectæ, e<lb></lb>runt ad invicem ut lineæ <emph type="italics"></emph>FG, GH, HI,<emph.end type="italics"></emph.end>eundemque ſervabunt or<lb></lb>dinem inter ſe. </s> <s>Idem vero ſic fit expeditius. <pb xlink:href="039/01/122.jpg" pagenum="94"></pb><arrow.to.target n="note70"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note70"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Producantur <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>K,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>BK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>HI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>DL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>GI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>; & jungatur <emph type="italics"></emph>KL<emph.end type="italics"></emph.end><lb></lb>occurrens rectæ <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>in <emph type="italics"></emph>i.<emph.end type="italics"></emph.end>Producatur <emph type="italics"></emph>iL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>M,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>iL<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HI,<emph.end type="italics"></emph.end>& agatur tum <emph type="italics"></emph>MQ<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>LB<emph.end type="italics"></emph.end>parallela rectæque <lb></lb><emph type="italics"></emph>AD<emph.end type="italics"></emph.end>occurrens in <emph type="italics"></emph>g,<emph.end type="italics"></emph.end>tum <emph type="italics"></emph>gi<emph.end type="italics"></emph.end>ſecans <emph type="italics"></emph>AB, BD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>f, h.<emph.end type="italics"></emph.end>Dico <lb></lb>factum. </s></p> <p type="main"> <s>Secet enim <emph type="italics"></emph>Mg<emph.end type="italics"></emph.end>rectam <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>rectam <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>in <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>& <lb></lb>agatur <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>quæ ſit ipſi <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>parallela & occurrat <emph type="italics"></emph>iL<emph.end type="italics"></emph.end>in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& <lb></lb>erunt <emph type="italics"></emph>gM<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Lh (gi<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>hi, Mi<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Li, GI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HI, AK<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>BK<emph.end type="italics"></emph.end>) & <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BL<emph.end type="italics"></emph.end>in eadem ratione. </s> <s>Secetur <emph type="italics"></emph>DL<emph.end type="italics"></emph.end>in <emph type="italics"></emph>R<emph.end type="italics"></emph.end>ut ſit <lb></lb><figure id="id.039.01.122.1.jpg" xlink:href="039/01/122/1.jpg"></figure><lb></lb><emph type="italics"></emph>DL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>RL<emph.end type="italics"></emph.end>in eadem illa ratione, & ob proportionales <emph type="italics"></emph>gS<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>gM, AS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AP,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DL<emph.end type="italics"></emph.end>; erit, ex æquo, ut <emph type="italics"></emph>gS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Lh<emph.end type="italics"></emph.end>ita <lb></lb><emph type="italics"></emph>AS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>RL<emph.end type="italics"></emph.end>; & mixtim, <emph type="italics"></emph>BL-RL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Lh-BL<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>AS-DS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>gS-AS.<emph.end type="italics"></emph.end>Id eſt <emph type="italics"></emph>BR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Bh<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ag<emph.end type="italics"></emph.end>ad<lb></lb>eoque ut <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="gq.">gque</expan><emph.end type="italics"></emph.end>Et viciſſim <emph type="italics"></emph>BR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Bh<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>gQ,<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>fh<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>fg.<emph.end type="italics"></emph.end>Sed ex conſtructione linea <emph type="italics"></emph>RL<emph.end type="italics"></emph.end>eadem ratione ſecta fuit <lb></lb>in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>R<emph.end type="italics"></emph.end>atque linea <emph type="italics"></emph>FI<emph.end type="italics"></emph.end>in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>H:<emph.end type="italics"></emph.end>ideoque eſt <emph type="italics"></emph>BR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BD<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>FH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FG.<emph.end type="italics"></emph.end>Ergo <emph type="italics"></emph>fh<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>FH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FG.<emph.end type="italics"></emph.end>Cum igitur <lb></lb>ſit etiam <emph type="italics"></emph>gi<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>hi<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Mi<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Li,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>GI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HI,<emph.end type="italics"></emph.end>patet li<lb></lb>neas <emph type="italics"></emph>FI, fi<emph.end type="italics"></emph.end>in <emph type="italics"></emph>g<emph.end type="italics"></emph.end>& <emph type="italics"></emph>h, G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ſimiliter ſectas eſſe. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/123.jpg" pagenum="95"></pb> <p type="main"> <s>In conſtructione Corollarii hujus poſtquam ducitur <emph type="italics"></emph>LK<emph.end type="italics"></emph.end>ſecans </s></p> <p type="main"> <s><arrow.to.target n="note71"></arrow.to.target><emph type="italics"></emph>CE<emph.end type="italics"></emph.end>in <emph type="italics"></emph>i,<emph.end type="italics"></emph.end>producere licet <emph type="italics"></emph>iE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>EV<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ei<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>FH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HI,<emph.end type="italics"></emph.end><lb></lb>& agere <emph type="italics"></emph>Vf<emph.end type="italics"></emph.end>parallelam ipſi <emph type="italics"></emph>BD.<emph.end type="italics"></emph.end>Eodem recidit ſi centro <emph type="italics"></emph>i,<emph.end type="italics"></emph.end>in<lb></lb>tervallo <emph type="italics"></emph>IH,<emph.end type="italics"></emph.end>deſcribatur circulus ſecans <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>X,<emph.end type="italics"></emph.end>& producatur <lb></lb><emph type="italics"></emph>iX<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Y,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>iY<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>IF,<emph.end type="italics"></emph.end>& agatur <emph type="italics"></emph>Yf<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>parallela. </s></p> <p type="margin"> <s><margin.target id="note71"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Problematis hujus ſolutiones alias <emph type="italics"></emph>Wrennus<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Walliſius<emph.end type="italics"></emph.end>olim ex<lb></lb>cogitarunt. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIX. PROBLEMA XXI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam ſpecie datam deſcribere, quæ a rectis quatuor poſitione <lb></lb>datis in partes ſecabitur, ordine, ſpecie & proportione datas.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p><figure id="id.039.01.123.1.jpg" xlink:href="039/01/123/1.jpg"></figure> <p type="main"> <s>Deſcribenda ſit Trajectoria <lb></lb><figure id="id.039.01.123.2.jpg" xlink:href="039/01/123/2.jpg"></figure><lb></lb><emph type="italics"></emph>fghi,<emph.end type="italics"></emph.end>quæ ſimilis ſit Lincæ curvæ <lb></lb><emph type="italics"></emph>FGHI,<emph.end type="italics"></emph.end>& cujus partes <emph type="italics"></emph>fg, gh, hi<emph.end type="italics"></emph.end><lb></lb>illius partibus <emph type="italics"></emph>FG, GH, HI<emph.end type="italics"></emph.end>ſi<lb></lb>miles & proportionales, rectis <lb></lb><emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AD, AD<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BD, BD<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>poſitione datis, prima pri<lb></lb>mis, ſecunda ſecundis, tertia ter<lb></lb>tiis interjaceant. </s> <s>Actis rectis <emph type="italics"></emph>FG, <lb></lb>GH, HI, FI,<emph.end type="italics"></emph.end>deſcribatur (per <lb></lb>Lem. </s> <s>XXVII.) Trapezium <emph type="italics"></emph>fghi<emph.end type="italics"></emph.end><lb></lb>quod ſit Trapezio <emph type="italics"></emph>FGHI<emph.end type="italics"></emph.end>ſimile & cujus anguli <emph type="italics"></emph>f, g, h, i<emph.end type="italics"></emph.end>tangant <lb></lb>rectas illas poſitione datas <emph type="italics"></emph>AB, AD, BD, CE,<emph.end type="italics"></emph.end>ſinguli ſingulas <lb></lb>dicto ordine. </s> <s>Dein circa hoc Trapezium deſcribatur Trajectoria <lb></lb>curvæ Lineæ <emph type="italics"></emph>FGHI<emph.end type="italics"></emph.end>conſimilis. <pb xlink:href="039/01/124.jpg" pagenum="96"></pb><arrow.to.target n="note72"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note72"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Conſtrui etiam poteſt hoc Problema ut ſequitur. </s> <s>Junctis <emph type="italics"></emph>FG, <lb></lb>GH, HI, FI<emph.end type="italics"></emph.end>produc <emph type="italics"></emph>GF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>jungeque <emph type="italics"></emph>FH, IG,<emph.end type="italics"></emph.end>& angulis <lb></lb><emph type="italics"></emph>FGH, VFH<emph.end type="italics"></emph.end>fac angulos <emph type="italics"></emph>CAK, DAL<emph.end type="italics"></emph.end>æquales. </s> <s>Concurrant <lb></lb><emph type="italics"></emph>AK, AL<emph.end type="italics"></emph.end>cum recta <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>& inde agantur <emph type="italics"></emph>KM, LN,<emph.end type="italics"></emph.end><lb></lb>quarum <emph type="italics"></emph>KM<emph.end type="italics"></emph.end>conſtituat angulum <emph type="italics"></emph>AKM<emph.end type="italics"></emph.end>æqualem angulo <emph type="italics"></emph>GHI,<emph.end type="italics"></emph.end><lb></lb>ſitque ad <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>LN<emph.end type="italics"></emph.end>conſtituat angulum <lb></lb><emph type="italics"></emph>ALN<emph.end type="italics"></emph.end>æqualem angulo <emph type="italics"></emph>FHI,<emph.end type="italics"></emph.end>ſitque ad <emph type="italics"></emph>AL<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FH.<emph.end type="italics"></emph.end>Du<lb></lb>cantur autem <emph type="italics"></emph>AK, KM, AL, LN<emph.end type="italics"></emph.end>ad eas partes linearum <emph type="italics"></emph>AD, <lb></lb>AK, AL,<emph.end type="italics"></emph.end>ut literæ <emph type="italics"></emph>CAKMC, ALKA, DALND<emph.end type="italics"></emph.end>eodem <lb></lb>ordine cum literis <emph type="italics"></emph>FGHIF<emph.end type="italics"></emph.end>in orbem redeant; & act <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>oc<lb></lb>currat rectæ <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>in <emph type="italics"></emph>i.<emph.end type="italics"></emph.end>Fac angulum <emph type="italics"></emph>iEP<emph.end type="italics"></emph.end>æqualem angulo <emph type="italics"></emph>IGF,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.124.1.jpg" xlink:href="039/01/124/1.jpg"></figure><lb></lb>ſitque <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ei<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GI;<emph.end type="italics"></emph.end>& per <emph type="italics"></emph>P<emph.end type="italics"></emph.end>agatur <emph type="italics"></emph>PQf,<emph.end type="italics"></emph.end>quæ <lb></lb>cum recta <emph type="italics"></emph>ADE<emph.end type="italics"></emph.end>contineat angulum <emph type="italics"></emph>PQE<emph.end type="italics"></emph.end>æqualem angulo <lb></lb><emph type="italics"></emph>FIG,<emph.end type="italics"></emph.end>rectæque <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>occurrat in <emph type="italics"></emph>f,<emph.end type="italics"></emph.end>& jungatur <emph type="italics"></emph>fi.<emph.end type="italics"></emph.end>Agantur au<lb></lb>rem <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad eas partes linearum <emph type="italics"></emph>CE, PE,<emph.end type="italics"></emph.end>ut literarum <lb></lb><emph type="italics"></emph>PEiP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PEQP<emph.end type="italics"></emph.end>idem ſit ordo circularis qui literarum <emph type="italics"></emph>FGHIF,<emph.end type="italics"></emph.end><lb></lb>& ſi ſuper linea <emph type="italics"></emph>fi<emph.end type="italics"></emph.end>eodem quoque literarum ordine conſtituatur <lb></lb>Trapezium <emph type="italics"></emph>fghi<emph.end type="italics"></emph.end>Trapezio <emph type="italics"></emph>FGHI<emph.end type="italics"></emph.end>ſimile, & circumſcribatur Tra<lb></lb>jectoria ſpecie data, ſolvetur Problema. </s></p> <p type="main"> <s>Hactenus de Orbibus inveniendis. </s> <s>Supereſt ut Motus corpo<lb></lb>rum in Orbibus inventis determinemus. <pb xlink:href="039/01/125.jpg" pagenum="97"></pb><arrow.to.target n="note73"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note73"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Inventione Motuum in Orbibus datis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXX. PROBLEMA XXII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Corporis in data Trajectoria Parabolica moti invenire locum ad <lb></lb>tempus aſſignatum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>S<emph.end type="italics"></emph.end>umbilicus & <emph type="italics"></emph>A<emph.end type="italics"></emph.end>vertex principa<lb></lb><figure id="id.039.01.125.1.jpg" xlink:href="039/01/125/1.jpg"></figure><lb></lb>lis Parabolæ, ſitque 4 <emph type="italics"></emph>ASXM<emph.end type="italics"></emph.end>æquale <lb></lb>areæ Parabolicæ abſcindendæ <emph type="italics"></emph>APS,<emph.end type="italics"></emph.end><lb></lb>quæ radio <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>vel poſt exceſſum cor<lb></lb>poris de vertice deſcripta fuit, vel an<lb></lb>te appulſum ejus ad verticem deſcri<lb></lb>benda eſt. </s> <s>Innoteſcit quantitas areæ il<lb></lb>lius abſcindendæ ex tempore ipſi pro<lb></lb>portionali. </s> <s>Biſeca <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>in <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>erigeque <lb></lb>perpendiculum <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>æquale 3 M, & <lb></lb>Circulus centro <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>HS<emph.end type="italics"></emph.end><lb></lb>deſcriptus ſecabit Parabolam in loco <lb></lb>quæſito <emph type="italics"></emph>P.<emph.end type="italics"></emph.end>Nam, demiſſa ad axem <lb></lb>perpendiculari <emph type="italics"></emph>PO<emph.end type="italics"></emph.end>& ducta <emph type="italics"></emph>PH,<emph.end type="italics"></emph.end>eſt <lb></lb><emph type="italics"></emph>AGq+GHq (=HP q=—AO-AG: quad.+—PO-GH: quad.)= <lb></lb>AOq+POq-2 <expan abbr="GAO-2GHXPO+AGq+GHq.">GAO-2GHXPO+AGq+GHque</expan><emph.end type="italics"></emph.end>Unde <lb></lb>2 <emph type="italics"></emph>GHXPO (=AOq+POq-2GAO)=AOq+1/4 <expan abbr="POq.">POque</expan><emph.end type="italics"></emph.end><lb></lb>Pro <emph type="italics"></emph>AOq<emph.end type="italics"></emph.end>ſcribe (<emph type="italics"></emph>AOXPOq/4AS<emph.end type="italics"></emph.end>); &, applicatis terminis omnibus ad <lb></lb>3<emph type="italics"></emph>PO<emph.end type="italics"></emph.end>ductiſQ.E.I. 2<emph type="italics"></emph>AS,<emph.end type="italics"></emph.end>fiet 4/3 <emph type="italics"></emph>GHXAS(=1/6AOXPO+1/2 ASXPO <lb></lb>=(AO+3AS/6)XPO=(4AO-3SO/6)XPO<emph.end type="italics"></emph.end>=areæ —<emph type="italics"></emph>APO-SPO)<emph.end type="italics"></emph.end><lb></lb>=areæ <emph type="italics"></emph>APS.<emph.end type="italics"></emph.end>Sed <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>erat 3 M, & inde 4/3 <emph type="italics"></emph>GHXAS<emph.end type="italics"></emph.end>eſt 4 <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>XM. </s> <s><lb></lb>Ergo area abſciſſa <emph type="italics"></emph>APS<emph.end type="italics"></emph.end>æqualis eſt abſcindendæ 4<emph type="italics"></emph>ASXM. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>AS,<emph.end type="italics"></emph.end>ut tempus quo corpùs deſcrip<lb></lb>ſit arcum <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad tempus quo corpus deſcripſit arcum inter verti<lb></lb>cem <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& perpendiculum ad axem ab umbilico <emph type="italics"></emph>S<emph.end type="italics"></emph.end>erectum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et Circulo <emph type="italics"></emph>ASP<emph.end type="italics"></emph.end>per corpus motum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>perpetuo tranſ<lb></lb>eunte, velocitas puncti <emph type="italics"></emph>H<emph.end type="italics"></emph.end>eſt ad velocitatem quam corpus habuit <pb xlink:href="039/01/126.jpg" pagenum="98"></pb><arrow.to.target n="note74"></arrow.to.target>in vertice <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut 3 ad 8; adeoQ.E.I. ea etiam ratione eſt linea <emph type="italics"></emph>GH<emph.end type="italics"></emph.end><lb></lb>ad lineam rectam quam corpus tempore motus ſui ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ea <lb></lb>cum velocitate quam habuit in vertice <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>deſcribere poſſet. </s></p> <p type="margin"> <s><margin.target id="note74"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Hinc etiam vice verſa inveniri poteſt tempus quo cor<lb></lb>pus deſcripſit arcum quemvis aſſignatum <emph type="italics"></emph>AP.<emph.end type="italics"></emph.end>Junge <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>& ad <lb></lb>medium ejus punctum erige perpendiculum rectæ <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>occur<lb></lb>rens in <emph type="italics"></emph>H.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Nulla extat Figura Ovalis cujus area, rectis pro lubitu abſciſſa, poſſit <lb></lb>per æquationes numero terminorum ac dimenſionum finitas genera<lb></lb>liter inveniri.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Intra Ovalem detur punctum quodvis, circa quod ceu polum re<lb></lb>volvatur perpetuo linea recta, uniformi cum motu, & interea in rec<lb></lb>ta illa exeat punctum mobile de polo, pergatque ſemper ea cum <lb></lb>velocitate, quæ ſit ut rectæ illius intra Ovalem quadratum. </s> <s>Hoc <lb></lb>motu punctum illud deſcribet Spiralem gyris infinitis. </s> <s>Jam ſi areæ <lb></lb>Ovalis a recta illa abſciſſæ incrementum per finitam æquationem <lb></lb>inveniri poteſt, invenietur etiam per eandem æquationem diſtantia <lb></lb>puncti a polo, quæ huic areæ proportionalis eſt, adeoque om<lb></lb>nia Spiralis puncta per æquationem finitam inveniri poſſunt: & <lb></lb>propterea rectæ cujuſvis poſitione datæ interſectio cum Spirali in<lb></lb>veniri etiam poteſt per æquationem finitam. </s> <s>Atqui recta omnis <lb></lb>infinite producta Spiralem ſecat in punctis numero infinitis, & æqua<lb></lb>tio, qua interſectio aliqua duarum linearum invenitur, exhibet ea<lb></lb>rum interſectiones omnes radicibus totidem, adeoque aſcendit ad <lb></lb>rot dimenſiones quot ſunt interſectiones. </s> <s>Quoniam Circuli duo ſe <lb></lb>mutuo ſecant in punctis duobus, interſectio una non invenietur <lb></lb>niſi per æquationem duarum dimenſionum, qua interſectio altera <lb></lb>etiam inveniatur. </s> <s>Quoniam duarum ſectionum Conicarum quatuor <lb></lb>eſſe poſſunt interſectiones, non poteſt aliqua earum generaliter in<lb></lb>veniri niſi per æquationem quatuor dimenſionum, qua omnes ſi<lb></lb>mul inveniantur. </s> <s>Nam ſi interſectiones illæ ſeorſim quærantur, quo<lb></lb>niam eadem eſt omnium lex & conditio, idem erit calculus in caſu <lb></lb>unoquoque & propterea eadem ſemper concluſio, quæ igitur de<lb></lb>bet omnes interſectiones ſimul complecti & indifferenter exhibere. <pb xlink:href="039/01/127.jpg" pagenum="99"></pb>Unde etiam interſectiones Sectionum Conicarum & Curvarum ter<lb></lb><arrow.to.target n="note75"></arrow.to.target>tiæ poteſtatis, eo quod ſex eſſe poſſunt, ſimul prodeunt per æqua<lb></lb>tiones ſex dimenſionum, & interſectiones duarum Curvarum tertiæ <lb></lb>poteſtatis, quia novem eſſe poſſunt, ſimul prodeunt per æqua<lb></lb>tiones dimenſionum novem. </s> <s>Id niſi neceſſario fieret, reducere licc<lb></lb>ret Problemata omnia Solida ad Plana, & pluſquam Solida ad Soli<lb></lb>da. </s> <s>Loquor hic de Curvis poteſtate irreducibilibus. </s> <s>Nam ſi æqua<lb></lb>tio per quam Curva definitur, ad inferiorem poteſtatem reduci <lb></lb>poſſit: Curva non erit unica, ſed ex duabus vel pluribus compoſi<lb></lb>ta, quarum interſectiones per calculos diverſos ſeorſim inveniri <lb></lb>poſſunt. </s> <s>Ad eundem modum interſectiones binæ rectarum & ſecti<lb></lb>onum Conicarum prodeunt ſemper per æquationes duarum dimen<lb></lb>ſionum; ternæ rectarum & Curvarum irreducibilium tertiæ poteſtatis <lb></lb>per æquationes trium, quaternæ rectarum & Curvarvm irreducibi<lb></lb>lium quartæ poteſtatis per æquationes dimenſionum quatuor, & ſic <lb></lb>in infinitum. </s> <s>Ergo rectæ & Spiralis interſectiones numero infinitæ, cum <lb></lb>Curva hæc ſit ſimplex & in Curvas plures irreducibilis, requirunt æ<lb></lb>quationes numero dimenſionum & radicum infinitas, quibus omnes <lb></lb>poſſunt ſimul exhiberi. </s> <s>Eſt enim eadem omnium lex & idem calculus. </s> <s><lb></lb>Nam ſi a polo in rectam illam ſecantem demittatur perpendiculum, <lb></lb>& perpendiculum illud una cum ſecante revolvatur circa polum, in<lb></lb>terſectiones Spiralis tranſibunt in ſe mutuo, quæque prima erat ſeu <lb></lb>proxima, poſt unam revolutionem ſecunda erit, poſt duas tertia, <lb></lb>& ſic deinceps: nec interea mutabitur æquatio niſi pro mutata mag<lb></lb>nitudine quantitatum per quas poſitio ſecantis determinatur. </s> <s>Unde <lb></lb>cum quantitates illæ poſt ſingulas revolutiones redeunt ad magNI<lb></lb>tudines primas, æquatio redibit ad formam primam, adeoque una <lb></lb>eademque exhibebit interſectiones omnes, & propterea radices ha<lb></lb>bebit numero infinitas, quibus omnes exhiberi poſſunt. </s> <s>Nequit <lb></lb>ergo interſectio rectæ & Spiralis per æquationem finitam generali<lb></lb>ter inveniri, & idcirco nulla extat Ovalis cujus area, rectis impe<lb></lb>ratis abſciſſa, poſſit per talem æquationem generaliter exhiberi. </s></p> <p type="margin"> <s><margin.target id="note75"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Eodem argumento, ſi intervallum poli & puncti, quo Spiralis de<lb></lb>ſcribitur, capiatur Ovalis perimetro abſciſſæ proportionale, pro<lb></lb>bari poteſt quod longitudo perimetri nequit per finitam æquatio<lb></lb>nem generaliter exhiberi. </s> <s>De Ovalibus autem hic loquor quæ non <lb></lb>tanguntur a figuris conjugatis in infinitum pergentibus. <pb xlink:href="039/01/128.jpg" pagenum="100"></pb><arrow.to.target n="note76"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note76"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hinc area Ellipſeos, quæ radio ab umbilico ad corpus mobile <lb></lb>ducto deſcribitur, non prodit ex dato tempore, per æquationem <lb></lb>finitam; & propterea per deſcriptionem Curvarum Geometrice ra<lb></lb>tionalium determinari nequit. </s> <s>Curvas Geometrice rationales ap<lb></lb>pello quarum puncta omnia per longitudines æquationibus defiNI<lb></lb>tas, id eſt, per longitudinum rationes complicatas, determinari <lb></lb>poſſunt; cæteraſque (ut Spirales, Quadratrices, Trochoides) Geo<lb></lb>metrice irrationales. </s> <s>Nam longitudines quæ ſunt vel non ſunt ut <lb></lb>numerus ad numerum (quemadmodum in decimo Elementorum) <lb></lb>ſunt Arithmetice rationales vel irrationales. </s> <s>Aream igitur Ellipſeos <lb></lb>tempori proportionalem abſcindo per Curvam Geometrice irratio<lb></lb>nalem ut ſequitur. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXI. PROBLEMA XXIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Corporis in data Trajectoria Elliptica moti invenire locum ad <lb></lb>tempus aſſignatum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ellipſeos <emph type="italics"></emph>APB<emph.end type="italics"></emph.end>ſit <emph type="italics"></emph>A<emph.end type="italics"></emph.end>vertex principalis, <emph type="italics"></emph>S<emph.end type="italics"></emph.end>umbilicus, & <emph type="italics"></emph>O<emph.end type="italics"></emph.end><lb></lb>centrum, ſitque <emph type="italics"></emph>P<emph.end type="italics"></emph.end>corporis locus inveniendus. </s> <s>Produc <emph type="italics"></emph>OA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>G,<emph.end type="italics"></emph.end><lb></lb>ut ſit <emph type="italics"></emph>OG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OA<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>OA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OS.<emph.end type="italics"></emph.end>Erige perpendiculum <emph type="italics"></emph>GH,<emph.end type="italics"></emph.end>centroque <lb></lb><figure id="id.039.01.128.1.jpg" xlink:href="039/01/128/1.jpg"></figure><lb></lb><emph type="italics"></emph>O<emph.end type="italics"></emph.end>& intervallo <emph type="italics"></emph>OG<emph.end type="italics"></emph.end>deſcribe circulum <emph type="italics"></emph>EFG,<emph.end type="italics"></emph.end>& ſuper regula <emph type="italics"></emph>GH,<emph.end type="italics"></emph.end><lb></lb>ceu fundo, progrediatur Rota <emph type="italics"></emph>GEF<emph.end type="italics"></emph.end>revolvendo circa axem <lb></lb>ſuum, & interea puncto ſuo <emph type="italics"></emph>A<emph.end type="italics"></emph.end>deſcribendo Trochoidem <emph type="italics"></emph>ALI.<emph.end type="italics"></emph.end><pb xlink:href="039/01/129.jpg" pagenum="101"></pb>Quo facto, cape <emph type="italics"></emph>GK<emph.end type="italics"></emph.end>in ratione ad Rotæ perimetrum <emph type="italics"></emph>GEFG,<emph.end type="italics"></emph.end>ut <lb></lb><arrow.to.target n="note77"></arrow.to.target>eſt tempus quo corpus progrediendo ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>deſcripſit arcum <emph type="italics"></emph>AP,<emph.end type="italics"></emph.end>ad <lb></lb>tempus revolutionis unius in Ellipſi. </s> <s>Erigatur perpendiculum <emph type="italics"></emph>KL<emph.end type="italics"></emph.end><lb></lb>occurrens Trochoidi in <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>& acta <emph type="italics"></emph>LP<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>KG<emph.end type="italics"></emph.end>parallela occurret <lb></lb>Ellipſi in corporis loco quæſito <emph type="italics"></emph>P.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note77"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Nam centro <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>OA<emph.end type="italics"></emph.end>deſcribatur ſemicirculus <emph type="italics"></emph>AQB,<emph.end type="italics"></emph.end><lb></lb>& arcui <emph type="italics"></emph>AQ<emph.end type="italics"></emph.end>occurrat <emph type="italics"></emph>LP<emph.end type="italics"></emph.end>producta in <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end>junganturque <emph type="italics"></emph>SQ, <expan abbr="Oq.">Oque</expan><emph.end type="italics"></emph.end><lb></lb>Arcui <emph type="italics"></emph>EFG<emph.end type="italics"></emph.end>occurrat <emph type="italics"></emph>OQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>& in eandem <emph type="italics"></emph>OQ<emph.end type="italics"></emph.end>demittatur per<lb></lb>pendiculum <emph type="italics"></emph>SR.<emph.end type="italics"></emph.end>Area <emph type="italics"></emph>APS<emph.end type="italics"></emph.end>eſt ut area <emph type="italics"></emph>AQS,<emph.end type="italics"></emph.end>id eſt, ut diffe<lb></lb>rentia inter ſectorem <emph type="italics"></emph>OQA<emph.end type="italics"></emph.end>& triangulum <emph type="italics"></emph>OQS,<emph.end type="italics"></emph.end>ſive ut differen<lb></lb>tia rectangulorum 1/2 <emph type="italics"></emph>OQXAQ<emph.end type="italics"></emph.end>& 1/2 <emph type="italics"></emph>OQXSR,<emph.end type="italics"></emph.end>hoc eſt, ob datam <lb></lb>1/2 <emph type="italics"></emph>OQ,<emph.end type="italics"></emph.end>ut differentia inter arcum <emph type="italics"></emph>AQ<emph.end type="italics"></emph.end>& rectam <emph type="italics"></emph>SR,<emph.end type="italics"></emph.end>adeoque (ob <lb></lb>æqualitatem datarum rationum <emph type="italics"></emph>SR<emph.end type="italics"></emph.end>ad ſinum arcus <emph type="italics"></emph>AQ, OS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OA, <lb></lb>OA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OG, AQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GF,<emph.end type="italics"></emph.end>& diviſim <emph type="italics"></emph>AQ-SR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GF<emph.end type="italics"></emph.end>-ſin. </s> <s>arc. <emph type="italics"></emph>AQ<emph.end type="italics"></emph.end>) <lb></lb>ut <emph type="italics"></emph>GK<emph.end type="italics"></emph.end>differentia inter arcum <emph type="italics"></emph>GF<emph.end type="italics"></emph.end>& ſinum arcus <emph type="italics"></emph><expan abbr="Aq.">Aque</expan> <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cæterum, cum difficilis ſit hujus Curvæ deſcriptio, præſtat ſolu<lb></lb>tionem vero proximam adhibere. </s> <s>Inveniatur tum angulus quidam <lb></lb>B, qui ſit ad angulum graduum 57,29578, quem arcus radio æqualis <lb></lb>ſubtendit, ut eſt umbilieorum diſtantia <emph type="italics"></emph>SH<emph.end type="italics"></emph.end>ad Ellipſeos diame<lb></lb>trum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; tum etiam longitudo quædam L, quæ ſit ad radium in <lb></lb>eadem ratione inverſe. </s> <s>Quibus ſemel inventis, Problema deinceps <lb></lb>confit per ſequentem Analyſin. </s> <s>Per conſtructionem quamvis (vel. </s> <s><lb></lb>utcunque conjec<lb></lb><figure id="id.039.01.129.1.jpg" xlink:href="039/01/129/1.jpg"></figure><lb></lb>turam faciendo) <lb></lb>cognoſcatur cor<lb></lb>poris locus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>pro<lb></lb>ximus vero ejus lo<lb></lb>co <emph type="italics"></emph>p.<emph.end type="italics"></emph.end>Demiſſaque ad <lb></lb>axem Ellipſeos or<lb></lb>dinatim applicata <lb></lb><emph type="italics"></emph>PR,<emph.end type="italics"></emph.end>ex propor<lb></lb>tione diametrorum <lb></lb>Ellipſeos, dabitur <lb></lb>Circuli circumſcri<lb></lb>pti <emph type="italics"></emph>AQB<emph.end type="italics"></emph.end>ordinatim applicata <emph type="italics"></emph>RQ,<emph.end type="italics"></emph.end>quæ ſinus eſt anguli <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>exi<lb></lb>ſtente <emph type="italics"></emph>AO<emph.end type="italics"></emph.end>radio. </s> <s>Sufficit angulum illum rudi calculo in numeris <lb></lb>proximis invenire. </s> <s>Cognoſcatur etiam angulus tempori propor-<pb xlink:href="039/01/130.jpg" pagenum="102"></pb><arrow.to.target n="note78"></arrow.to.target>tionalis, id eſt, qui ſit ad quatuor rectos, ut eſt tempus quo corpus <lb></lb>deſcripſit arcum <emph type="italics"></emph>Ap,<emph.end type="italics"></emph.end>ad tempus revolutionis unius in Ellipſi. </s> <s>Sit <lb></lb>angulus iſte N. </s> <s>Tum capiatur & angulus D ad angulum B, ut <lb></lb>eſt ſinus iſte anguli <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>ad radium, & angulus E ad angulum <lb></lb>N-<emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>+D, ut eſt longitudo L ad longitudinem eandem L <lb></lb>coſinu anguli <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>diminutam, ubi angulus iſte recto minor eſt, <lb></lb>auctam ubi major. </s> <s>Poſtea capiatur tum angulus F ad angulum B, <lb></lb>ut eſt ſinus anguli <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>+E ad radium, tum angulus G ad angu<lb></lb>lum N-<emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>-E+F ut eſt longitudo L ad longitudinem ean<lb></lb>dem coſinu anguli <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>+E diminutam ubi angulus iſte recto mi<lb></lb>nor eſt, auctam ubi major. </s> <s>Tertia vice capiatur angulus H ad an<lb></lb>gulum B, ut eſt ſinus anguli <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>+E+G ad radium; & angu<lb></lb>lus I ad angulum N-<emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>-E-G+H, ut eſt longitudo L ad <lb></lb>eandem longitudinem coſinu anguli <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>+E+G diminutam, <lb></lb>ubi angulus iſte re<lb></lb><figure id="id.039.01.130.1.jpg" xlink:href="039/01/130/1.jpg"></figure><lb></lb>cto minor eſt, auc<lb></lb>tam ubi major. </s> <s>Et <lb></lb>ſic pergere licet in <lb></lb>infinitum. </s> <s>DeNI<lb></lb>que capiatur angu<lb></lb>lus <emph type="italics"></emph>AOq<emph.end type="italics"></emph.end>æqualis <lb></lb>angulo <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>+E <lb></lb>+G+I+&c. </s> <s>e t <lb></lb>ex coſinu ejus <emph type="italics"></emph>Or<emph.end type="italics"></emph.end><lb></lb>& ordinata <emph type="italics"></emph>pr,<emph.end type="italics"></emph.end>quæ <lb></lb>eſt ad ſinum ejus <lb></lb><emph type="italics"></emph>qr<emph.end type="italics"></emph.end>ut Ellipſeos axis minor ad axem majorem, habebitur corporis <lb></lb>locus correctus <emph type="italics"></emph>p.<emph.end type="italics"></emph.end>Si quando angulus N-<emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>+D negativus <lb></lb>eſt, debet ſignum+ipſius E ubique mutari in-, & ſignum-in+. <lb></lb>Idem intelligendum eſt de ſignis ipſorum G & I, ubi anguli <lb></lb>N-<emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>-E+F, & N-<emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>-E-G+H negativi prodeunt. </s> <s><lb></lb>Convergit autem ſeries infinita <emph type="italics"></emph>AOQ<emph.end type="italics"></emph.end>+E+G+I+&c. </s> <s>quam <lb></lb>celerrime, adeo ut vix unquam opus fuerit ultra progredi quam <lb></lb>ad terminum ſecundum E. </s> <s>Et fundatur calculus in hoc Theore<lb></lb>mate, quod area <emph type="italics"></emph>APS<emph.end type="italics"></emph.end>ſit ut differentia inter arcum <emph type="italics"></emph>AQ<emph.end type="italics"></emph.end>& <lb></lb>rectam ab umbilico <emph type="italics"></emph>S<emph.end type="italics"></emph.end>in Radium <emph type="italics"></emph>OQ<emph.end type="italics"></emph.end>perpendiculariter de<lb></lb>miſſam. </s></p> <p type="margin"> <s><margin.target id="note78"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Non diſſimili calculo conficitur Problema in Hyperbola. </s> <s>Sit <lb></lb>ejus Centrum <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>Vertex <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>Umbilicus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>& Aſymptotos <emph type="italics"></emph>OK.<emph.end type="italics"></emph.end>Cog-<pb xlink:href="039/01/131.jpg" pagenum="103"></pb>noſcatur quantitas areæ abſcindendæ tempori proportionalis. </s> <s>Sit ea <lb></lb><arrow.to.target n="note79"></arrow.to.target>A, & fiat conjectura de poſitione rectæ <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>quæ aream <emph type="italics"></emph>APS<emph.end type="italics"></emph.end><lb></lb>abſcindat veræ proximam. </s> <s>Jun<lb></lb><figure id="id.039.01.131.1.jpg" xlink:href="039/01/131/1.jpg"></figure><lb></lb>gatur <emph type="italics"></emph>OP,<emph.end type="italics"></emph.end>& ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ad <lb></lb>Aſymptoton agantur <emph type="italics"></emph>AI, PK<emph.end type="italics"></emph.end><lb></lb>Aſymptoto alteri parallelæ, & per <lb></lb>Tabulam Logarithmorum dabi<lb></lb>tur Area <emph type="italics"></emph>AIKP,<emph.end type="italics"></emph.end>eique æqualis <lb></lb>area <emph type="italics"></emph>OPA,<emph.end type="italics"></emph.end>quæ ſubducta de tri<lb></lb>angulo <emph type="italics"></emph>OPS<emph.end type="italics"></emph.end>relinquet aream ab<lb></lb>ſciſſam <emph type="italics"></emph>APS.<emph.end type="italics"></emph.end>Applicando areæ <lb></lb>abſcindendæ A & abſciſſæ <emph type="italics"></emph>APS<emph.end type="italics"></emph.end><lb></lb>differentiam duplam 2 <emph type="italics"></emph>APS<emph.end type="italics"></emph.end>-2 A <lb></lb>vel 2 A-2 <emph type="italics"></emph>APS<emph.end type="italics"></emph.end>ad lineam <emph type="italics"></emph>SN,<emph.end type="italics"></emph.end>quæ ab umbilico <emph type="italics"></emph>S<emph.end type="italics"></emph.end>in tangentem <lb></lb><emph type="italics"></emph>PT<emph.end type="italics"></emph.end>perpendicularis eſt, orietur longitudo chordæ <emph type="italics"></emph><expan abbr="Pq.">Pque</expan><emph.end type="italics"></emph.end>Inſcri<lb></lb>batur autem chorda illa <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>inter <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſi area abſciſſa <emph type="italics"></emph>APS<emph.end type="italics"></emph.end><lb></lb>major ſit area abſcindenda A, ſecus ad puncti <emph type="italics"></emph>P<emph.end type="italics"></emph.end>contrarias partes: <lb></lb>& punctum <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>erit locus corporis accuratior. </s> <s>Et computatione <lb></lb>repetita invenietur idem accuratior in perpetuum. </s></p> <p type="margin"> <s><margin.target id="note79"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Atque his calculis Problema generaliter confit Analytice. </s> <s>Ve<lb></lb>rum uſibus Aſtronomicis accommodatior eſt calculus particularis <lb></lb>qui ſequitur. </s> <s>Exiſtentibus <emph type="italics"></emph>AO, OB, OD<emph.end type="italics"></emph.end>ſemiaxibus Ellipſeos, & <lb></lb>L ipſius latere recto, ac D differentia inter ſemiaxem minorem <emph type="italics"></emph>OD<emph.end type="italics"></emph.end><lb></lb>& lateris recti ſemiſſem 1/2 L; quære tum angulum Y, cujus ſinus <lb></lb>ſit ad Radium ut eſt rectangu<lb></lb><figure id="id.039.01.131.2.jpg" xlink:href="039/01/131/2.jpg"></figure><lb></lb>lum ſub differentia illa D, & <lb></lb>ſemiſumma axium <emph type="italics"></emph>AO+OD<emph.end type="italics"></emph.end><lb></lb>ad quadratum axis majoris <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; <lb></lb>tum angulum Z, cujus ſinus <lb></lb>ſit ad Radium ut eſt duplum <lb></lb>rectangulum ſub umbilieorum <lb></lb>diſtantia <emph type="italics"></emph>SH<emph.end type="italics"></emph.end>& differentia <lb></lb>illa D ad triplum quadratum <lb></lb>ſemiaxis majoris <emph type="italics"></emph>AO.<emph.end type="italics"></emph.end>His <lb></lb>angulis ſemel inventis; locus corporis ſic deinceps determinabitur. </s> <s><lb></lb>Sume angulum T proportionalem tempori quo arcus <emph type="italics"></emph>BP<emph.end type="italics"></emph.end>deſcrip<lb></lb>tus eſt, ſcu motui medio (ut loquuntur) æqualem; & angulum <lb></lb>V (primam medii motus æquationem) ad angulum Y (æquatio<lb></lb>nem maximam primam) ut eſt ſinus dupli anguli T ad Radium; <pb xlink:href="039/01/132.jpg" pagenum="104"></pb><arrow.to.target n="note80"></arrow.to.target>atque angulum X (æquationem ſecundam) ad angulum Z (æqua<lb></lb>tionem maximam ſecundam) ut eſt cubus ſinus anguli T ad cubum <lb></lb>Radii. </s> <s>Angulorum T, V, X vel ſummæ T+X+V, ſi angulus <lb></lb>T recto minor eſt, vel differentiæ T+X-V, ſi is recto major eſt <lb></lb>rectiſQ.E.D.obus minor, æqualem cape angulum <emph type="italics"></emph>BHP<emph.end type="italics"></emph.end>(motum <lb></lb>medium æquatum;) &, ſi <emph type="italics"></emph>HP<emph.end type="italics"></emph.end>occurrat Ellipſi in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>acta <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ab<lb></lb>ſcindet aream <emph type="italics"></emph>BSP<emph.end type="italics"></emph.end>tempori proportionalem quamproxime. </s> <s>Hæc <lb></lb>Praxis ſatis expedita videtur, <lb></lb><figure id="id.039.01.132.1.jpg" xlink:href="039/01/132/1.jpg"></figure><lb></lb>propterea quod angulorum per<lb></lb>exiguorum V & X (in minutis <lb></lb>ſecundis, ſi placet, poſitorum) <lb></lb>figuras duas terſve primas in<lb></lb>venire ſufficit. </s> <s>Sed & ſatis ac<lb></lb>curata eſt ad Theoriam Planeta<lb></lb>rum. </s> <s>Nam in Orbe vel Martis <lb></lb>ipſius, cujus Æquatio centri ma<lb></lb>xima eſt graduum decem, error <lb></lb>vix ſuperabit minutum unum <lb></lb>ſecundum. </s> <s>Invento autem angulo motus medii æquati <emph type="italics"></emph>BHP,<emph.end type="italics"></emph.end>an<lb></lb>gulus veri motus <emph type="italics"></emph>BSP<emph.end type="italics"></emph.end>& diſtantia <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>in promptu ſunt per <lb></lb><emph type="italics"></emph>Wardi<emph.end type="italics"></emph.end>methodum notiſſimam. </s></p> <p type="margin"> <s><margin.target id="note80"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Hactenus de Motu corporum in lineis Curvis. </s> <s>Fieri autem po<lb></lb>teſt ut mobile recta deſcendat vel recta aſcendat, & quæ ad iſtiuſ<lb></lb>modi Motus ſpectant, pergo jam exponere. <pb xlink:href="039/01/133.jpg" pagenum="105"></pb><arrow.to.target n="note81"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note81"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Corporum Aſcenſu & Deſcenſu Rectilineo.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXII. PROBLEMA XXIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſito quod Vis centripeta ſit reciproce proportionalis quadrato di<lb></lb>ſtantiæ loeorum a centro, Spatia definire quæ corpus recta cadendo <lb></lb>datis temporibus deſcribit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Si Corpus non cadit perpendicu<lb></lb><figure id="id.039.01.133.1.jpg" xlink:href="039/01/133/1.jpg"></figure><lb></lb>lariter deſcribet id, per Corol. </s> <s>1. Prop. </s> <s>XIII, <lb></lb>Sectionem aliquam Conicam cujus umbili<lb></lb>cus congruit cum centro virium. </s> <s>Sit Sec<lb></lb>tio illa Conica <emph type="italics"></emph>ARPB<emph.end type="italics"></emph.end>& umbilicus ejus <emph type="italics"></emph>S.<emph.end type="italics"></emph.end><lb></lb>Et primo ſi Figura Ellipſis eſt, ſuper hu<lb></lb>jus axe majore <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>deſcribatur Semicirculus <lb></lb><emph type="italics"></emph>ADB,<emph.end type="italics"></emph.end>& per corpus decidens tranſeat rec<lb></lb>ta <emph type="italics"></emph>DPC<emph.end type="italics"></emph.end>perpendicularis ad axem; actiſque <lb></lb><emph type="italics"></emph>DS, PS<emph.end type="italics"></emph.end>erit area <emph type="italics"></emph>ASD<emph.end type="italics"></emph.end>areæ <emph type="italics"></emph>ASP<emph.end type="italics"></emph.end>at<lb></lb>que adeo etiam tempori proportionalis. </s> <s>Ma<lb></lb>nente axe <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>minuatur perpetuo latitudo <lb></lb>Ellipſeos, & ſemper manebit area <emph type="italics"></emph>ASD<emph.end type="italics"></emph.end><lb></lb>tempori proportionalis. </s> <s>Minuatur latitudo <lb></lb>illa in infinitum: &, Orbe <emph type="italics"></emph>APB<emph.end type="italics"></emph.end>jam coin<lb></lb>cidente cum axe <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& umbilico <emph type="italics"></emph>S<emph.end type="italics"></emph.end>cum <lb></lb>axis termino <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>deſcendet corpus in recta <lb></lb><emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>& area <emph type="italics"></emph>ABD<emph.end type="italics"></emph.end>evadet tempori pro<lb></lb>portionalis. </s> <s>Dabitur itaque Spatium <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end><lb></lb>quod corpus de loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>perpendiculariter <lb></lb>cadendo tempore dato deſcribit, ſi modo tempori proportiona<lb></lb>lis capiatur area <emph type="italics"></emph>ABD,<emph.end type="italics"></emph.end>& a puncto <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ad rectam <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>demit<lb></lb>tatur perpendicularis <emph type="italics"></emph>DC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end><pb xlink:href="039/01/134.jpg" pagenum="106"></pb><arrow.to.target n="note82"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note82"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Si Figura illa <emph type="italics"></emph>RPB<emph.end type="italics"></emph.end>Hyperbola eſt, deſcribatur ad ean<lb></lb>dem diametrum principalem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>Hyperbola rectangula <emph type="italics"></emph>BED:<emph.end type="italics"></emph.end><lb></lb>& quoniam areæ <emph type="italics"></emph>CSP, CBfP, SPfB<emph.end type="italics"></emph.end>ſunt ad areas <emph type="italics"></emph>CSD, <lb></lb>CBED, SDEB,<emph.end type="italics"></emph.end>ſingulæ ad ſingulas, in data ratione altitudi<lb></lb>num <emph type="italics"></emph>CP, CD<emph.end type="italics"></emph.end>; & area <emph type="italics"></emph>SPfB<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.134.1.jpg" xlink:href="039/01/134/1.jpg"></figure><lb></lb>proportionalis eſt tempori quo <lb></lb>corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>movebitur per arcum <lb></lb><emph type="italics"></emph>PfB<emph.end type="italics"></emph.end>; erit etiam area <emph type="italics"></emph>SDEB<emph.end type="italics"></emph.end>ei<lb></lb>dem tempori proportionalis. </s> <s><lb></lb>Minuatur latus rectum Hyper<lb></lb>bolæ <emph type="italics"></emph>RPB<emph.end type="italics"></emph.end>in infinitum ma<lb></lb>nente latere tranſverſo, & coibit <lb></lb>arcus <emph type="italics"></emph>PB<emph.end type="italics"></emph.end>cum recta <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>& um<lb></lb>bilicus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>cum vertice <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& recta <lb></lb><emph type="italics"></emph>SD<emph.end type="italics"></emph.end>cum recta <emph type="italics"></emph>BD.<emph.end type="italics"></emph.end>Proinde a<lb></lb>rea <emph type="italics"></emph>BDEB<emph.end type="italics"></emph.end>proportionalis erit <lb></lb>tempori quo corpus <emph type="italics"></emph>C<emph.end type="italics"></emph.end>recto <lb></lb>deſcenſu deſcribit lineam <emph type="italics"></emph>CB. <lb></lb><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Et ſimili argumento ſi <lb></lb>Figura <emph type="italics"></emph>RPB<emph.end type="italics"></emph.end>Parabola eſt, & <lb></lb>eodem vertice principali <emph type="italics"></emph>B<emph.end type="italics"></emph.end>de<lb></lb>ſcribatur alia Parabola <emph type="italics"></emph>BED,<emph.end type="italics"></emph.end><lb></lb>quæ ſemper maneat data interea <lb></lb>dum Parabola prior in cujus perimetro corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>movetur, dimi<lb></lb>nuto & in nihilum redacto ejus latere recto, conveniat cum linea <lb></lb><emph type="italics"></emph>CB<emph.end type="italics"></emph.end>; fiet ſegmentum Parabolicum <emph type="italics"></emph>BDEB<emph.end type="italics"></emph.end>proportionale tempori <lb></lb>quo corpus illud <emph type="italics"></emph>P<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>C<emph.end type="italics"></emph.end>deſcendet ad centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>B. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXIII. THEOREMA IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſitis jam inventis, dico quod corporis cadentis Velocitas in loco quo<lb></lb>vis<emph.end type="italics"></emph.end>C <emph type="italics"></emph>est ad velocitatem corporis centro<emph.end type="italics"></emph.end>B <emph type="italics"></emph>intervallo<emph.end type="italics"></emph.end>BC <emph type="italics"></emph>Circu<lb></lb>lum deſcribentis, in ſubduplicata ratione quam<emph.end type="italics"></emph.end>AC, <emph type="italics"></emph>diſtantia cor<lb></lb>poris a Circuli vel Hyperbolæ rect angulæ vertice ulteriore<emph.end type="italics"></emph.end>A, <emph type="italics"></emph>habet <lb></lb>ad Figuræ ſemidiametrum principalem<emph.end type="italics"></emph.end>1/2 AB. </s></p> <p type="main"> <s>Biſecetur <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>communis utriuſque Figuræ <emph type="italics"></emph>RPB, DEB<emph.end type="italics"></emph.end>dia<lb></lb>meter, in <emph type="italics"></emph>O<emph.end type="italics"></emph.end>; & agatur recta <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>quæ tangat Figuram <emph type="italics"></emph>RPB<emph.end type="italics"></emph.end>in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>atque <pb xlink:href="039/01/135.jpg" pagenum="107"></pb>etiam ſecet communem illam diametrum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>(ſi opus eſt productam) </s></p> <p type="main"> <s><arrow.to.target n="note83"></arrow.to.target>in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>; ſitque <emph type="italics"></emph>SY<emph.end type="italics"></emph.end>ad hanc rectam, & <emph type="italics"></emph>BQ<emph.end type="italics"></emph.end>ad <lb></lb><figure id="id.039.01.135.1.jpg" xlink:href="039/01/135/1.jpg"></figure><lb></lb>hanc diametrum perpendicularis, atque Figu<lb></lb>ræ <emph type="italics"></emph>RPB<emph.end type="italics"></emph.end>latus rectum ponatur L. </s> <s>Conſtat <lb></lb>per Cor. </s> <s>9. Prop. </s> <s>XVI, quod corporis in <lb></lb>linea <emph type="italics"></emph>RPB<emph.end type="italics"></emph.end>circa centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>moventis velo<lb></lb>citas in loco quovis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ſit ad velocitatem cor<lb></lb>poris intervallo <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>circa idem centrum Cir<lb></lb>culum deſcribentis in ſubduplicata ratione rec<lb></lb>tanguli 1/2 LX<emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SY<emph.end type="italics"></emph.end>quadratum. </s> <s>Eſt au<lb></lb>tem ex Conicis <emph type="italics"></emph>ACB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CPq<emph.end type="italics"></emph.end>ut 2 <emph type="italics"></emph>AO<emph.end type="italics"></emph.end>ad L, <lb></lb>adeoque (2<emph type="italics"></emph>CPqXAO/ACB<emph.end type="italics"></emph.end>) æquale L. </s> <s>Ergo ve<lb></lb>locitates illæ ſunt ad invicem in ſubduplicata <lb></lb>ratione (<emph type="italics"></emph>CPqXAOXSP/ACB<emph.end type="italics"></emph.end>) ad <emph type="italics"></emph>SY quad.<emph.end type="italics"></emph.end>Por<lb></lb>ro ex Conicis eſt <emph type="italics"></emph>CO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BO<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>BO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TO,<emph.end type="italics"></emph.end><lb></lb>& compoſite vel diviſim ut <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BT.<emph.end type="italics"></emph.end><lb></lb>Unde vel dividendo vel componendo fit <lb></lb><emph type="italics"></emph>BO<emph.end type="italics"></emph.end>-vel+<emph type="italics"></emph>CO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BO<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BT,<emph.end type="italics"></emph.end>id eſt <lb></lb><emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AO<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BQ<emph.end type="italics"></emph.end>; indeque (<emph type="italics"></emph>CPqXAOXSP/ACB<emph.end type="italics"></emph.end>) æquale eſt <lb></lb>(<emph type="italics"></emph>BQqXACXSP/AOXBC.<emph.end type="italics"></emph.end>) Minuatur jam in infinitum Figuræ <emph type="italics"></emph>RPB<emph.end type="italics"></emph.end>latitu<lb></lb>do <emph type="italics"></emph>CP,<emph.end type="italics"></emph.end>ſic ut punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>coeat cum puncto <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>punctumque <emph type="italics"></emph>S<emph.end type="italics"></emph.end>cum <lb></lb>puncto <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& linea <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>cum linea <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>lineaque <emph type="italics"></emph>SY<emph.end type="italics"></emph.end>cum linea <emph type="italics"></emph>BQ<emph.end type="italics"></emph.end>; <lb></lb>& corporis jam recta deſcendentis in linea <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>velocitas fiet ad <lb></lb>velocitatem corporis centro <emph type="italics"></emph>B<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>Circulum deſcribentis, <lb></lb>in ſubduplicata ratione ipſius (<emph type="italics"></emph>BQqXACXSP/AOXBC<emph.end type="italics"></emph.end>) ad <emph type="italics"></emph>SYq,<emph.end type="italics"></emph.end>hoc eſt (neg<lb></lb>lectis æqualitatis rationibus <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BQq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SYq<emph.end type="italics"></emph.end>) in ſub<lb></lb>duplicata ratione <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AO<emph.end type="italics"></emph.end>ſive 1/2 <emph type="italics"></emph>AB. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note83"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Punctis <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end>coeuntibus, fit <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TS<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AC<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>AO.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Corpus ad datam a centro diſtantiam in Circulo quo<lb></lb>vis revolvens, motu ſuo ſurſum verſo aſcendet ad duplam ſuam a <lb></lb>centro diſtantiam. <pb xlink:href="039/01/136.jpg" pagenum="108"></pb><arrow.to.target n="note84"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note84"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXIV. THEOREMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Figura<emph.end type="italics"></emph.end>BED <emph type="italics"></emph>Parabola eſt, dico<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.136.1.jpg" xlink:href="039/01/136/1.jpg"></figure><lb></lb><emph type="italics"></emph>quod corporis cadentis Veloci<lb></lb>tas in loco quovis<emph.end type="italics"></emph.end>C <emph type="italics"></emph>æqualis eſt <lb></lb>velocitati qua corpus centro<emph.end type="italics"></emph.end>B <lb></lb><emph type="italics"></emph>dimidio intervalli ſui<emph.end type="italics"></emph.end>BC <emph type="italics"></emph>Cir<lb></lb>culum uniformiter deſcribere <lb></lb>potest.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam corporis Parabolam <lb></lb><emph type="italics"></emph>RPB<emph.end type="italics"></emph.end>circa centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>deſcri<lb></lb>bentis velocitas in loco quovis <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>(per Corol. </s> <s>7. Prop. </s> <s>XVI) æ<lb></lb>qualis eſt velocitati corporis di<lb></lb>midio intervalli <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>Circulum cir<lb></lb>ca idem centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>uniformiter <lb></lb>deſcribentis. </s> <s>Minuatur Parabolæ <lb></lb>latitudo <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>in infinitum eo, ut <lb></lb>arcus Parabolicus <emph type="italics"></emph>PfB<emph.end type="italics"></emph.end>cum rec<lb></lb>ta <emph type="italics"></emph>CB,<emph.end type="italics"></emph.end>centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>cum vertice <emph type="italics"></emph>B,<emph.end type="italics"></emph.end><lb></lb>& intervallum <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>cum intervallo <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>coincidat, & conſtabit Pro<lb></lb>poſitio. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXV. THEOREMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, dico quod area Figuræ<emph.end type="italics"></emph.end>DES, <emph type="italics"></emph>radio indefinito<emph.end type="italics"></emph.end>SD <emph type="italics"></emph>de<lb></lb>ſcripta, æqualis ſit areæ quam corpus, radio dimidium lateris recti <lb></lb>Figuræ<emph.end type="italics"></emph.end>DES <emph type="italics"></emph>æquante, circa centrum<emph.end type="italics"></emph.end>S <emph type="italics"></emph>uniformiter gyrando, eo<lb></lb>dem tempore deſcribere potest.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam concipe corpus <emph type="italics"></emph>C<emph.end type="italics"></emph.end>quam minima temporis particula lineo<lb></lb>lam <emph type="italics"></emph>Cc<emph.end type="italics"></emph.end>cadendo deſcribere, & interea corpus aliud <emph type="italics"></emph>K,<emph.end type="italics"></emph.end>uniformi<lb></lb>ter in Circulo <emph type="italics"></emph>OKk<emph.end type="italics"></emph.end>circa centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>gyrando, arcum <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>deſcri<lb></lb>bere. </s> <s>Erigantur perpendicula <emph type="italics"></emph>CD, cd<emph.end type="italics"></emph.end>occurrentia Figuræ <emph type="italics"></emph>DES<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>D, d.<emph.end type="italics"></emph.end>Jungantur <emph type="italics"></emph>SD, Sd, SK, Sk<emph.end type="italics"></emph.end>& ducatur <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>axi <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>oc<lb></lb>rens in <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>& ad eam demittatur perpendiculum <emph type="italics"></emph>SY.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/137.jpg" pagenum="109"></pb> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>1. Jam ſi Figura <emph type="italics"></emph>DES<emph.end type="italics"></emph.end>Circulus eſt vel Hyperbola, biſece<lb></lb><arrow.to.target n="note85"></arrow.to.target>tur ejus tranſverſa diameter <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>in <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>& erit <lb></lb><figure id="id.039.01.137.1.jpg" xlink:href="039/01/137/1.jpg"></figure><lb></lb><emph type="italics"></emph>SO<emph.end type="italics"></emph.end>dimidium lateris recti. </s> <s>Et quoniam eſt <lb></lb><emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Cc<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>TD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TS<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SY,<emph.end type="italics"></emph.end>erit ex æquo <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TS<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>CDXCc<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SYXDd.<emph.end type="italics"></emph.end>Sed per Corol. </s> <s>1. Prop. </s> <s><lb></lb>XXXIII, eſt <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TS<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AO,<emph.end type="italics"></emph.end>puta ſi <lb></lb>in coitu punctorum <emph type="italics"></emph>D, d<emph.end type="italics"></emph.end>capiantur linearum <lb></lb>rationes ultimæ. </s> <s>Ergo <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>eſt ad (<emph type="italics"></emph>AO<emph.end type="italics"></emph.end>ſeu) <emph type="italics"></emph>SK<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>CDXCc<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SYXDd.<emph.end type="italics"></emph.end>Porro corporis <lb></lb>deſcendentis velocitas in <emph type="italics"></emph>C<emph.end type="italics"></emph.end>eſt ad velocitatem <lb></lb>corporis Circulum intervallo <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>circa cen<lb></lb>trum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>deſcribentis in ſubduplicata ratione <lb></lb><emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad (<emph type="italics"></emph>AO<emph.end type="italics"></emph.end>vel) <emph type="italics"></emph>SK<emph.end type="italics"></emph.end>(per Prop. </s> <s>XXXIII.) Et <lb></lb>hæc velocitas ad velocitatem corporis deſcri<lb></lb>bentis Circulum <emph type="italics"></emph>OKk<emph.end type="italics"></emph.end>in ſubduplicata ratione <lb></lb><emph type="italics"></emph>SK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>per Cor. </s> <s>6. Prop. </s> <s>IV, & ex æquo velo<lb></lb>citas prima ad ultimam, hoc eſt lineola <emph type="italics"></emph>Cc<emph.end type="italics"></emph.end>ad <lb></lb>arcum <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>in ſubduplicata ratione <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SC,<emph.end type="italics"></emph.end><lb></lb>id eſt in ratione <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CD.<emph.end type="italics"></emph.end>Quare eſt <emph type="italics"></emph>CDXCc<emph.end type="italics"></emph.end><lb></lb>æquale <emph type="italics"></emph>ACXKk,<emph.end type="italics"></emph.end>& propterea <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SK<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>ACXKk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SYXDd,<emph.end type="italics"></emph.end><expan abbr="indeq;">indeque</expan> <emph type="italics"></emph>SKXKk<emph.end type="italics"></emph.end>æqua<lb></lb>le <emph type="italics"></emph>SYXDd,<emph.end type="italics"></emph.end>& 1/2 <emph type="italics"></emph>SKXKk<emph.end type="italics"></emph.end>æquale 1/2 <emph type="italics"></emph>SYXDd,<emph.end type="italics"></emph.end><lb></lb>id eſt area <emph type="italics"></emph>KSk<emph.end type="italics"></emph.end>æqualis areæ <emph type="italics"></emph>SDd.<emph.end type="italics"></emph.end>Singulis <lb></lb>igitur temporis particulis generantur arearum <lb></lb>duarum particulæ <emph type="italics"></emph>KSk,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SDd,<emph.end type="italics"></emph.end>quæ, ſi mag<lb></lb>nitudo earum minuatur & numerus augeatur in infinitum, ratio<lb></lb>nem obtinent æqualitatis, & propterea (per Corollarium Lem<lb></lb>matis IV) areæ totæ ſimul genitæ ſunt ſemper æquales, <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note85"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>2. Quod ſi Figura <emph type="italics"></emph>DES<emph.end type="italics"></emph.end>Parabola ſit, invenietur eſſe ut ſu<lb></lb>pra <emph type="italics"></emph>CDXCc<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SYXDd<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TS,<emph.end type="italics"></emph.end>hoc eſt ut 2 ad 1, ad<lb></lb>eoque 1/4 <emph type="italics"></emph>CDXCc<emph.end type="italics"></emph.end>æquale eſſe 1/2 <emph type="italics"></emph>SYXDd.<emph.end type="italics"></emph.end>Sed corporis caden<lb></lb>tis velocitas in <emph type="italics"></emph>C<emph.end type="italics"></emph.end>æqualis eſt velocitati qua Circulus intervallo 1/2 <emph type="italics"></emph>SC<emph.end type="italics"></emph.end><lb></lb>uniformiter deſcribi poſſit (per Prop. </s> <s>XXXIV) Et hæc velocitas ad ve<lb></lb>locitatem qua Circulus radio <emph type="italics"></emph>SK<emph.end type="italics"></emph.end>deſcribi poſſit, hoc eſt, lineola <lb></lb><emph type="italics"></emph>Cc<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>(per Corol. </s> <s>6. Prop. </s> <s>IV) eſt in ſubduplicata ratione <lb></lb><emph type="italics"></emph>SK<emph.end type="italics"></emph.end>ad 1/2 <emph type="italics"></emph>SC,<emph.end type="italics"></emph.end>id eſt, in ratione <emph type="italics"></emph>SK<emph.end type="italics"></emph.end>ad 1/2 <emph type="italics"></emph>CD.<emph.end type="italics"></emph.end>Quare eſt 1/2 <emph type="italics"></emph>SKXKk<emph.end type="italics"></emph.end><lb></lb>æquale 1/4 <emph type="italics"></emph>CDXCc,<emph.end type="italics"></emph.end>adeoque æquale 1/2 <emph type="italics"></emph>SYXDd,<emph.end type="italics"></emph.end>hoc eſt, area <emph type="italics"></emph>KSk<emph.end type="italics"></emph.end><lb></lb>æqualis areæ <emph type="italics"></emph>SDd,<emph.end type="italics"></emph.end>ut ſupra. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/138.jpg" pagenum="110"></pb><arrow.to.target n="note86"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note86"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXVI. PROBLEMA XXV.<emph.end type="center"></emph.end></s></p><figure id="id.039.01.138.1.jpg" xlink:href="039/01/138/1.jpg"></figure> <p type="main"> <s><emph type="italics"></emph>Corporis de loco dato<emph.end type="italics"></emph.end>A <emph type="italics"></emph>cadentis determinare Tem<lb></lb>pora deſcenſus.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Super diametro <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>(diſtantia corporis a cen<lb></lb>tro ſub initio) deſcribe Semicirculum <emph type="italics"></emph>ADS,<emph.end type="italics"></emph.end>ut & <lb></lb>huic æqualem Semicirculum <emph type="italics"></emph>OKH<emph.end type="italics"></emph.end>circa centrum <lb></lb><emph type="italics"></emph>S.<emph.end type="italics"></emph.end>De corporis loco quovis <emph type="italics"></emph>C<emph.end type="italics"></emph.end>erige ordinatim ap<lb></lb>plicatam <emph type="italics"></emph>CD.<emph.end type="italics"></emph.end>Junge <emph type="italics"></emph>SD,<emph.end type="italics"></emph.end>& areæ <emph type="italics"></emph>ASD<emph.end type="italics"></emph.end>æqua<lb></lb>lem conſtitue ſectorem <emph type="italics"></emph>OSK.<emph.end type="italics"></emph.end>Patet per Prop.<lb></lb>XXXV, quod corpus cadendo deſcribet ſpatium <emph type="italics"></emph>AC<emph.end type="italics"></emph.end><lb></lb>eodem Tempore quo corpus aliud uniformiter cir<lb></lb>ca centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>gyrando, deſcribere poteſt arcum <lb></lb><emph type="italics"></emph>OK. <expan abbr="q.">que</expan> E. F.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXVII. PROBLEMA XXVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Corporis de loco dato ſurſum vel deorſum projecti definire Tempora <lb></lb>aſcenſus vel deſcenſus.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Exeat corpus de loco dato <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ſecundum <lb></lb><figure id="id.039.01.138.2.jpg" xlink:href="039/01/138/2.jpg"></figure><lb></lb>lineam <emph type="italics"></emph>ASG<emph.end type="italics"></emph.end>cum velocitate quacunque. </s> <s><lb></lb>In duplicata ratione hujus velocitatis ad <lb></lb>uniformem in Circulo velocitatem, qua cor<lb></lb>pus ad intervallum datum <emph type="italics"></emph>SG<emph.end type="italics"></emph.end>circa centrum <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end>revolvi poſſet, cape <emph type="italics"></emph>GA<emph.end type="italics"></emph.end>ad 1/2 <emph type="italics"></emph>AS.<emph.end type="italics"></emph.end><lb></lb>Si ratio illa eſt numeri binarii ad unita<lb></lb>tem, punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>infinite diſtat, quo ca<lb></lb>ſu Parabola vertice <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>axe <emph type="italics"></emph>SC,<emph.end type="italics"></emph.end>latere quo<lb></lb>vis recto deſcribenda eſt. </s> <s>Patet hoc per <lb></lb>Prop. </s> <s>XXXIV. </s> <s>Sin ratio illa minor vel ma<lb></lb>jor eſt quam 2 ad 1, priore caſu Circulus, <lb></lb>poſteriore Hyperbola rectangula ſuper di<lb></lb>ametro <emph type="italics"></emph>SA<emph.end type="italics"></emph.end>deſcribi debet. </s> <s>Patet per <lb></lb>Prop. </s> <s>XXXIII. </s> <s>Tum centro <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>intervallo <lb></lb>æquante dimidium lateris recti, deſcribatur <lb></lb>Circulus <emph type="italics"></emph>HKk,<emph.end type="italics"></emph.end>& ad corporis aſcenden<lb></lb>tis vel deſcendentis loca duo quævis <emph type="italics"></emph>G, C,<emph.end type="italics"></emph.end><lb></lb>erigantur perpendicula <emph type="italics"></emph>GI, CD<emph.end type="italics"></emph.end>occurren<lb></lb>tia Conicæ Sectioni vel Circulo in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/139.jpg" pagenum="111"></pb>Dein junctis <emph type="italics"></emph>SI, SD,<emph.end type="italics"></emph.end>fiant ſegmentis <emph type="italics"></emph>SEIS, SEDS,<emph.end type="italics"></emph.end>ſec<lb></lb><arrow.to.target n="note87"></arrow.to.target>tores <emph type="italics"></emph>HSK, HSk<emph.end type="italics"></emph.end>æquales, & per Prop. </s> <s>XXXV, corpus <emph type="italics"></emph>G<emph.end type="italics"></emph.end>deſcri<lb></lb>bet ſpatium <emph type="italics"></emph>GC<emph.end type="italics"></emph.end>eodem Tempore quo corpus <emph type="italics"></emph>K<emph.end type="italics"></emph.end>deſcribere po<lb></lb>teſt arcum <emph type="italics"></emph>Kk. </s> <s><expan abbr="q.">que</expan> E. F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note87"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXVIII. THEOREMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſito quod Vis centripeta proportionalis ſit altitudini ſeu diſtantiæ lo<lb></lb>eorum a centro, dico quod cadentium Tempora, Velocitates & Spa<lb></lb>tia deſcripta ſunt arcubus, arcuumque finibus rectis & ſinibus <lb></lb>verſis reſpective proportionalia.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Cadat corpus de loco quovis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſecun<lb></lb><figure id="id.039.01.139.1.jpg" xlink:href="039/01/139/1.jpg"></figure><lb></lb>dum rectam <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>; & centro virium <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>in<lb></lb>tervallo <emph type="italics"></emph>AS,<emph.end type="italics"></emph.end>deſcribatur Circuli quadrans <lb></lb><emph type="italics"></emph>AE,<emph.end type="italics"></emph.end>ſitque <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ſinus rectus arcus cujuſ<lb></lb>vis <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>; & corpus <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>Tempore <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>ca<lb></lb>dendo deſcribet Spatium <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>inque loco <lb></lb><emph type="italics"></emph>C<emph.end type="italics"></emph.end>acquiret Velocitatem <emph type="italics"></emph>CD.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Demonſtratur eodem modo ex Propoſi<lb></lb>tione X, quo Propoſitio XXXII, ex Propo<lb></lb>ſitione XI demonſtrata fuit. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc æqualia ſunt Tempora quibus corpus unum de loco <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>cadendo pervenit ad centrum <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>& corpus aliud revolvendo de<lb></lb>ſcribit arcum quadrantalem <emph type="italics"></emph>ADE.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Proinde æqualia ſunt Tempora omnia quibus corpora de <lb></lb>locis quibuſvis ad uſque centrum cadunt. </s> <s>Nam revolventium tem<lb></lb>pora omnia periodica (per Corol. </s> <s>3. Prop. </s> <s>IV.) æquantur. <pb xlink:href="039/01/140.jpg" pagenum="112"></pb><arrow.to.target n="note88"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note88"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXIX. PROBLEMA XXVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſita cujuſcunque generis Vi centripeta, & conceſſis figurarum <lb></lb>curvilinearum quadraturis, requiritu, corporis recta aſcenden<lb></lb>tis vel deſcendentis tum Velocitas in locis ſingulis, tum Tempus <lb></lb>quo corpus ad locum quemvis perveniet: Et contra.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>De loco quovis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in recta <emph type="italics"></emph>ADEC<emph.end type="italics"></emph.end>cadat corpus <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>deque loco <lb></lb>ejus <emph type="italics"></emph>E<emph.end type="italics"></emph.end>erigatur ſemper perpendicularis <emph type="italics"></emph>EG,<emph.end type="italics"></emph.end>vi centripetæ in loco <lb></lb>illo ad centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>tendenti proportio<lb></lb><figure id="id.039.01.140.1.jpg" xlink:href="039/01/140/1.jpg"></figure><lb></lb>nalis: Sitque <emph type="italics"></emph>BFG<emph.end type="italics"></emph.end>linea curva quam <lb></lb>punctum <emph type="italics"></emph>G<emph.end type="italics"></emph.end>perpetuo tangit. </s> <s>Coinci<lb></lb>dat autem <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>ipſo motus initio cum <lb></lb>perpendiculari <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>& erit corporis Ve<lb></lb>locitas in loco quovis <emph type="italics"></emph>E<emph.end type="italics"></emph.end>ut areæ cur<lb></lb>vilineæ <emph type="italics"></emph>ABGE<emph.end type="italics"></emph.end>latus quadratum. <lb></lb><emph type="italics"></emph><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>In <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>EM<emph.end type="italics"></emph.end>lateri quadra<lb></lb>to areæ <emph type="italics"></emph>ABGE<emph.end type="italics"></emph.end>reciproce proportio<lb></lb>nalis, & ſit <emph type="italics"></emph>ALM<emph.end type="italics"></emph.end>linea curva quam <lb></lb>punctum <emph type="italics"></emph>M<emph.end type="italics"></emph.end>perpetuotangit, & erit Tem<lb></lb>pus quo corpus cadendo deſcribit li<lb></lb>neam <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>ut area curvilinea <emph type="italics"></emph>ALME. <lb></lb><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Etenim in recta <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>capiatur linea <lb></lb>quam minima <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>datæ longitudinis, <lb></lb>ſitque <emph type="italics"></emph>DLF<emph.end type="italics"></emph.end>locus lineæ <emph type="italics"></emph>EMG<emph.end type="italics"></emph.end>ubi <lb></lb>corpus verſabatur in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>; & ſi ea ſit vis centripeta, ut areæ <emph type="italics"></emph>ABGE<emph.end type="italics"></emph.end><lb></lb>latus quadratum ſit ut deſcendentis velocitas, erit area ipſa in du<lb></lb>plicata ratione velocitatis, id eſt, ſi pro velocitatibus in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>E<emph.end type="italics"></emph.end><lb></lb>ſcribantur V & V+I, erit area <emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>ut VV, & area <emph type="italics"></emph>ABGE<emph.end type="italics"></emph.end>ut <lb></lb>VV+2 VI+II, & diviſim area <emph type="italics"></emph>DFGE<emph.end type="italics"></emph.end>ut 2 VI+II, adeoque <lb></lb>(<emph type="italics"></emph>DFGE/DE<emph.end type="italics"></emph.end>) ut (2VI+II/<emph type="italics"></emph>DE<emph.end type="italics"></emph.end>), id eſt, ſi primæ quantitatum naſcentium <lb></lb>rationes ſumantur, longitudo <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>ut quantitas (2VI/<emph type="italics"></emph>DE<emph.end type="italics"></emph.end>), adeoque e<lb></lb>tiam ut quantitatis hujus dimidium (IXV/<emph type="italics"></emph>DE<emph.end type="italics"></emph.end>). Eſt autem tempus quo <pb xlink:href="039/01/141.jpg" pagenum="113"></pb>corpus cadendo deſcribit lineolam <emph type="italics"></emph>DE,<emph.end type="italics"></emph.end>ut lineola illa directe & <lb></lb><arrow.to.target n="note89"></arrow.to.target>velocitas V inverſe, eſtque vis ut velocitatis incrementum I directe <lb></lb>& tempus inverſe, adeoque ſi primæ naſcentium rationes ſuman<lb></lb>tur, ut (IXV/<emph type="italics"></emph>DE<emph.end type="italics"></emph.end>), hoc eſt, ut longitudo <emph type="italics"></emph>DF.<emph.end type="italics"></emph.end>Ergo Vis ipſi <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>EG<emph.end type="italics"></emph.end><lb></lb>proportionalis facit ut corpus ea cum Velocitate deſcendat quæ ſit <lb></lb>ut areæ <emph type="italics"></emph>ABGE<emph.end type="italics"></emph.end>latus quadratum. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note89"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Porro cum tempus, quo quælibet longitudinis datæ lineola <emph type="italics"></emph>DE<emph.end type="italics"></emph.end><lb></lb>deſcribatur, ſit ut velocitas inverſe adeoque ut areæ <emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>latus <lb></lb>quadratum inverſe; ſitque <emph type="italics"></emph>DL,<emph.end type="italics"></emph.end>atque adeo area naſcens <emph type="italics"></emph>DLME,<emph.end type="italics"></emph.end><lb></lb>ut idem latus quadratum inverſe: erit tempus ut area <emph type="italics"></emph>DLME,<emph.end type="italics"></emph.end>& <lb></lb>ſumma omnium temporum ut ſumma omnium arearum, hoc eſt <lb></lb>(per Corol. </s> <s>Lem. </s> <s>IV) Tempus totum quo linea <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>deſcribitur ut <lb></lb>area tota <emph type="italics"></emph>AME. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Si <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ſit locus de quo corpus cadere debet, ut, urgen<lb></lb>te aliqua uniformi vi centripeta nota (qualis vulgo ſupponitur <lb></lb>Gravitas) velocitatem acquirat in loco <emph type="italics"></emph>D<emph.end type="italics"></emph.end>æqualem velocitati <lb></lb>quam corpus aliud vi quacunque cadens acquiſivit eodem loco <emph type="italics"></emph>D,<emph.end type="italics"></emph.end><lb></lb>& in perpendiculari <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>DR,<emph.end type="italics"></emph.end>quæ ſit ad <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>ut vis illa <lb></lb>uniformis ad vim alteram in loco <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>& compleatur rectangulum <lb></lb><emph type="italics"></emph>PDRQ,<emph.end type="italics"></emph.end>eique æqualis abſcindatur area <emph type="italics"></emph>ABFD;<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>A<emph.end type="italics"></emph.end>locus <lb></lb>de quo corpus alterum cecidit. </s> <s>Namque completo rectangulo <lb></lb><emph type="italics"></emph>DRSE,<emph.end type="italics"></emph.end>cum ſit area <emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>ad aream <emph type="italics"></emph>DFGE<emph.end type="italics"></emph.end>ut VV ad <lb></lb>2VI, adeoque ut 1/2 V ad I, id eſt, ut ſemiſſis velocitatis totius <lb></lb>ad incrementum velocitatis corporis vi inæquabili cadentis; & ſi<lb></lb>militer area <emph type="italics"></emph>PQRD<emph.end type="italics"></emph.end>ad aream <emph type="italics"></emph>DRSE<emph.end type="italics"></emph.end>ut ſemiſſis velocitatis to<lb></lb>tius ad incrementum velocitatis corporis uniformi vi cadentis; <lb></lb>ſintQ.E.I.crementa illa (ob æqualitatem temporum naſcentium) <lb></lb>ut vires generatrices, id eſt, ut ordinatim applicatæ <emph type="italics"></emph>DF, DR,<emph.end type="italics"></emph.end><lb></lb>adeoque ut areæ naſcentes <emph type="italics"></emph>DFGE, DRSE<emph.end type="italics"></emph.end>; erunt (ex æquo) <lb></lb>areæ totæ <emph type="italics"></emph>ABFD, PQRD<emph.end type="italics"></emph.end>ad invicem ut ſemiſſes totarum ve<lb></lb>locitatum, & propterea (ob æqualitatem velocitatum) æquantur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Unde ſi corpus quodlibet de loco quocunque <emph type="italics"></emph>D<emph.end type="italics"></emph.end>data <lb></lb>cum velocitate vel ſurſum vel deorſum projiciatur, & detur lex vis <lb></lb>centripetæ, invenietur velocitas ejus in alio quovis loco <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>erigen<lb></lb>do ordinatam <emph type="italics"></emph>eg,<emph.end type="italics"></emph.end>& capiendo velocitatem illam ad velocitatem in <lb></lb>loco <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ut eſt latus quadratum rectanguli <emph type="italics"></emph>PQRD<emph.end type="italics"></emph.end>area curvili<lb></lb>nea <emph type="italics"></emph>DFge<emph.end type="italics"></emph.end>vel aucti, ſi locus <emph type="italics"></emph>e<emph.end type="italics"></emph.end>eſt loco <emph type="italics"></emph>D<emph.end type="italics"></emph.end>inferior, vel diminuti, <lb></lb>ſi is ſuperior eſt, ad latus quadratum rectanguli ſolius <emph type="italics"></emph>PQRD,<emph.end type="italics"></emph.end>id <lb></lb>eſt, ut √<emph type="italics"></emph>PQRD<emph.end type="italics"></emph.end>+vel-<emph type="italics"></emph>DFge<emph.end type="italics"></emph.end>ad √<emph type="italics"></emph>PQRD.<emph.end type="italics"></emph.end><pb xlink:href="039/01/142.jpg" pagenum="114"></pb><arrow.to.target n="note90"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note90"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Tempus quoQ.E.I.noteſcet erigendo ordinatam <emph type="italics"></emph>em<emph.end type="italics"></emph.end>re<lb></lb>ciproce proportionalem lateri quadrato ex <emph type="italics"></emph>PQRD<emph.end type="italics"></emph.end>+vel-<emph type="italics"></emph>DFge,<emph.end type="italics"></emph.end><lb></lb>& capiendo tempus quo corpus deſcripſit lineam <emph type="italics"></emph>De<emph.end type="italics"></emph.end>ad tempus <lb></lb>quo corpus alterum vi uniformi cecidit a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& cadendo pervenit ad <lb></lb><emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ut area curvilinea <emph type="italics"></emph>DLme<emph.end type="italics"></emph.end>ad rectangulum 2<emph type="italics"></emph>PDXDL.<emph.end type="italics"></emph.end>Nam<lb></lb>que tempus quo corpus vi uniformi deſcendens deſcripſit lineam <lb></lb><emph type="italics"></emph>PD<emph.end type="italics"></emph.end>eſt ad tempus quo corpus idem deſcripſit lineam <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>in ſub<lb></lb>duplicata ratione <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PE,<emph.end type="italics"></emph.end>id eſt (lineola <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>jamjam naſcen<lb></lb>te) in ratione <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>+1/2 <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>ſeu 2<emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>PD+DE,<emph.end type="italics"></emph.end><lb></lb>& diviſim, ad tempus quo corpus idem deſcripſit lineolam <emph type="italics"></emph>DE<emph.end type="italics"></emph.end><lb></lb>ut 2<emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DE,<emph.end type="italics"></emph.end>adeoque ut rectangulum 2<emph type="italics"></emph>PDXDL<emph.end type="italics"></emph.end>ad aream <lb></lb><emph type="italics"></emph>DLME<emph.end type="italics"></emph.end>; eſtque tempus quo corpus utrumQ.E.D.ſcripſit lineo<lb></lb>lam <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>ad tempus quo corpus alterum inæquabili motu deſcrip<lb></lb>ſit lineam <emph type="italics"></emph>De<emph.end type="italics"></emph.end>ut area <emph type="italics"></emph>DLME<emph.end type="italics"></emph.end>ad aream <emph type="italics"></emph>DLme,<emph.end type="italics"></emph.end>& ex æquo <lb></lb>tempus primum ad tempus ultimum ut rectangulum 2<emph type="italics"></emph>PDXDL<emph.end type="italics"></emph.end><lb></lb>ad aream <emph type="italics"></emph>DLme.<emph.end type="italics"></emph.end></s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"></emph>SECTIO VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Inventione Orbium in quibus corpora Viribus quibuſcunque cen<lb></lb>tripetis agitata revolvuntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XL. THEOREMA XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpus, cogente Vi quacunque centripeta, moveatur utcunque, & <lb></lb>corpus aliud recta aſcendat vel deſcendat, ſintque eorum Velocita<lb></lb>tes in aliquo æqualium altitudinum caſu æquales, Velocitates eorum <lb></lb>in omnibus æqualibus altitudinibus erunt æquales.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Deſcendat corpus aliquod ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>per <emph type="italics"></emph>D, E,<emph.end type="italics"></emph.end>ad centrum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>& <lb></lb>moveatur corpus aliud a <emph type="italics"></emph>V<emph.end type="italics"></emph.end>in linea curva <emph type="italics"></emph>VIKk,<emph.end type="italics"></emph.end>Centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>in<lb></lb>tervallis quibuſvis deſcribantur circuli concentrici <emph type="italics"></emph>DI, EK<emph.end type="italics"></emph.end>rectæ <lb></lb><emph type="italics"></emph>AC<emph.end type="italics"></emph.end>in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>curvæque <emph type="italics"></emph>VIK<emph.end type="italics"></emph.end>in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end>occurrentes. </s> <s>Junga<lb></lb>tur <emph type="italics"></emph>IC<emph.end type="italics"></emph.end>occurrens ipſi <emph type="italics"></emph>KE<emph.end type="italics"></emph.end>in <emph type="italics"></emph>N;<emph.end type="italics"></emph.end>& in <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>demittatur perpendi<lb></lb>culum <emph type="italics"></emph>NT<emph.end type="italics"></emph.end>; ſitque circumferentiarum circulorum intervallum <emph type="italics"></emph>DE<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>IN<emph.end type="italics"></emph.end>quam minimum, & habeant corpora in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I<emph.end type="italics"></emph.end>velocita-<pb xlink:href="039/01/143.jpg" pagenum="115"></pb>tes æquales. </s> <s>Quoniam diſtantiæ <emph type="italics"></emph>CD, CI<emph.end type="italics"></emph.end>æquantur, erunt vi</s></p> <p type="main"> <s><arrow.to.target n="note91"></arrow.to.target>res centripetæ in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I<emph.end type="italics"></emph.end>æquales. </s> <s>Exponantur hæ vires per æ<lb></lb>quales lineolas <emph type="italics"></emph>DE, IN<emph.end type="italics"></emph.end>; & ſi vis una <emph type="italics"></emph>IN<emph.end type="italics"></emph.end>(per Legum Corol. </s> <s>2.) <lb></lb>reſolvatur in duas <emph type="italics"></emph>NT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IT,<emph.end type="italics"></emph.end>vis <emph type="italics"></emph>NT,<emph.end type="italics"></emph.end>agendo ſecundum lineam <lb></lb><emph type="italics"></emph>NT<emph.end type="italics"></emph.end>corporis curſui <emph type="italics"></emph>ITK<emph.end type="italics"></emph.end>perpendicularem, nil mutabit velocita<lb></lb>tem corporis in curſu illo, ſed retrahet ſolummodo corpus a cur<lb></lb>ſu rectilineo, facietQ.E.I.ſum de Orbis tangente perpetuo deflecte<lb></lb>re, inque via curvilinea <emph type="italics"></emph>ITKk<emph.end type="italics"></emph.end>progredi. </s> <s>In hoc effectu produ<lb></lb>cendo vis illa tota conſumetur: vis autem altera <emph type="italics"></emph>IT,<emph.end type="italics"></emph.end>ſecundum <lb></lb>corporis curſum agendo, tota accelerabit illud, ac dato tem<lb></lb>pore quam minimo accelerationem generabit ſibi ipſi proportiona<lb></lb>lem. </s> <s>Proinde corporum in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I<emph.end type="italics"></emph.end>accelerationes æqualibus tem<lb></lb>poribus factæ (ſi ſumantur linearum naſcentium <emph type="italics"></emph>DE, IN, IK, <lb></lb>IT, NT<emph.end type="italics"></emph.end>rationes primæ) ſunt ut lineæ <emph type="italics"></emph>DE, IT:<emph.end type="italics"></emph.end>temporibus au<lb></lb>tem inæqualibus ut lineæ illæ & tempora conjunctim. </s> <s>Tempora <lb></lb>autem quibus <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>deſcribuntur, ob æqualitatem velocita<lb></lb><figure id="id.039.01.143.1.jpg" xlink:href="039/01/143/1.jpg"></figure><lb></lb>tum ſunt ut viæ deſcriptæ <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IK,<emph.end type="italics"></emph.end>adeoque accelerationes, in <lb></lb>curſu corporum per lineas <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IK,<emph.end type="italics"></emph.end>funt ut <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IT, DE<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>IK<emph.end type="italics"></emph.end>conjunctim, id eſt ut <emph type="italics"></emph>DE quad<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ITXIK rectangulum.<emph.end type="italics"></emph.end>Sed <lb></lb><emph type="italics"></emph>rectangulum ITXIK<emph.end type="italics"></emph.end>æquale eſt <emph type="italics"></emph>IN quadrato,<emph.end type="italics"></emph.end>hoc eſt, æquale <lb></lb><emph type="italics"></emph>DE quadrato;<emph.end type="italics"></emph.end>& propterea accelerationes in tranſitu corporum a <lb></lb><emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>E<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end>æquales generantur. </s> <s>Æquales igitur ſunt cor-<pb xlink:href="039/01/144.jpg" pagenum="116"></pb><arrow.to.target n="note92"></arrow.to.target>porum velocitates in <emph type="italics"></emph>E<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& eodem argumento ſemper reperi<lb></lb>entur æquales in ſubſequentibus æqualibus diſtantiis. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note91"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="margin"> <s><margin.target id="note92"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Sed & eodem argumento corpora æquivelocia & æqualiter a cen<lb></lb>tro diſtantia, in aſcenſu ad æquales diſtantias æqualiter retarda<lb></lb>buntur. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi corpus vel funipendulum oſcilletur, vel im<lb></lb>pedimento quovis politiſſimo & perfecte lubrico cogatur in li<lb></lb>nea curva moveri, & corpus aliud recta aſcendat vel deſcendat, <lb></lb>ſintque velocitates eorum in eadem quacunque altitudine æquales: <lb></lb>erunt velocitates eorum in aliis quibuſcunque æqualibus altitudi<lb></lb>nibus æquales. </s> <s>NamQ.E.I.pedimento vaſis abſolute lubrici idem <lb></lb>præſtatur quod vi tranſverſa <emph type="italics"></emph>NT.<emph.end type="italics"></emph.end>Corpus eo non retardatur, <lb></lb>non acceleratur, ſed tantum cogitur de curſu rectilineo diſcedere. </s></p><figure id="id.039.01.144.1.jpg" xlink:href="039/01/144/1.jpg"></figure> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Hinc etiam ſi quantitas P ſit maxima a centro diſtan<lb></lb>tia, ad quam corpus vel oſcillans vel in Trajectoria quacunque re<lb></lb>volvens, deque quovis Trajectoriæ puncto, ea quam ibi habet <lb></lb>velocitate ſurſum projectum aſcendere poſſit; ſitque quantitas A <lb></lb>diſtantia corporis a centro in alio quovis Orbitæ puncto, & vis <lb></lb>centripeta ſemper ſit ut ipſius A dignitas quælibet A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>, cujus <lb></lb>Index <emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1 eſt numerus quilibet <emph type="italics"></emph>n<emph.end type="italics"></emph.end>unitate diminutus; velocitas <lb></lb>corporis in omni altitudine A erit ut √P<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>-A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, atque adeo da<lb></lb>tur. </s> <s>Namque velocitas recta aſcendentis ac deſcendentis (per Prop. </s> <s><lb></lb>XXXIX) eſt in hac ipſa ratione. <pb xlink:href="039/01/145.jpg" pagenum="117"></pb><arrow.to.target n="note93"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note93"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLI. PROBLEMA XXVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſita cujuſcunque generis Vi centripeta & conceſſis Figurarum <lb></lb>curvilinearum quadraturis, requiruntur tum Trajectoriæ in qui<lb></lb>bus corpora movebuntur, tum Tempora motuum in Trajectoriis <lb></lb>inventis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Tendat vis quælibet ad centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& invenienda ſit Trajectoria <lb></lb><emph type="italics"></emph>VITKk.<emph.end type="italics"></emph.end>Detur Circulus <emph type="italics"></emph>VXY<emph.end type="italics"></emph.end>centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>intervallo quovis <emph type="italics"></emph>CV<emph.end type="italics"></emph.end><lb></lb>deſcriptus, centroque eodem deſcribantur alii quivis circuli <emph type="italics"></emph>ID, <lb></lb>KE<emph.end type="italics"></emph.end>Trajectoriam ſecantes in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end>rectamque <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>E.<emph.end type="italics"></emph.end><lb></lb>Age tum rectam <emph type="italics"></emph>CNIX<emph.end type="italics"></emph.end>ſecantem circulos <emph type="italics"></emph>KE, VY<emph.end type="italics"></emph.end>in <emph type="italics"></emph>N<emph.end type="italics"></emph.end>& <emph type="italics"></emph>X,<emph.end type="italics"></emph.end><lb></lb>tum rectam <emph type="italics"></emph>CKY<emph.end type="italics"></emph.end>occurrentem circulo <emph type="italics"></emph>VXY<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Y.<emph.end type="italics"></emph.end>Sint autem <lb></lb>puncta <emph type="italics"></emph>I<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end>ſibi invicem viciniſſima, & pergat corpus ab <emph type="italics"></emph>V<emph.end type="italics"></emph.end>per <lb></lb><emph type="italics"></emph>I, T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>k;<emph.end type="italics"></emph.end>ſitque punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>locus ille de quo corpus aliud <lb></lb>cadere debet ut in loco <emph type="italics"></emph>D<emph.end type="italics"></emph.end>velocitatem acquirat æqualem veloci<lb></lb>tati corporis prioris in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>; & ſtantibus quæ in Propoſitione XXXIX, <lb></lb>lineola <emph type="italics"></emph>IK,<emph.end type="italics"></emph.end>dato tempore quam minimo deſcripta, erit ut ve<lb></lb>locitas atque adeo ut latus quadratum areæ <emph type="italics"></emph>ABFD,<emph.end type="italics"></emph.end>& triangu<lb></lb>lum <emph type="italics"></emph>ICK<emph.end type="italics"></emph.end>tempori proportionale dabitur, adeoque <emph type="italics"></emph>KN<emph.end type="italics"></emph.end>erit reci<lb></lb>proce ut altitudo <emph type="italics"></emph>IC,<emph.end type="italics"></emph.end>id eſt, ſi detur quantitas aliqua Q, & alti<lb></lb>tudo <emph type="italics"></emph>IC<emph.end type="italics"></emph.end>nominetur A, ut Q/A. </s> <s>Hanc quantitatem Q/A nominemus Z, <lb></lb>& ponamus eam eſſe magnitudinem ipſius Q ut ſit in aliquo <lb></lb>caſu √ <emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>ad Z ut eſt <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KN,<emph.end type="italics"></emph.end>& erit in omni caſu <lb></lb>√<emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>ad Z ut <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KN,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>ad ZZ ut <emph type="italics"></emph><expan abbr="IKq.">IKque</expan><emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="KNq.">KNque</expan><emph.end type="italics"></emph.end><lb></lb>& diviſim <emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>-ZZ ad ZZ ut <emph type="italics"></emph>IN quad<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KN quad,<emph.end type="italics"></emph.end>ad<lb></lb>eoque √<emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>-ZZ ad (Z ſeu)Q/A ut <emph type="italics"></emph>IN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KN,<emph.end type="italics"></emph.end>& propterea <lb></lb>AX<emph type="italics"></emph>KN<emph.end type="italics"></emph.end>æquale (QX<emph type="italics"></emph>IN/√ABFD<emph.end type="italics"></emph.end>-ZZ). Unde cum <emph type="italics"></emph>YXXXC<emph.end type="italics"></emph.end>ſit ad <lb></lb>AX<emph type="italics"></emph>KN<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CXq<emph.end type="italics"></emph.end>ad AA, erit rectangulum <emph type="italics"></emph>YXXXC<emph.end type="italics"></emph.end>æquale <lb></lb>(QX<emph type="italics"></emph>INXCX quad.<emph.end type="italics"></emph.end>/AA√<emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>-ZZ). Igitur ſi in perpendiculo <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>capiantur <lb></lb>ſemper <emph type="italics"></emph>Db, Dc<emph.end type="italics"></emph.end>ipſis (Q/2√<emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>-ZZ) & (QX<emph type="italics"></emph>CX quad.<emph.end type="italics"></emph.end>/2AA√<emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>-ZZ) <lb></lb>æquales reſpective, & deſcribantur curvæ lineæ <emph type="italics"></emph>ab, cd<emph.end type="italics"></emph.end>quas <pb xlink:href="039/01/146.jpg" pagenum="118"></pb><arrow.to.target n="note94"></arrow.to.target>puncta <emph type="italics"></emph>b, c<emph.end type="italics"></emph.end>perpetuo tangunt; deque puncto <emph type="italics"></emph>V<emph.end type="italics"></emph.end>ad lineam <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>eri<lb></lb>gatur perpendiculum <emph type="italics"></emph>Vad<emph.end type="italics"></emph.end>abſcindens areas curvilineas <emph type="italics"></emph>VDba, <lb></lb>VDcd,<emph.end type="italics"></emph.end>& erigantur etiam ordinatæ <emph type="italics"></emph>Ez, Ex:<emph.end type="italics"></emph.end>quoniam rectan<lb></lb>gulum <emph type="italics"></emph>DbXIN<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>DbzE<emph.end type="italics"></emph.end>æquale eſt dimidio rectanguli <lb></lb>AX<emph type="italics"></emph>KN,<emph.end type="italics"></emph.end>ſeu triangulo <emph type="italics"></emph>ICK<emph.end type="italics"></emph.end>; & rectangulum <emph type="italics"></emph>DcXIN<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>DcxE<emph.end type="italics"></emph.end>æquale eſt dimidio rectanguli <emph type="italics"></emph>YXXXC,<emph.end type="italics"></emph.end>ſeu triangulo <lb></lb><emph type="italics"></emph>XCY;<emph.end type="italics"></emph.end>hoc eſt, quoniam arearum <emph type="italics"></emph>VDba, VIC<emph.end type="italics"></emph.end>æquales ſemper <lb></lb>ſunt naſcentes particulæ <emph type="italics"></emph>DbzE, ICK,<emph.end type="italics"></emph.end>& arearum <emph type="italics"></emph>VDcd, <lb></lb>VCX<emph.end type="italics"></emph.end>æquales ſemper ſunt naſcentes particulæ <emph type="italics"></emph>DcxE, XCY,<emph.end type="italics"></emph.end><lb></lb>erit area genita <emph type="italics"></emph>VDba<emph.end type="italics"></emph.end>æqualis areæ genitæ <emph type="italics"></emph>VIC,<emph.end type="italics"></emph.end>adeoque tem<lb></lb>pori proportionalis, & area genita <emph type="italics"></emph>VDcd<emph.end type="italics"></emph.end>æqualis Sectori ge<lb></lb>nito <emph type="italics"></emph>VCX.<emph.end type="italics"></emph.end>Dato igitur tempore quovis ex quo corpus diſceſ<lb></lb>ſit de loco <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>dabitur area ipſi proportionalis <emph type="italics"></emph>VDba,<emph.end type="italics"></emph.end>& inde <lb></lb>dabitur corporis altitudo <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>CI<emph.end type="italics"></emph.end>; & area <emph type="italics"></emph>VDcd,<emph.end type="italics"></emph.end>eique <lb></lb>æqualis Sector <emph type="italics"></emph>VCX<emph.end type="italics"></emph.end>una cum ejus angulo <emph type="italics"></emph>VCI.<emph.end type="italics"></emph.end>Datis autem <lb></lb>angulo <emph type="italics"></emph>VCI<emph.end type="italics"></emph.end>& altitudine <emph type="italics"></emph>CI<emph.end type="italics"></emph.end>datur locus <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>in quo corpus com<lb></lb>pleto illo tempore reperietur. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note94"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc maximæ minimæque corporum altitudines, id eſt <lb></lb>Apſides Trajectoriarum expedite inveniri poſſunt. </s> <s>Sunt enim <lb></lb>Apſides puncta illa in quibus recta <emph type="italics"></emph>IC<emph.end type="italics"></emph.end>per centrum ducta incidit <lb></lb>perpendiculariter in Trajectoriam <emph type="italics"></emph>VIK:<emph.end type="italics"></emph.end>id quod ſit ubi rectæ <emph type="italics"></emph>IK<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>NK<emph.end type="italics"></emph.end>æquantur, adeoque ubi area <emph type="italics"></emph>ABFD<emph.end type="italics"></emph.end>æqualis eſt ZZ. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Sed & angulus <emph type="italics"></emph>KIN,<emph.end type="italics"></emph.end>in quo Trajectoria alibi ſecat <lb></lb>lineam illam <emph type="italics"></emph>IC,<emph.end type="italics"></emph.end>ex data corporis altitudine <emph type="italics"></emph>IC<emph.end type="italics"></emph.end>expedite inveNI<lb></lb>tur; nimirum capiendo ſinum ejus ad radium ut <emph type="italics"></emph>KN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>IK,<emph.end type="italics"></emph.end>id <lb></lb>eſt, ut Z ad latus quadratum areæ <emph type="italics"></emph>ABFD.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& vertice principali <emph type="italics"></emph>V<emph.end type="italics"></emph.end>deſcribatur Sectio quæ<lb></lb>libet Conica <emph type="italics"></emph>VRS,<emph.end type="italics"></emph.end>& a quovis ejus puncto <emph type="italics"></emph>R<emph.end type="italics"></emph.end>agatur Tangens <emph type="italics"></emph>RT<emph.end type="italics"></emph.end><lb></lb>occurrens axi infinite producto <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>in puncto <emph type="italics"></emph>T;<emph.end type="italics"></emph.end>dein juncta <emph type="italics"></emph>CR<emph.end type="italics"></emph.end><lb></lb>ducatur recta <emph type="italics"></emph>CP,<emph.end type="italics"></emph.end>quæ æqualis ſit abſciſſæ <emph type="italics"></emph>CT,<emph.end type="italics"></emph.end>angulumque <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end><lb></lb>Sectori <emph type="italics"></emph>VCR<emph.end type="italics"></emph.end>proportionalem conſtituat; tendat autem ad centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end><lb></lb>Vis centripeta Cubo diſtantiæ loeorum a centro reciproce propor<lb></lb>tionalis, & exeat corpus de loco <emph type="italics"></emph>V<emph.end type="italics"></emph.end>juſta cum Velocitate ſecundum <lb></lb>lineam rectæ <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>perpendicularem: progredietur corpus illud in <lb></lb>Trajectoria quam punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>perpetuo tangit; adeoque ſi Conica <lb></lb>ſectio <emph type="italics"></emph>CVRS<emph.end type="italics"></emph.end>Hyperbola ſit, deſcendet idem ad centrum: Sin <lb></lb>ea Ellipſis ſit, aſcendet illud perpetuo & abibit in infinitum. </s> <s>Et con<lb></lb>tra, ſi corpus quacunque cum Velocitate exeat de loco <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>& perin<lb></lb>de ut incæperit vel obliQ.E.D.ſcendere ad centrum, vel ab eo ob-<pb xlink:href="039/01/147.jpg" pagenum="119"></pb>lique aſcendere, Figura <emph type="italics"></emph>CVRS<emph.end type="italics"></emph.end>vel Hyperbola ſit vel Ellipſis, in<lb></lb><arrow.to.target n="note95"></arrow.to.target>veniri poteſt Trajectoria augendo vel minuendo angulum <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end><lb></lb>in data aliqua ratione. </s> <s>Sed &, Vi centripeta in centrifugam verſa, <lb></lb><figure id="id.039.01.147.1.jpg" xlink:href="039/01/147/1.jpg"></figure><lb></lb>aſcendet corpus obliQ.E.I. Trajectoria <emph type="italics"></emph>VPQ<emph.end type="italics"></emph.end>quæ invenitur capi<lb></lb>endo angulum <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>Sectori Elliptico <emph type="italics"></emph>CVRC<emph.end type="italics"></emph.end>proportionalem, & <lb></lb>longitudinem <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>longitudini <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>æqualem ut ſupra. </s> <s>Conſequun<lb></lb>tur hæc omnia ex Propoſitione præcedente, per Curvæ cujuſdam <lb></lb>quadraturam, cujus inventionem, ut ſatis facilem, brevitatis gratia <lb></lb>miſſam facio. </s></p> <p type="margin"> <s><margin.target id="note95"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLII. PROBLEMA XXIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Data lege Vis centripetæ, requiritur motus corporis de loco dato <lb></lb>data cum Velocitate ſecundum datam rectam egreſſi.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Stantibus quæ in tribus Propoſitionibus præcedentibus: exeat <lb></lb>corpus de loco <emph type="italics"></emph>I<emph.end type="italics"></emph.end>ſecundum lineolam <emph type="italics"></emph>IT,<emph.end type="italics"></emph.end>ea cum Velocitate quam <lb></lb>corpus aliud, vi aliqua uniformi centripeta, de loco <emph type="italics"></emph>P<emph.end type="italics"></emph.end>cadendo ac<lb></lb>quirere poſſet in <emph type="italics"></emph>D:<emph.end type="italics"></emph.end>ſitque hæc vis uniformis ad vim qua corpus <pb xlink:href="039/01/148.jpg" pagenum="120"></pb><arrow.to.target n="note96"></arrow.to.target>primum urgetur in <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DF.<emph.end type="italics"></emph.end>Pergat autem corpus verſus <lb></lb><emph type="italics"></emph>k;<emph.end type="italics"></emph.end>centroque <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& intervallo <emph type="italics"></emph>Ck<emph.end type="italics"></emph.end>deſcribatur circulus <emph type="italics"></emph>ke<emph.end type="italics"></emph.end>occurrens <lb></lb>rectæ <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>& erigantur curvarum <emph type="italics"></emph>ALMm, BFGg, abzv, dcxw<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.148.1.jpg" xlink:href="039/01/148/1.jpg"></figure><lb></lb>ordinatim applicatæ <emph type="italics"></emph>em, eg, ev, ew.<emph.end type="italics"></emph.end>Ex dato rectangulo <emph type="italics"></emph>PDRQ,<emph.end type="italics"></emph.end><lb></lb>dataque lege vis centripetæ qua corpus primum agitatur, dantur cur<lb></lb>væ lineæ <emph type="italics"></emph>BFGg, ALMm,<emph.end type="italics"></emph.end>per conſtructionem Problematis XXVII, <lb></lb>& ejus Corol. </s> <s>1. Deinde ex dato angulo <emph type="italics"></emph>CIT<emph.end type="italics"></emph.end>datur proportio naſcen<lb></lb>tium <emph type="italics"></emph>IK, KN,<emph.end type="italics"></emph.end>& inde, per conſtructionem Prob. </s> <s>XXVIII, datur <lb></lb>quantitas Q, una cum curvis lineis <emph type="italics"></emph>abzv, dcxw:<emph.end type="italics"></emph.end>adeoque com<lb></lb>pleto tempore quovis <emph type="italics"></emph>Dbve,<emph.end type="italics"></emph.end>datur tum corporis altitudo <emph type="italics"></emph>Ce<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Ck,<emph.end type="italics"></emph.end><lb></lb>tum area <emph type="italics"></emph>Dcwe,<emph.end type="italics"></emph.end>eique æqualis Sector <emph type="italics"></emph>XCy,<emph.end type="italics"></emph.end>anguluſque <emph type="italics"></emph>ICk<emph.end type="italics"></emph.end>& <lb></lb>locus <emph type="italics"></emph>k<emph.end type="italics"></emph.end>in quo corpus tunc verſabitur. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note96"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Supponimus autem in his Propoſitionibus Vim centripetam in <lb></lb>receſſu quidem a centro variari ſecundum legem quamcunque quam <lb></lb>quis imaginari poteſt, in æqualibus autem a centro diſtantiis eſſe <lb></lb>undeque eandem. </s> <s>Atque hactenus Motum corporum in Orbibus <lb></lb>immobilibus conſideravimus. </s> <s>Supereſt ut de Motu eorum in Orbi<lb></lb>bus qui circa centrum virium revolvuntur adjiciamus pauca. <pb xlink:href="039/01/149.jpg" pagenum="121"></pb><arrow.to.target n="note97"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note97"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Motu Corporum in Orbibus mobilibus, deque motu Apſidum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLIII. PROBLEMA XXX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Efficiendum est ut corpus in Trajectoria quacunque circa centrum <lb></lb>Virium revolvente perinde moveri poſſit, atque corpus aliud in <lb></lb>eadem Trajectoria quieſcente.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>In Orbe <emph type="italics"></emph>VPK<emph.end type="italics"></emph.end>po<lb></lb><figure id="id.039.01.149.1.jpg" xlink:href="039/01/149/1.jpg"></figure><lb></lb>ſitione dato revolvatur <lb></lb>corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>pergendo a <lb></lb><emph type="italics"></emph>V<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>K.<emph.end type="italics"></emph.end>A centro <lb></lb><emph type="italics"></emph>C<emph.end type="italics"></emph.end>agatur ſemper <emph type="italics"></emph>Cp,<emph.end type="italics"></emph.end><lb></lb>quæ ſit ipſi <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>æqualis, <lb></lb>angulumque <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>an<lb></lb>gulo <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>proportio<lb></lb>nalem conſtituat; & a<lb></lb>rea quam linea <emph type="italics"></emph>Cp<emph.end type="italics"></emph.end>de<lb></lb>ſcribit erit ad aream <lb></lb><emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>quam linea <emph type="italics"></emph>CP<emph.end type="italics"></emph.end><lb></lb>ſimul deſcribit, ut velo<lb></lb>citas lineæ deſcribentis <lb></lb><emph type="italics"></emph>Cp<emph.end type="italics"></emph.end>ad velocitatem li<lb></lb>neæ deſcribentis <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>; <lb></lb>hoc eſt, ut angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>VCP,<emph.end type="italics"></emph.end>adeoQ.E.I. data ra<lb></lb>tione, & propterea tempori proportionalis. </s> <s>Cum area tempori <lb></lb>proportionalis ſit quam linea <emph type="italics"></emph>Cp<emph.end type="italics"></emph.end>in plano immobili deſcribit, ma<lb></lb>nifeſtum eſt quod corpus, cogente juſtæ quantitatis Vi centripeta, <lb></lb>revolvi poſſit una cum puncto <emph type="italics"></emph>p<emph.end type="italics"></emph.end>in Curva illa linea quam punctum <lb></lb>idem <emph type="italics"></emph>p<emph.end type="italics"></emph.end>ratione jam expoſita deſcribit in plano immobili. </s> <s>Fiat angu<lb></lb>lus <emph type="italics"></emph>VCu<emph.end type="italics"></emph.end>angulo <emph type="italics"></emph>PCp,<emph.end type="italics"></emph.end>& linea <emph type="italics"></emph>Cu<emph.end type="italics"></emph.end>lineæ <emph type="italics"></emph>CV,<emph.end type="italics"></emph.end>atque Figura <emph type="italics"></emph>uCp<emph.end type="italics"></emph.end>Fi<lb></lb>guræ <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>æqualis, & corpus in <emph type="italics"></emph>p<emph.end type="italics"></emph.end>ſemper exiſtens movebitur in <pb xlink:href="039/01/150.jpg" pagenum="122"></pb><arrow.to.target n="note98"></arrow.to.target>perimetro Figuræ revolventis <emph type="italics"></emph>uCp,<emph.end type="italics"></emph.end>eodemque tempore deſcribet <lb></lb>arcum ejus <emph type="italics"></emph>up<emph.end type="italics"></emph.end>quo corpus aliud <emph type="italics"></emph>P<emph.end type="italics"></emph.end>arcum ipſi ſimilem & æqualem <lb></lb><emph type="italics"></emph>VP<emph.end type="italics"></emph.end>in Figura quieſcente <emph type="italics"></emph>VPK<emph.end type="italics"></emph.end>deſcribere poteſt. </s> <s>Quæratur igi<lb></lb>tur, per Corollarium quintum propoſitionis VI, Vis centripeta qua <lb></lb>corpus revolvi poſſit in Curva illa linea quam punctum <emph type="italics"></emph>p<emph.end type="italics"></emph.end>deſcribit <lb></lb>in plano immobili, & ſolvetur Problema. <emph type="italics"></emph>q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note98"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLIV. THEOREMA XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Differentia Virium, quibus corpus in Orbe quieſcente, & corpus a<lb></lb>liud in eodem Orbe revolvente æqualiter moveri poſſunt, est <lb></lb>in triplicata ratione communis altitudinis inverſe.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Partibus Orbis quie<lb></lb><figure id="id.039.01.150.1.jpg" xlink:href="039/01/150/1.jpg"></figure><lb></lb>ſcentis <emph type="italics"></emph>VP, PK<emph.end type="italics"></emph.end>ſunto <lb></lb>ſimiles & æquales Or<lb></lb>bis revolventis partes <lb></lb><emph type="italics"></emph>up, pk<emph.end type="italics"></emph.end>; & punctorum <lb></lb><emph type="italics"></emph>P, K<emph.end type="italics"></emph.end>diſtantia intelli<lb></lb>gatur eſſe quam miNI<lb></lb>ma. </s> <s>A puncto <emph type="italics"></emph>k<emph.end type="italics"></emph.end>in re<lb></lb>ctam <emph type="italics"></emph>pC<emph.end type="italics"></emph.end>demitte per<lb></lb>pendiculum <emph type="italics"></emph>kr,<emph.end type="italics"></emph.end>idem<lb></lb>que produc ad <emph type="italics"></emph>m,<emph.end type="italics"></emph.end>ut ſit <lb></lb><emph type="italics"></emph>mr<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>kr<emph.end type="italics"></emph.end>ut angulus <lb></lb><emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>VCP.<emph.end type="italics"></emph.end><lb></lb>Quoniam corporum al<lb></lb>titudines <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pC, KC<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>kC<emph.end type="italics"></emph.end>ſemper æquan<lb></lb>tur, manifeſtum eſt quod linearum <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pC<emph.end type="italics"></emph.end>incrementa vel <lb></lb>decrementa ſemper ſint æqualia, ideoque ſi corporum in locis <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>p<emph.end type="italics"></emph.end>exiſtentium diſtinguantur motus ſinguli (per Legum <lb></lb>Corol. </s> <s>2.) in binos, quorum hi verſus centrum, ſive ſecundum <lb></lb>lineas <emph type="italics"></emph>PC, pC<emph.end type="italics"></emph.end>determinentur, & alteri prioribus tranſverſi ſint, <lb></lb>& ſecundum lineas ipſis <emph type="italics"></emph>PC, pC<emph.end type="italics"></emph.end>perpendiculares directionem <lb></lb>habeant; motus verſus centrum erunt æquales, & motus tranſ<lb></lb>verſus corporis <emph type="italics"></emph>p<emph.end type="italics"></emph.end>erit ad motum tranſverſum corporis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ut mo<lb></lb>tus angularis lineæ <emph type="italics"></emph>pC,<emph.end type="italics"></emph.end>ad motum angularem lineæ <emph type="italics"></emph>PC,<emph.end type="italics"></emph.end>id eſt, <pb xlink:href="039/01/151.jpg" pagenum="123"></pb>ut angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>VCP.<emph.end type="italics"></emph.end>Igitur eodem tempore quo <lb></lb><arrow.to.target n="note99"></arrow.to.target>corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>motu ſuo utroque pervenit ad punctum <emph type="italics"></emph>K,<emph.end type="italics"></emph.end>corpus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>æ<lb></lb>quali in centrum motu æqualiter movebitur a <emph type="italics"></emph>p<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>adeoque <lb></lb>completo illo tempore reperietur alicubi in linea <emph type="italics"></emph>mkr,<emph.end type="italics"></emph.end>quæ per <lb></lb>punctum <emph type="italics"></emph>k<emph.end type="italics"></emph.end>in lineam <emph type="italics"></emph>pC<emph.end type="italics"></emph.end>perpendicularis eſt; & motu tranſverſo <lb></lb>acquiret diſtantiam a linea <emph type="italics"></emph>pC,<emph.end type="italics"></emph.end>quæ ſit ad diſtantiam quam cor<lb></lb>pus alterum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>acquirit a linea <emph type="italics"></emph>PC,<emph.end type="italics"></emph.end>ut eſt motus tranſverſus cor<lb></lb>poris <emph type="italics"></emph>p<emph.end type="italics"></emph.end>ad motum tranſverſum corporis alterius <emph type="italics"></emph>P.<emph.end type="italics"></emph.end>Quare cum <lb></lb><emph type="italics"></emph>kr<emph.end type="italics"></emph.end>æqualis ſit diſtantiæ quam corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>acquirit a linea <emph type="italics"></emph>PC,<emph.end type="italics"></emph.end>ſitque <lb></lb><emph type="italics"></emph>mr<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>kr<emph.end type="italics"></emph.end>ut angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>VCP,<emph.end type="italics"></emph.end>hoc eſt, ut motus <lb></lb>tranſverſus corporis <emph type="italics"></emph>p<emph.end type="italics"></emph.end>ad motum tranſverſum corporis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>manife<lb></lb>ſtum eſt quod corpus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>completo illo tempore reperietur in loco <lb></lb><emph type="italics"></emph>m.<emph.end type="italics"></emph.end>Hæc ita ſe habebunt ubi corpora <emph type="italics"></emph>p<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>æqualiter ſecundum <lb></lb>lineas <emph type="italics"></emph>pC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>moventur, adeoque æqualibus Viribus ſecundum <lb></lb>lineas illas urgentur. </s> <s>Capiatur autem angulum <emph type="italics"></emph>pCn<emph.end type="italics"></emph.end>ad angulum <lb></lb><emph type="italics"></emph>pCk<emph.end type="italics"></emph.end>ut eſt angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulus <emph type="italics"></emph>VCP,<emph.end type="italics"></emph.end>ſitque <emph type="italics"></emph>nC<emph.end type="italics"></emph.end>æqualis <lb></lb><emph type="italics"></emph>kC,<emph.end type="italics"></emph.end>& corpus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>completo illo tempore revera reperietur in <emph type="italics"></emph>n<emph.end type="italics"></emph.end>; ad<lb></lb>eoque Vi majore urgetur quam corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſi modo angulus <emph type="italics"></emph>mCp<emph.end type="italics"></emph.end><lb></lb>angulo <emph type="italics"></emph>kCp<emph.end type="italics"></emph.end>major eſt, id eſt ſi Orbis <emph type="italics"></emph>upk<emph.end type="italics"></emph.end>vel movetur in con<lb></lb>ſequentia, vel movetur in antecedentia majore celeritate quam <lb></lb>ſit dupla ejus qua linea <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>in conſequentia fertur; & Vi mino<lb></lb>re ſi Orbis tardius movetur in antecedentia. </s> <s>Eſtque Virium dif<lb></lb>ferentia ut loeorum intervallum <emph type="italics"></emph>mn,<emph.end type="italics"></emph.end>per quod corpus illud <emph type="italics"></emph>p<emph.end type="italics"></emph.end><lb></lb>ipſius actione, dato illo temporis ſpatio, transferri debet. </s> <s>Centro <lb></lb><emph type="italics"></emph>C<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>Cn<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Ck<emph.end type="italics"></emph.end>deſcribi intelligatur Circulus ſecans <lb></lb>lineas <emph type="italics"></emph>mr, mn<emph.end type="italics"></emph.end>productas in <emph type="italics"></emph>s<emph.end type="italics"></emph.end>& <emph type="italics"></emph>t,<emph.end type="italics"></emph.end>& erit rectangulum <emph type="italics"></emph>mnXmt<emph.end type="italics"></emph.end>æ<lb></lb>quale rectangulo <emph type="italics"></emph>mkXms,<emph.end type="italics"></emph.end>adeoque <emph type="italics"></emph>mn<emph.end type="italics"></emph.end>æquale (<emph type="italics"></emph>mkXms/mt<emph.end type="italics"></emph.end>). Cum <lb></lb>autem triangula <emph type="italics"></emph>pCk, pCn<emph.end type="italics"></emph.end>dentur magnitudine, ſunt <emph type="italics"></emph>kr<emph.end type="italics"></emph.end>& <emph type="italics"></emph>mr,<emph.end type="italics"></emph.end><lb></lb>earumQ.E.D.fferentia <emph type="italics"></emph>mk<emph.end type="italics"></emph.end>& ſumma <emph type="italics"></emph>ms<emph.end type="italics"></emph.end>reciproce ut altitudo <emph type="italics"></emph>pC,<emph.end type="italics"></emph.end><lb></lb>adeoque rectangulum <emph type="italics"></emph>mkXms<emph.end type="italics"></emph.end>eſt reciproce ut quadratum altitudi<lb></lb>nis <emph type="italics"></emph>pC.<emph.end type="italics"></emph.end>Eſt & <emph type="italics"></emph>mt<emph.end type="italics"></emph.end>directe ut 1/2 <emph type="italics"></emph>mt,<emph.end type="italics"></emph.end>id eſt, ut altitudo <emph type="italics"></emph>pC.<emph.end type="italics"></emph.end>Hæ <lb></lb>ſunt primæ rationes linearum naſcentium; & hinc fit (<emph type="italics"></emph>mkXms/mt<emph.end type="italics"></emph.end>), id <lb></lb>eſt lineola naſcens <emph type="italics"></emph>mn,<emph.end type="italics"></emph.end>eique proportionalis Virium differentia reci<lb></lb>proce ut cubus altitudinis <emph type="italics"></emph>pC. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note99"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc differentia virium in locis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>p<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>k,<emph.end type="italics"></emph.end>eſt <lb></lb>ad vim qua corpus motu Circulari revolvi poſſit ab <emph type="italics"></emph>R<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>K<emph.end type="italics"></emph.end>eodem <lb></lb>tempore quo corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Orbe immobili deſcribit arcum <emph type="italics"></emph>PK,<emph.end type="italics"></emph.end>ut <lb></lb>lineola naſcens <emph type="italics"></emph>mn<emph.end type="italics"></emph.end>ad ſinum verſum arcus naſcentis <emph type="italics"></emph>RK,<emph.end type="italics"></emph.end>id eſt <pb xlink:href="039/01/152.jpg" pagenum="124"></pb><arrow.to.target n="note100"></arrow.to.target>ut (<emph type="italics"></emph>mkXms/mt<emph.end type="italics"></emph.end>) ad (<emph type="italics"></emph>rkq/2kC<emph.end type="italics"></emph.end>), vel ut <emph type="italics"></emph>mkXms<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>rk<emph.end type="italics"></emph.end>quadratum; hoc eſt, ſi <lb></lb>capiantur datæ quantitates F, G in ea ratione ad invicem quam <lb></lb>habet angulus <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>VCp,<emph.end type="italics"></emph.end>ut GG-FF ad FF. </s> <s>Et <lb></lb>propterea, ſi centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>intervallo quovis <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Cp<emph.end type="italics"></emph.end>deſcribatur <lb></lb>Sector circularis æqualis areæ toti <emph type="italics"></emph>VPC,<emph.end type="italics"></emph.end>quam corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>tempore <lb></lb>quovis in Orbe immobili revolvens radio ad centrum ducto de<lb></lb>ſcrip ſit: differentia virium, quibus corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Orbe immobili & <lb></lb>corpus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>in Orbe mobili revolvuntur, erit ad vim centripetam qua <lb></lb>corpus aliquod radio ad centrum ducto Sectorem illum, eodem tem<lb></lb>pore quo deſcripta ſit area <emph type="italics"></emph>VPC<emph.end type="italics"></emph.end>uniformiter deſeribere potuiſſet, <lb></lb>ut GG-FF ad FF. </s> <s>Namque Sector ille & area <emph type="italics"></emph>pCk<emph.end type="italics"></emph.end>ſunt ad in<lb></lb>vicem ut tempora quibus deſcribuntur. </s></p> <p type="margin"> <s><margin.target id="note100"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si Orbis <emph type="italics"></emph>VPK<emph.end type="italics"></emph.end>Ellipſis ſit umbilicum habens <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& Ap<lb></lb>ſidem ſummam <emph type="italics"></emph>V;<emph.end type="italics"></emph.end>eique ſimilis & æqualis ponatur Ellipſis <emph type="italics"></emph>upk,<emph.end type="italics"></emph.end><lb></lb>ita ut ſit ſemper <emph type="italics"></emph>pC<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>PC,<emph.end type="italics"></emph.end>& angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ſit ad angulum <lb></lb><emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>in data ratione G ad F; pro altitudine autem <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>pC<emph.end type="italics"></emph.end><lb></lb>ſcribatur A, & pro Ellipſeos latere recto ponatur 2 R: erit vis qua <lb></lb>corpus in Ellipſi mobili revolvi poteſt, ut (FF/AA)+(RGG-RFF/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) <lb></lb>& contra. </s> <s>Exponatur enim vis qua corpus revolvatur in immota <lb></lb>Ellipſi per quantitatem (FF/AA), & vis in <emph type="italics"></emph>V<emph.end type="italics"></emph.end>erit (FF/<emph type="italics"></emph>CV quad.<emph.end type="italics"></emph.end>). Vis au<lb></lb>tem qua corpus in Circulo ad diſtantiam <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>ea cum velocitate <lb></lb>revolvi poſſet quam corpus in Ellipſi revolvens habet in <emph type="italics"></emph>V,<emph.end type="italics"></emph.end><lb></lb>eſt ad vim qua corpus in Ellipſi revolvens urgetur in Apſide <emph type="italics"></emph>V,<emph.end type="italics"></emph.end><lb></lb>ut dimidium lateris recti Ellipſeos. </s> <s>ad Circuli ſemidiametrum <emph type="italics"></emph>CV,<emph.end type="italics"></emph.end><lb></lb>adeoque valet (RFF/<emph type="italics"></emph>CV cub.<emph.end type="italics"></emph.end>): & vis quæ ſit ad hanc ut GG-FF ad <lb></lb>FF, valet (RGG-RFF/<emph type="italics"></emph>CV cub.<emph.end type="italics"></emph.end>): eſtque hæc vis (per hujus Corol. </s> <s>1.) <lb></lb>differentia virium in <emph type="italics"></emph>V<emph.end type="italics"></emph.end>quibus corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Ellipſi immota <emph type="italics"></emph>VPK,<emph.end type="italics"></emph.end><lb></lb>& corpus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>in Ellipſi mobili <emph type="italics"></emph>upk<emph.end type="italics"></emph.end>revolvuntur. </s> <s>Unde cum (per <lb></lb>hanc Prop.) differentia illa in alia quavis altitudine A ſit ad ſe<lb></lb>ipſam in altitudine <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>ut (1/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) ad (1/<emph type="italics"></emph>CV cub.<emph.end type="italics"></emph.end>), eadem differentia <lb></lb>in omni altitudine. </s> <s>A valebit (RGG-RFF/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>). Igitur ad vim (FF/AA) <lb></lb>qua corpus revolvi poteſt in Ellipſi immobili <emph type="italics"></emph>VPK,<emph.end type="italics"></emph.end>addatur ex<lb></lb>ceſſus (RGG-RFF/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) & componetur vis tota (FF/AA)+(RGG-RFF/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) <pb xlink:href="039/01/153.jpg" pagenum="125"></pb>qua corpus in Ellipſi mobili <emph type="italics"></emph>upk<emph.end type="italics"></emph.end>iiſdem temporibus revolvi <lb></lb><arrow.to.target n="note101"></arrow.to.target>poſſit. </s></p> <p type="margin"> <s><margin.target id="note101"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Ad eundem modum colligetur quod, ſi Orbis immo<lb></lb>bilis <emph type="italics"></emph>VPK<emph.end type="italics"></emph.end>Ellipſis ſit centrum habens in virium centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>; ei<lb></lb>que ſimilis, æqualis & concentrica ponatur Ellipſis mobilis <emph type="italics"></emph>upk;<emph.end type="italics"></emph.end><lb></lb>ſitque 2 R Ellipſeos hujus latus rectum principale, & 2T latus <lb></lb>tranſverſum ſive axis major, atque angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ſemper ſit ad <lb></lb>angulum <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>ut G ad F; vires quibus corpora in Ellipſi im<lb></lb>mobili & mobili temporibus æqualibus revolvi poſſunt, erunt ut <lb></lb>(FFA/T <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) & (FFA/T <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>)+(RGG-RFF/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) reſpective. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Et univerſaliter, ſi corporis altitudo maxima <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>no<lb></lb>minetur T, & radius curvaturæ quam Orbis <emph type="italics"></emph>VPK<emph.end type="italics"></emph.end>habet in <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>id <lb></lb>eſt radius Circuli æqualiter curvi, nominetur R, & vis centripeta <lb></lb>qua corpus in Trajectoria quacunQ.E.I.mobili <emph type="italics"></emph>VPK<emph.end type="italics"></emph.end>revolvi po<lb></lb>teſt, in loco <emph type="italics"></emph>V<emph.end type="italics"></emph.end>dicatur (VFF/TT), atque aliis in locis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>indefinite dica<lb></lb>tur X, altitudine <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>nominata A, & capiatur G ad F in data <lb></lb>ratione anguli <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>VCP:<emph.end type="italics"></emph.end>erit vis centripeta qua <lb></lb>corpus idem eoſdem motus in eadem Trajectoria <emph type="italics"></emph>upk<emph.end type="italics"></emph.end>circula<lb></lb>riter mota temporibus iiſdem peragere poteſt, ut ſumma virium <lb></lb>X+(VRGG-VRFF/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>). </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Dato igitur motu corporis in Orbe quocunQ.E.I.mo<lb></lb>bili, augeri vel minui poteſt ejus motus angularis circa centrum <lb></lb>virium in ratione data, & inde inveniri novi Orbes immobiles in <lb></lb>quibus corpora novis viribus centripetis gyrentur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Igitur ſi ad rectam <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>po<lb></lb><figure id="id.039.01.153.1.jpg" xlink:href="039/01/153/1.jpg"></figure><lb></lb>ſitione datam erigatur perpendiculum <lb></lb><emph type="italics"></emph>VP<emph.end type="italics"></emph.end>longitudinis indeterminatæ, jun<lb></lb>gaturque <emph type="italics"></emph>CP,<emph.end type="italics"></emph.end>& ipſi æqualis agatur <lb></lb><emph type="italics"></emph>Cp,<emph.end type="italics"></emph.end>conſtituens angulum <emph type="italics"></emph>VCp,<emph.end type="italics"></emph.end>qui ſit <lb></lb>ad angulum <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>in data ratione; <lb></lb>vis qua corpus gyrari poteſt in Curva <lb></lb>illa <emph type="italics"></emph>Vpk<emph.end type="italics"></emph.end>quam punctum <emph type="italics"></emph>p<emph.end type="italics"></emph.end>perpetuo <lb></lb>tangit, erit reciproce ut cubus altitu<lb></lb>dinis <emph type="italics"></emph>Cp.<emph.end type="italics"></emph.end>Nam corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>per vim inertiæ, nulla alia vi urgente, <lb></lb>uniformiter progredi poteſt in recta <emph type="italics"></emph>VP.<emph.end type="italics"></emph.end>Addatur vis in centrum <lb></lb><emph type="italics"></emph>C,<emph.end type="italics"></emph.end>cubo altitudinis <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Cp<emph.end type="italics"></emph.end>reciproce proportionalis, & (per <lb></lb>jam demonſtrata) detorQ.E.I.ur motus ille rectilineus in lineam <pb xlink:href="039/01/154.jpg" pagenum="126"></pb><arrow.to.target n="note102"></arrow.to.target>curvam <emph type="italics"></emph>Vpk.<emph.end type="italics"></emph.end>Eſt autem hæc Curva <emph type="italics"></emph>Vpk<emph.end type="italics"></emph.end>eadem cum Curva illa <lb></lb><emph type="italics"></emph>VPQ<emph.end type="italics"></emph.end>in Corol. </s> <s>3. Prop. </s> <s>XLI inventa, in qua ibi diximus corpora <lb></lb>hujuſmodi viribus attracta oblique aſcendere. </s></p> <p type="margin"> <s><margin.target id="note102"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLV. PROBLEMA XXXI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Orbium qui ſunt Circulis maxime finitimi requiruntur motus Ap<lb></lb>ſidum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Problema ſolvitur Arithmetice faciendo ut Orbis, quem corpus <lb></lb>in Ellipſi mobili (ut in Propoſitionis ſuperioris Corol. </s> <s>2, vel 3) <lb></lb>revolvens deſcribit in plano immobili, accedat ad formam Orbis <lb></lb>cujus Apſides requiruatur, & quærendo Apſides Orbis quem cor<lb></lb>pus illud in plano immobili deſcribit. </s> <s>Orbes autem eandem ac<lb></lb>quirent formam, ſi vires centripetæ quibus deſcribuntur, inter ſe <lb></lb>collatæ, in æqualibus altitudinibus reddantur proportionales. </s> <s>Sit <lb></lb>punctum <emph type="italics"></emph>V<emph.end type="italics"></emph.end>Apſis ſumma, & ſcribantur T pro altitudine maxima <lb></lb><emph type="italics"></emph>CV,<emph.end type="italics"></emph.end>A pro altitudine quavis alia <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Cp,<emph.end type="italics"></emph.end>& X pro alti<lb></lb>titudinum differentia <emph type="italics"></emph>CV-CP<emph.end type="italics"></emph.end>; & vis qua corpus in Ellipſi <lb></lb>circa umbilicum ſuum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>(ut in Corollario 2.) revolvente move<lb></lb>tur, quæQ.E.I. Corollario 2. erat ut (FF/AA)+(RGG-RFF/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>), id eſt <lb></lb>ut (FFA+RGG-RFF/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>), ſubſtituendo T-X pro A, erit ut <lb></lb>(RGG-RFF+TFF-FFX/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>). Reducenda ſimiliter eſt vis alia <lb></lb>quævis centripeta ad fractionem cujus denominator ſit A <emph type="italics"></emph>cub.,<emph.end type="italics"></emph.end>& <lb></lb>numeratores, facta homologorum terminorum collatione, ſtatuendi <lb></lb>ſunt analogi. </s> <s>Res Exemplis patebit. </s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>1. Ponamus vim centripetam uniformem eſſe, adeoque <lb></lb>ut (A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>), ſive (ſcribendo T-X pro A in Numeratore) ut <lb></lb>(T <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>-3TTX+3TXX-X <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>); & collatis Numeratorum ter<lb></lb>minis correſpondentibus, nimirum datis cum datis & non datis <lb></lb>cum non datis, fiet RGG-RFF+TFF ad T <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>ut-FFX ad <lb></lb>-3TTX+3TXX-X<emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>ſive ut-FF ad-3TT+3TX <lb></lb>-XX. </s> <s>Jam cum Orbis ponatur Circulo quam maxime finitimus, <lb></lb>coeat Orbis cum Circulo; & ob factas R, T æquales, atque X in infi-<pb xlink:href="039/01/155.jpg" pagenum="127"></pb>nitum diminutam, rationes ultimæ erunt RGG ad T <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>ut-FF <lb></lb><arrow.to.target n="note103"></arrow.to.target>ad-3TT ſeu GG ad TT ut FF ad 3TT & viciſſim GG ad <lb></lb>FF ut TT ad 3 TT id eſt, ut 1 ad 3; adeoque G ad F, <lb></lb>hoc eſt angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>VCP,<emph.end type="italics"></emph.end>ut 1 ad √3. Er<lb></lb>go cum corpus in Ellipſi immobili, ab Apſide ſumma ad Ap<lb></lb>ſidem imam deſcendendo conficiat angulum <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>(ut ita di<lb></lb>cam) gradum 180; corpus aliud in Ellipſi mobili, atque adeo in <lb></lb>Orbe immobili de quo agimus, ab Apſide ſumma ad Apſidem <lb></lb>imam deſcendendo conficiet angulum <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>gradum (180/√3): id <lb></lb>adeo ob ſimilitudinem Orbis hujus, quem corpus agente uniformi <lb></lb>vi centripeta deſcribit, & Orbis illius quem corpus in Ellipſi re<lb></lb>volvente gyros peragens deſcribit in plano quieſcente. </s> <s>Per ſu<lb></lb>periorem terminorum collationem ſimiles redduntur hi Orbes, non <lb></lb>univerſaliter, ſed tunc cum ad formam circularem quam maxime <lb></lb>appropinquant. </s> <s>Corpus igitur uniformi cum vi centripeta in <lb></lb>Orbe propemodum circulari revolvens, inter Apſidem ſummam <lb></lb>& Apſidem imam conficiet ſemper angulum (180/√3) graduum, ſeu <lb></lb>103 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>55 <emph type="italics"></emph>m.<emph.end type="italics"></emph.end>23 <emph type="italics"></emph>ſec.<emph.end type="italics"></emph.end>ad centrum; perveniens ab Apſide ſumma ad <lb></lb>Apſidem imam ubi ſemel confecit hunc angulum, & inde ad Apſi<lb></lb>dem ſummam rediens ubi iterum confecit eundem angulum; & <lb></lb>ſic deinceps in infinitum. </s></p> <p type="margin"> <s><margin.target id="note103"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>2. Ponamus vim centripetam eſſe ut altitudinis A dig<lb></lb>nitas quælibet A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-3<emph.end type="sup"></emph.end> ſeu (A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>/A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>): ubi <emph type="italics"></emph>n<emph.end type="italics"></emph.end>-3 & <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ſignificant digNI<lb></lb>tatum indices quoſcunQ.E.I.tegros vel fractos, rationales vel irratio<lb></lb>nales, affirmativos vel negativos. </s> <s>Numerator ille A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> ſeu —T-X<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end><lb></lb>in ſeriem indeterminatam per Methodum noſtram Serierum conver<lb></lb>gentium reducta, evadit T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>-<emph type="italics"></emph>n<emph.end type="italics"></emph.end>XT<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>+(<emph type="italics"></emph>nn-n<emph.end type="italics"></emph.end>/2)XXT<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-2<emph.end type="sup"></emph.end> &c. </s> <s><lb></lb>Et collatis hujus terminis cum terminis Numeratoris alterius <lb></lb>RGG-RFF+TFF-FFX, fit RGG-RFF+TFF ad T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end><lb></lb>ut-FF ad-<emph type="italics"></emph>n<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>+(<emph type="italics"></emph>nn-n<emph.end type="italics"></emph.end>/2)XT<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-2<emph.end type="sup"></emph.end> &c. </s> <s>Et ſumendo ratio<lb></lb>nes ultimas ubi Orbes ad formam circularem accedunt, fit RGG <lb></lb>ad T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> ut-FF ad-<emph type="italics"></emph>n<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>, ſeu GG ad T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end> ut FF ad <emph type="italics"></emph>n<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>, <lb></lb>& viciſſim GG ad FF ut T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end> ad <emph type="italics"></emph>n<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end> id eſt ut 1 ad <emph type="italics"></emph>n<emph.end type="italics"></emph.end>; <lb></lb>adeoque G ad F, id eſt angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>VCP,<emph.end type="italics"></emph.end><pb xlink:href="039/01/156.jpg" pagenum="128"></pb><arrow.to.target n="note104"></arrow.to.target>ut 1 ad √<emph type="italics"></emph>n.<emph.end type="italics"></emph.end>Quare cum angulus <emph type="italics"></emph>VCP,<emph.end type="italics"></emph.end>in deſcenſu corporis <lb></lb>ab Apſide ſumma ad Apſidem imam in Ellipſi confectus, ſit <lb></lb>graduum 180; conficietur angulus <emph type="italics"></emph>VCp,<emph.end type="italics"></emph.end>in deſcenſu corporis <lb></lb>ab Apſide ſumma ad Apſidem imam, in Orbe propemodum Cir<lb></lb>culari quem corpus quodvis vi centripeta dignitati A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-3<emph.end type="sup"></emph.end> pro<lb></lb>portionali deſcribit, æqualis angulo graduum (180/√<emph type="italics"></emph>n<emph.end type="italics"></emph.end>); & hoc angulo <lb></lb>repetito corpus redibit ab Apſide ima ad Apſidem ſummam, & <lb></lb>ſic deinceps in infinitum. </s> <s>Ut ſi vis centripeta ſit ut diſtantia cor<lb></lb>poris a centro, id eſt, ut A ſeu (A<emph type="sup"></emph>4<emph.end type="sup"></emph.end>/A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>), erit <emph type="italics"></emph>n<emph.end type="italics"></emph.end>æqualis 4 & √<emph type="italics"></emph>n<emph.end type="italics"></emph.end>æqualis 2; <lb></lb>adeoque angulus inter Apſidem ſummam & Apſidem imam æ<lb></lb>qualis (180/2) <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>ſeu 90 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>Completa igitur quarta parte revolutio<lb></lb>nis unius corpus perveniet ad Apſidem imam, & completa alia <lb></lb>quarta parte ad Apſidem ſummam, & ſic deinceps per vices in <lb></lb>infinitum. </s> <s>Id quod etiam ex Propoſitione x. </s> <s>manifeſtum eſt. </s> <s>Nam <lb></lb>corpus urgente hac vi centripeta revolvetur in Ellipſi immobili, <lb></lb>cujus centrum eſt in centro virium. </s> <s>Quod ſi vis centripeta ſit reci<lb></lb>proce ut diſtantia, id eſt directe ut 1/A ſeu (A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>/A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>), erit <emph type="italics"></emph>n<emph.end type="italics"></emph.end>æqualis 2, ad<lb></lb>eoQ.E.I.ter Apſidem ſummam & imam angulus erit graduum (180/√2) <lb></lb>ſeu 127 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>16 <emph type="italics"></emph>m.<emph.end type="italics"></emph.end>45 <emph type="italics"></emph>ſec.<emph.end type="italics"></emph.end>& propterea corpus tali vi revolvens, perpe<lb></lb>tua anguli hujus repetitione, vicibus alternis ab Apſide ſumma ad <lb></lb>imam & ab ima ad ſummam perveniet in æternum. </s> <s>Porro ſi vis <lb></lb>centripeta ſit reciproce ut latus quadrato-quadratum undecimæ <lb></lb>dignitatis altitudinis, id eſt reciproce ut A (11/4), adeoQ.E.D.recte ut <lb></lb>(1/A<emph type="sup"></emph>11/4<emph.end type="sup"></emph.end>) ſeu ut (A<emph type="sup"></emph>1/4<emph.end type="sup"></emph.end>/A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>) erit <emph type="italics"></emph>n<emph.end type="italics"></emph.end>æqualis 1/4, & (180/√<emph type="italics"></emph>n<emph.end type="italics"></emph.end>) <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>æqualis 360 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>& prop<lb></lb>terea corpus de Apſide ſumma diſcedens & ſubinde perpetuo de<lb></lb>ſcendens, perveniet ad Apſidem imam ubi complevit revolutionem <lb></lb>integram, dein perpetuo aſcenſu complendo aliam revolutionem in<lb></lb>regram, redibit ad Apſidem ſummam: & ſic per vices in æternum. </s></p> <p type="margin"> <s><margin.target id="note104"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>3. Aſſumentes <emph type="italics"></emph>m<emph.end type="italics"></emph.end>& <emph type="italics"></emph>n<emph.end type="italics"></emph.end>pro quibuſvis indicibus dignitatum <lb></lb>Altitudinis, & <emph type="italics"></emph>b, c<emph.end type="italics"></emph.end>pro numeris quibuſvis datis, ponamus vim cen<lb></lb>tripetam eſſe ut (<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>), id eſt, ut (<emph type="italics"></emph>b<emph.end type="italics"></emph.end>in —T-X<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>in —T-X<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) <lb></lb>ſeu (per eandem Methodum noſtram Serierum convergentium) ut <lb></lb>(<emph type="italics"></emph>b<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>-<emph type="italics"></emph>mb<emph.end type="italics"></emph.end>XT<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>-<emph type="italics"></emph>nc<emph.end type="italics"></emph.end>XT<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>+(<emph type="italics"></emph>mm-mb<emph.end type="italics"></emph.end>/2)XXT<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-2<emph.end type="sup"></emph.end>+(<emph type="italics"></emph>nn-nc<emph.end type="italics"></emph.end>/2)XXT<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-2<emph.end type="sup"></emph.end> <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end>/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) <pb xlink:href="039/01/157.jpg" pagenum="129"></pb>& collatis numeratorum terminis, fiet RGG-RFF+TFF <lb></lb><arrow.to.target n="note105"></arrow.to.target>ad <emph type="italics"></emph>b<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, ut -FF ad -<emph type="italics"></emph>mb<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>-<emph type="italics"></emph>nc<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end><lb></lb>+(<emph type="italics"></emph>mm-m<emph.end type="italics"></emph.end>/2)<emph type="italics"></emph>b<emph.end type="italics"></emph.end>XT<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-2<emph.end type="sup"></emph.end>+(<emph type="italics"></emph>nn-n<emph.end type="italics"></emph.end>/2)<emph type="italics"></emph>c<emph.end type="italics"></emph.end>XT<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-2<emph.end type="sup"></emph.end> &c. </s> <s>Et ſumendo rationes ulti<lb></lb>mas quæ prodeunt ubi Orbes ad formam circularem accedunt, fit <lb></lb>GG ad <emph type="italics"></emph>b<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>, ut FF ad <emph type="italics"></emph>mb<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>+<emph type="italics"></emph>nc<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>, & <lb></lb>viciſſim GG ad FF ut <emph type="italics"></emph>b<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end> ad <emph type="italics"></emph>mb<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>+<emph type="italics"></emph>nc<emph.end type="italics"></emph.end>T<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>. </s> <s><lb></lb>Quæ proportio, exponendo altitudinem maximam <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>ſeu T Arith<lb></lb>metice per Unitatem, fit GG ad FF ut <emph type="italics"></emph>b+c<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>mb+nc,<emph.end type="italics"></emph.end>adeoque ut <lb></lb>1 ad (<emph type="italics"></emph>mb+nc/b+c<emph.end type="italics"></emph.end>). Unde eſt G ad F, id eſt angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>ad angulum <lb></lb><emph type="italics"></emph>VCP,<emph.end type="italics"></emph.end>ut 1 ad √(<emph type="italics"></emph>mb+nc/b+c<emph.end type="italics"></emph.end>). Et propterea cum angulus <emph type="italics"></emph>VCP<emph.end type="italics"></emph.end>inter <lb></lb>Apſidem ſummam & Apſidem imam in Ellipſi immobili ſit 180 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end><lb></lb>erit angulus <emph type="italics"></emph>VCp<emph.end type="italics"></emph.end>inter eaſdem Apſides, in Orbe quem corpus vi <lb></lb>centripeta quantitati (<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>) proportionali deſcribit, æqua<lb></lb>lis angulo graduum 180 √(<emph type="italics"></emph>b+c/mb+nc<emph.end type="italics"></emph.end>). Et eodem argumento ſi vis cen<lb></lb>tripeta ſit ut (<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>-<emph type="italics"></emph>c<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>), angulus inter Apſides invenietur graduum <lb></lb>180 √(<emph type="italics"></emph>b-c/mb-nc<emph.end type="italics"></emph.end>). Nec ſecus reſolvetur Problema in caſibus diffi<lb></lb>cilioribus. </s> <s>Quantitas cui vis centripeta proportionalis eſt, re<lb></lb>ſolvi ſemper debet in Series convergentes denominatorem ha<lb></lb>bentes A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>Dein pars data numeratoris qui ex illa operatione <lb></lb>provenit ad ipſius partem alteram non datam, & pars data nu<lb></lb>meratoris hujus RGG-RFF+TFF-FFX ad ipſius partem <lb></lb>alteram non datam in eadem ratione ponendæ ſunt: Et quantitates <lb></lb>ſuperfluas delendo, ſcribendoque Unitatem pro T, obtinebitur <lb></lb>proportio G ad F. </s></p> <p type="margin"> <s><margin.target id="note105"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi vis centripeta ſit ut aliqua altitudinis digNI<lb></lb>tas, inveniri poteſt dignitas illa ex motu Apſidum; & contra. </s> <s><lb></lb>Nimirum ſi motus totus angularis, quo corpus redit ad Apſidem <lb></lb>eandem, ſit ad motum angularem revolutionis unius, ſeu graduum <lb></lb>360, ut numerus aliquis <emph type="italics"></emph>m<emph.end type="italics"></emph.end>ad numerum alium <emph type="italics"></emph>n,<emph.end type="italics"></emph.end>& altitudo no<lb></lb>minetur A: erit vis ut altitudinis dignitas illa A<emph type="sup"></emph>(<emph type="italics"></emph>nn/mm<emph.end type="italics"></emph.end>)-3<emph.end type="sup"></emph.end>, cujus In-<pb xlink:href="039/01/158.jpg" pagenum="130"></pb><arrow.to.target n="note106"></arrow.to.target>dex eſt (<emph type="italics"></emph>nn/mm<emph.end type="italics"></emph.end>)-3. Id quod per Exempla ſecunda manifeſtum eſt. </s> <s><lb></lb>Unde liquet vim illam in majore quam triplicata altitudinis ratione, <lb></lb>in receſſu a centro, decreſcere non poſſe: Corpus tali vi revolvens <lb></lb>deque Apſide diſcedens, ſi cæperit deſcendere nunquam perveniet <lb></lb>ad Apſidem imam ſeu altitudinem minimam, ſed deſcendet uſque ad <lb></lb>centrum, deſcribens Curvam illam lineam de qua egimus in Cor. </s> <s>3. <lb></lb>Prop. </s> <s>XLI. </s> <s>Sin cæperit illud, de Apſide diſcedens, vel minimum <lb></lb>aſcendere; aſcendet in infinitum, neque unquam perveniet ad Ap<lb></lb>ſidem ſummam. </s> <s>Deſcribet enim Curvam illam lineam de qua ac<lb></lb>tum eſt in eodem Corol. </s> <s>& in Corol. </s> <s>6, Prop. </s> <s>XLIV. </s> <s>Sic & ubi <lb></lb>vis, in receſſu a centro, decreſcit in majore quam triplicata ratione <lb></lb>altitudinis, corpus de Apſide diſcedens, perinde ut cæperit deſcen<lb></lb>dere vel aſcendere, vel deſcendet ad centrum uſque vel aſcendet <lb></lb>in infinitum. </s> <s>At ſi vis, in receſſu a centro, vel decreſcat in minore <lb></lb>quam triplicata ratione altitudinis, vel creſcat in altitudinis ratione <lb></lb>quacunque; corpus nunquam deſcendet ad centrum uſque, ſed ad <lb></lb>Apſidem imam aliquando perveniet: & contra, ſi corpus de Apſi<lb></lb>de ad Apſidem alternis vicibus deſcendens & aſcendens nunquam <lb></lb>appellat ad centrum; vis in receſſu a centro aut augebitur, aut in <lb></lb>minore quam triplicata altitudinis ratione decreſcet: & quo ci<lb></lb>tius corpus de Apſide ad Apſidem redierit, eo longius ratio virium <lb></lb>recedet a ratione illa triplicata. </s> <s>Ut ſi corpus revolutionibus 8 vel <lb></lb>4 vel 2 vel 1 1/2 de Apſide ſumma ad Apſidem ſummam alterno de<lb></lb>ſcenſu & aſcenſu redierit; hoc eſt, ſi fuerit <emph type="italics"></emph>m<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ut 8 vel 4 vel <lb></lb>2 vel 1 1/2 ad 1, adeoque (<emph type="italics"></emph>nn/mm<emph.end type="italics"></emph.end>)-3 valeat (1/64)-3 vel (1/16) -3 vel 1/4-3 <lb></lb>vel 4/9-3: erit vis ut A<emph type="sup"></emph>(1/64)-3<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>(1/16)-3<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>1/4-3<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>4/9-3<emph.end type="sup"></emph.end>, <lb></lb>id eſt, reciproce ut A<emph type="sup"></emph>3-(1/64)<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>3-(1/16)<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>3-1/4<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>3-4/9<emph.end type="sup"></emph.end>. </s> <s><lb></lb>Si corpus ſingulis revolutionibus redierit ad Apſidem eandem immo<lb></lb>tam; erit <emph type="italics"></emph>m<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ut 1 ad 1, adeoque A (<emph type="italics"></emph>nn/mm<emph.end type="italics"></emph.end>)-3 æqualis A<emph type="sup"></emph>-2<emph.end type="sup"></emph.end> ſeu (1/AA<gap></gap>) <lb></lb>& propterea decrementum virium in ratione duplicata altitudinis, <lb></lb>ut in præcedentibus demonſtratum eſt. </s> <s>Si corpus partibus revo<lb></lb>lutionis unius vel tribus quartis, vel duabus tertiis, vel una ter<lb></lb>tia, vel una quarta, ad Apſidem eandem redierit; erit <emph type="italics"></emph>m<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ut <lb></lb>1/4 vel 2/3 vel 1/3 vel 1/4 ad 1, adeoque A(<emph type="italics"></emph>nn/mm<emph.end type="italics"></emph.end>)-3 æqualis A<emph type="sup"></emph>(16/9)-3<emph.end type="sup"></emph.end> vel <lb></lb>A<emph type="sup"></emph>9/4-3<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>9-3<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>16-3<emph.end type="sup"></emph.end>; & propterea vis aut reciproce ut <pb xlink:href="039/01/159.jpg" pagenum="131"></pb>A<emph type="sup"></emph>(11/9)<emph.end type="sup"></emph.end> vel A<emph type="sup"></emph>1/4<emph.end type="sup"></emph.end>, aut directe ut A<emph type="sup"></emph>6<emph.end type="sup"></emph.end> vel A <emph type="sup"></emph>13<emph.end type="sup"></emph.end>. </s> <s>Denique ſi corpus pergendo <lb></lb><arrow.to.target n="note107"></arrow.to.target>ab Apſide ſumma ad Apſidem ſummam confecerit revolutionem in<lb></lb>tegram, & præterea gradus tres, adeoque Apſis illa ſingulis corporis <lb></lb>revolutionibus confecerit in conſequentia gradus tres; erit <emph type="italics"></emph>m<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ut <lb></lb>363 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>ad 360<emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>ſive ut 121 ad 120, adeoque A<emph type="sup"></emph>(<emph type="italics"></emph>nn/mm<emph.end type="italics"></emph.end>)-3<emph.end type="sup"></emph.end> erit æquale <lb></lb>A<emph type="sup"></emph>-(29523/14641)<emph.end type="sup"></emph.end>; & propterea vis centripeta reciproce ut A <emph type="sup"></emph>(29523/14641)<emph.end type="sup"></emph.end> ſeu re<lb></lb>ciproce ut A<emph type="sup"></emph>2 (4/2+3)<emph.end type="sup"></emph.end> proxime. </s> <s>Decreſcit igitur vis centripeta in ratio<lb></lb>ne paulo majore quam duplicata, ſed quæ vicibus 59 3/4 propius ad <lb></lb>duplicatam quam ad triplicatam accedit. </s></p> <p type="margin"> <s><margin.target id="note106"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="margin"> <s><margin.target id="note107"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Hinc etiam ſi corpus, vi centripeta quæ ſit reciproce <lb></lb>ut quadratum altitudinis, revolvatur in Ellipſi umbilicum haben<lb></lb>te in centro virium, & huic vi centripetæ addatur vel auferatur <lb></lb>vis alia quævis extranea; cognoſci poteſt (per Exempla tertia) <lb></lb>motus Apſidum qui ex vi illa extranea orietur: & contra. </s> <s>Ut ſi <lb></lb>vis qua corpus revolvitur in Ellipſi ſit ut (1/AA), & vis extranea ab<lb></lb>lata ut <emph type="italics"></emph>c<emph.end type="italics"></emph.end>A, adeoque vis reliqua ut (A-<emph type="italics"></emph>c<emph.end type="italics"></emph.end>A<emph type="sup"></emph>4<emph.end type="sup"></emph.end>/A <emph type="italics"></emph>cub.<emph.end type="italics"></emph.end>); erit (in Exemplis ter<lb></lb>tiis) <emph type="italics"></emph>b<emph.end type="italics"></emph.end>æqualis 1, <emph type="italics"></emph>m<emph.end type="italics"></emph.end>æqualis 1, <emph type="italics"></emph>n<emph.end type="italics"></emph.end>æqualis 4, adeoque angulus revo<lb></lb>lutionis inter Apſides æqualis angulo graduum 180 √(1-<emph type="italics"></emph>c<emph.end type="italics"></emph.end>/1-4<emph type="sup"></emph><emph type="italics"></emph>c<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>). Po<lb></lb>natur vim illam extraneam eſſe 357,<emph type="sup"></emph>45<emph.end type="sup"></emph.end> partibus minorem quam vis <lb></lb>altera qua corpus revolvitur in Ellipſi, id eſt <emph type="italics"></emph>c<emph.end type="italics"></emph.end>eſſe (100/35745), exiſtente A <lb></lb>vel T æquali 1; & 180 √(1-<emph type="italics"></emph>c<emph.end type="italics"></emph.end>/1-4<emph type="sup"></emph><emph type="italics"></emph>c<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>) evadet 180 √(35645/35345), ſeu 180, 7623, <lb></lb>id eſt, 180 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>45 <emph type="italics"></emph>m.<emph.end type="italics"></emph.end>44 <emph type="italics"></emph>ſ.<emph.end type="italics"></emph.end>Igitur corpus de Apſide ſumma diſce<lb></lb>dens, motu angulari 180 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>45 <emph type="italics"></emph>m.<emph.end type="italics"></emph.end>44. <emph type="italics"></emph>ſ.<emph.end type="italics"></emph.end>perveniet ad Apſidem <lb></lb>imam, & hoc motu duplicato ad Apſidem ſummam redibit: adeo<lb></lb>que Apſis ſumma ſingulis revolutionibus progrediendo conficiet <lb></lb>1 <emph type="italics"></emph>gr.<emph.end type="italics"></emph.end>31 <emph type="italics"></emph>m.<emph.end type="italics"></emph.end>28 <emph type="italics"></emph>ſec.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Hactenus de Motu corporum in Orbibus quorum plana per <lb></lb>centrum Virium tranſeunt. </s> <s>Supereſt ut Motus etiam determine<lb></lb>mus in planis excentricis. </s> <s>Nam Scriptores qui Motum gravium <lb></lb>tractant, conſiderare ſolent aſcenſus & deſcenſus ponderum, <lb></lb>tam obliquos in planis quibuſcunQ.E.D.tis, quam perpendicu<lb></lb>lares: & pari jure Motus corporum Viribus quibuſcunque cen-<pb xlink:href="039/01/160.jpg" pagenum="132"></pb><arrow.to.target n="note108"></arrow.to.target>tra petentium, & planis excentricis innitentium hic conſiderandus <lb></lb>venit. </s> <s>Plana autem ſupponimus eſſe politiſſima & abſolute lubrica <lb></lb>ne corpora retardent. </s> <s>Quinimo, in his demonſtrationibus, vi<lb></lb>ce planorum quibus corpora incumbunt quæque tangunt incum<lb></lb>bendo, uſurpamus plana his parallela, in quibus centra corpo<lb></lb>rum moventur & Orbitas movendo deſcribunt. </s> <s>Et eadem lege <lb></lb>Motus corporum in ſuperficiebus Curvis peractos ſubinde de<lb></lb>terminamus. </s></p> <p type="margin"> <s><margin.target id="note108"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"></emph>SECTIO X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Motu Corporum in Superficiebus datis, deque Funipendulorum <lb></lb>Motu reciproco.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLVI. PROBLEMA XXXII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſita cujuſcunque generis Vi centripeta, datoque tum Virium cen<lb></lb>tro tum Plano quocunQ.E.I. quo corpus revolvitur, & conceſ<lb></lb>ſis Figurarum curvilinearum quadraturis: requiritur Motus cor<lb></lb>poris de loco dato, data cum Velocitate, ſecundum rectam in <lb></lb>Plano illo datam egreſſi.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>S<emph.end type="italics"></emph.end>centrum Virium, <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>diſtantia minima centri hujus a Plano <lb></lb>dato, <emph type="italics"></emph>P<emph.end type="italics"></emph.end>corpus de loco <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ſecundum rectam <emph type="italics"></emph>PZ<emph.end type="italics"></emph.end>egrediens, <emph type="italics"></emph>Q<emph.end type="italics"></emph.end><lb></lb>corpus idem in Trajectoria ſua revolvens, & <emph type="italics"></emph>PQR<emph.end type="italics"></emph.end>Trajectoria <lb></lb>illa, in Plano dato deſcripta, quam invenire oportet. </s> <s>Jungantur <emph type="italics"></emph>CQ <lb></lb>QS,<emph.end type="italics"></emph.end>& ſi in <emph type="italics"></emph>QS<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>SV<emph.end type="italics"></emph.end>proportionalis vi centripetæ qua <lb></lb>corpus trahitur verſus centrum <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>& agatur <emph type="italics"></emph>VT<emph.end type="italics"></emph.end>quæ fit parallela <lb></lb><emph type="italics"></emph>CQ<emph.end type="italics"></emph.end>& occurrat <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T:<emph.end type="italics"></emph.end>Vis <emph type="italics"></emph>SV<emph.end type="italics"></emph.end>reſolvetur (per Legum Corol. </s> <s>2.) <lb></lb>in vires <emph type="italics"></emph>ST, TV;<emph.end type="italics"></emph.end>quarum <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>trahendo corpus ſecundum lineam <lb></lb>plano perpendicularem, nil mutat motum ejus in hoc plano. </s> <s>Vis <lb></lb>autem altera <emph type="italics"></emph>TV,<emph.end type="italics"></emph.end>agendo ſecundum poſitionem plani, trahit cor<lb></lb>pus directe verſus punctum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>in plano datum, adeoque facit illud <lb></lb>in hoc plano perinde moveri ac ſi vis <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>tolleretur, & corpus vi <lb></lb>ſola <emph type="italics"></emph>TV<emph.end type="italics"></emph.end>revolveretur circa centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>in ſpatio libero. </s> <s>Data autem <pb xlink:href="039/01/161.jpg" pagenum="133"></pb>vi centripeta <emph type="italics"></emph>TV<emph.end type="italics"></emph.end>qua corpus <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>in ſpatio libero circa centrum <lb></lb><arrow.to.target n="note109"></arrow.to.target>datum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>revolvitur, datur per Prop. </s> <s>XLII, tum Trajectoria <emph type="italics"></emph>PQR<emph.end type="italics"></emph.end><lb></lb>quam corpus deſcribit, tum locus <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>in quo corpus ad datum quod<lb></lb>vis tempus verſabitur, tum denique velocitas corporis in loco illo <lb></lb><emph type="italics"></emph>Q<emph.end type="italics"></emph.end>; & contra. <emph type="italics"></emph><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note109"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLVII. THEOREMA XV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſito quod Vis centripeta proportionalis ſit diſtantiæ corporis a <lb></lb>centro; corpora omnia in planis quibuſcunque revolventia de<lb></lb>ſcribent Ellipſes, & revolutiones Temporibus æqualibus peragent; <lb></lb>quæque moventur in lineis rectis, ultro citroQ.E.D.ſcurrendo, <lb></lb>ſingulas eundi & redeundi periodos iiſdem Temporibus abſol<lb></lb>vent.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam, ſtantibus quæ <lb></lb><figure id="id.039.01.161.1.jpg" xlink:href="039/01/161/1.jpg"></figure><lb></lb>in ſuperiore Propoſitio<lb></lb>ne, vis <emph type="italics"></emph>SV<emph.end type="italics"></emph.end>qua corpus <lb></lb><emph type="italics"></emph>Q<emph.end type="italics"></emph.end>in plano quovis <emph type="italics"></emph>PQR<emph.end type="italics"></emph.end><lb></lb>revolvens trahitur ver<lb></lb>ſus centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>eſt ut di<lb></lb>ſtantia <emph type="italics"></emph><expan abbr="Sq;">Sque</expan><emph.end type="italics"></emph.end>atque adeo <lb></lb>ob proportionales <emph type="italics"></emph>SV<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>SQ, TV<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CQ,<emph.end type="italics"></emph.end>vis <lb></lb><emph type="italics"></emph>TV<emph.end type="italics"></emph.end>qua corpus trahi<lb></lb>tur verſus punctum <emph type="italics"></emph>C<emph.end type="italics"></emph.end><lb></lb>in Orbis plano datum, <lb></lb>eſt ut diſtantia <emph type="italics"></emph>C Q.<emph.end type="italics"></emph.end>Vi<lb></lb>res igitur, quibus cor<lb></lb>pora in plano <emph type="italics"></emph>PQR<emph.end type="italics"></emph.end><lb></lb>verſantia trahuntur ver<lb></lb>ſus punctum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>ſunt pro <lb></lb>ratione diſtantiarum æquales viribus quibus corpora undiquaque <lb></lb>trahuntur verſus centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>; & propterea corpora movebuntur iiſ<lb></lb>dem Temporibus, in iiſdem Figuris, in plano quovis <emph type="italics"></emph>PQR<emph.end type="italics"></emph.end>circa <lb></lb>punctum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>atQ.E.I. ſpatiis liberis circa centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>; adeoque (per <lb></lb>Corol. </s> <s>2. Prop. </s> <s>X, & Corol. </s> <s>2. Prop. </s> <s>XXXVIII) Temporibus ſemper <pb xlink:href="039/01/162.jpg" pagenum="134"></pb><arrow.to.target n="note110"></arrow.to.target>æqualibus, vel deſcribent Ellipſes in plano illo circa centrum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end><lb></lb>vel periodos movendi ultro citroQ.E.I. lineis rectis per centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end><lb></lb>in plano illo ductis, complebunt. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note110"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>His affines ſunt aſcenſus ac deſcenſus corporum in ſuperficiebus <lb></lb>curvis. </s> <s>Concipe lineas curvas in plano deſcribi, dein circa axes <lb></lb>quoſvis datos per centrum Virium tranſeuntes revolvi, & ea revo<lb></lb>lutione ſuperficies curvas deſcribere; tum corpora ita moveri ut <lb></lb>eorum centra in his ſuperficiebus perpetuo reperiantur. </s> <s>Si cor<lb></lb>pora illa oblique aſcendendo & deſcendendo currant ultro citroque <lb></lb>peragentur eorum motus in planis per axem tranſeuntibus, atque <lb></lb>adeo in lineis curvis quarum revolutione curvæ illæ ſuperficies ge<lb></lb>nitæ ſunt. </s> <s>Iſtis igitur in caſibus ſufficit motum in his lineis cur<lb></lb>vis conſiderare. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLVIII. THEOREMA XVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Rota Globo extrinſecus ad angulos rectos inſiſtat, & more ro<lb></lb>tarum revolvendo progrediatur in circulo maximo; longitudo <lb></lb>Itineris curvilinei, quod punctum quodvis in Rotæ perimetro da<lb></lb>tum, ex quo Globum tetigit, confecit, (quodque Cycloidem vel <lb></lb>Epicycloidem nominare licet) erit ad duplicatum ſinum verſum <lb></lb>arcus dimidii qui Globum ex eo tempore inter eundum tetigit, <lb></lb>ut ſumma diametrorum Globi & Rotæ ad ſemidiametrum Globi.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLIX. THEOREMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Rota Globo concavo ad rectos angulos intrinſecus inſiſtat & re<lb></lb>volvendo progrediatur in circulo maximo; longitudo Itineris <lb></lb>curvilinei quod punctum quodvis in Rotæ perimetro datum, ex <lb></lb>quo Globum tetigit, confecit, erit ad duplicatum ſinum verſum <lb></lb>arcus dimidii qui Globum toto hoc tempore inter eundum teti<lb></lb>git, ut differentia diametrorum Globi & Rotæ ad ſemidiame<lb></lb>trum Globi.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/163.jpg" pagenum="135"></pb> <p type="main"> <s>Sit <emph type="italics"></emph>ABL<emph.end type="italics"></emph.end>Globus, <emph type="italics"></emph>C<emph.end type="italics"></emph.end>centrum ejus, <emph type="italics"></emph>BPV<emph.end type="italics"></emph.end>Rota ei inſiſtens, <emph type="italics"></emph>E<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note111"></arrow.to.target>centrum Rotæ, <emph type="italics"></emph>B<emph.end type="italics"></emph.end>punctum contactus, & <emph type="italics"></emph>P<emph.end type="italics"></emph.end>punctum datum in pe<lb></lb>rimetro Rotæ. </s> <s>Concipe hanc Rotam pergere in circulo maximo <lb></lb><emph type="italics"></emph>ABL<emph.end type="italics"></emph.end>ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>per <emph type="italics"></emph>B<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>& inter eundum ita revolvi ut ar<lb></lb>cus <emph type="italics"></emph>AB, PB<emph.end type="italics"></emph.end>ſibi invicem ſemper æquentur, atque punctum illud <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>in perimetro Rotæ datum interea deſcribere Viam curvilineam <lb></lb><emph type="italics"></emph>AP.<emph.end type="italics"></emph.end>Sit autem <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>Via tota curvilinea deſcripta ex quo Rota <lb></lb>Globum tetigit in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>& erit Viæ hujus longitudo <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad duplum <lb></lb><figure id="id.039.01.163.1.jpg" xlink:href="039/01/163/1.jpg"></figure><lb></lb>ſinum verſum arcus 1/2 <emph type="italics"></emph>PB,<emph.end type="italics"></emph.end>ut 2 <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CB.<emph.end type="italics"></emph.end>Nam recta <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>(ſi <lb></lb>opus eſt producta) occurrat Rotæ in <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>junganturque <emph type="italics"></emph>CP, BP, <lb></lb>EP, VP,<emph.end type="italics"></emph.end>& in <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>productam demittatur normalis <emph type="italics"></emph>VF.<emph.end type="italics"></emph.end>Tan<lb></lb>gant <emph type="italics"></emph>PH, VH<emph.end type="italics"></emph.end>Circulum in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>V<emph.end type="italics"></emph.end>concurrentes in <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>ſecetque <lb></lb><emph type="italics"></emph>PH<emph.end type="italics"></emph.end>ipſam <emph type="italics"></emph>VF<emph.end type="italics"></emph.end>in <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>& ad <emph type="italics"></emph>VP<emph.end type="italics"></emph.end>demittantur normales <emph type="italics"></emph>GI, HK.<emph.end type="italics"></emph.end><pb xlink:href="039/01/164.jpg" pagenum="136"></pb><arrow.to.target n="note112"></arrow.to.target>Centro item <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& intervallo quovis deſcribatur circulus <emph type="italics"></emph>nom<emph.end type="italics"></emph.end>ſe<lb></lb>cans rectam <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>in <emph type="italics"></emph>n,<emph.end type="italics"></emph.end>Rotæ perimetrum <emph type="italics"></emph>BP<emph.end type="italics"></emph.end>&c. </s> <s>in <emph type="italics"></emph>o,<emph.end type="italics"></emph.end>& Viam curvi<lb></lb>lineam <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>in <emph type="italics"></emph>m;<emph.end type="italics"></emph.end>centroque <emph type="italics"></emph>V<emph.end type="italics"></emph.end>& intervallo <emph type="italics"></emph>Vo<emph.end type="italics"></emph.end>deſcribatur circu<lb></lb>lus ſecans <emph type="italics"></emph>VP<emph.end type="italics"></emph.end>productam in <emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note111"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="margin"> <s><margin.target id="note112"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Quoniam Rota eundo ſemper revolvitur circa punctum con<lb></lb>tactus <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>manifeſtum eſt quod recta <emph type="italics"></emph>BP<emph.end type="italics"></emph.end>perpendicularis eſt ad <lb></lb><figure id="id.039.01.164.1.jpg" xlink:href="039/01/164/1.jpg"></figure><lb></lb>lineam illam curvam <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>quam Rotæ punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>deſcribit, atque <lb></lb>adeo quod recta <emph type="italics"></emph>VP<emph.end type="italics"></emph.end>tanget hanc curvam in puncto <emph type="italics"></emph>P.<emph.end type="italics"></emph.end>Circuli <lb></lb><emph type="italics"></emph>nom<emph.end type="italics"></emph.end>radius ſenſim auctus vel diminutus æquetur tandem diſtantiæ <lb></lb><emph type="italics"></emph>CP<emph.end type="italics"></emph.end>; &, ob ſimilitudinem Figuræ evaneſcentis <emph type="italics"></emph>Pnomq<emph.end type="italics"></emph.end>& Figuræ <lb></lb><emph type="italics"></emph>PFGVI,<emph.end type="italics"></emph.end>ratio ultima lineolarum evaneſcentium <emph type="italics"></emph>Pm, Pn, Po, Pq,<emph.end type="italics"></emph.end><pb xlink:href="039/01/165.jpg" pagenum="137"></pb>id eſt, ratio mutationum momentanearum curvæ <emph type="italics"></emph>AP,<emph.end type="italics"></emph.end>rectæ <lb></lb><arrow.to.target n="note113"></arrow.to.target><emph type="italics"></emph>CP,<emph.end type="italics"></emph.end>arcus circularis <emph type="italics"></emph>BP,<emph.end type="italics"></emph.end>ac rectæ <emph type="italics"></emph>VP,<emph.end type="italics"></emph.end>eadem erit quæ linea<lb></lb>rum <emph type="italics"></emph>PV, PF, PG, PI<emph.end type="italics"></emph.end>reſpective. </s> <s>Cum autem <emph type="italics"></emph>VF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CF<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>VH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>perpendiculares ſunt, angulique <emph type="italics"></emph>HVG, VCF<emph.end type="italics"></emph.end>prop<lb></lb>terea æquales; & angulus <emph type="italics"></emph>VHG<emph.end type="italics"></emph.end>(ob angulos quadrilateri <emph type="italics"></emph>HVEP<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>V<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>rectos) angulo <emph type="italics"></emph>CEP<emph.end type="italics"></emph.end>æqualis eſt, ſimilia erunt tri<lb></lb>angula <emph type="italics"></emph>VHG, CEP<emph.end type="italics"></emph.end>; & inde fiet ut <emph type="italics"></emph>EP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ita <emph type="italics"></emph>HG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HV<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>HP<emph.end type="italics"></emph.end>& ita <emph type="italics"></emph>KI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KP,<emph.end type="italics"></emph.end>& compoſite vel diviſim ut <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ita <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PK,<emph.end type="italics"></emph.end>& duplicatis conſequentibus ut <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>ad 2 <emph type="italics"></emph>CE<emph.end type="italics"></emph.end><lb></lb>ita <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PV,<emph.end type="italics"></emph.end>atQ.E.I.a adeo <emph type="italics"></emph>Pq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Pm.<emph.end type="italics"></emph.end>Eſt igitur decremen<lb></lb>tum lineæ <emph type="italics"></emph>VP,<emph.end type="italics"></emph.end>id eſt, incrementum lineæ <emph type="italics"></emph>BV-VP<emph.end type="italics"></emph.end>ad incremen<lb></lb>tum lineæ curvæ <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>in data ratione <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>ad 2 <emph type="italics"></emph>CE,<emph.end type="italics"></emph.end>& prop<lb></lb>terea (per Corol. </s> <s>Lem. </s> <s>IV.) longitudines <emph type="italics"></emph>BV-VP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AP,<emph.end type="italics"></emph.end>in<lb></lb>crementis illis genitæ, ſunt in eadem ratione. </s> <s>Sed, exiſtente <emph type="italics"></emph>BV<emph.end type="italics"></emph.end>ra<lb></lb>dio, eſt <emph type="italics"></emph>VP<emph.end type="italics"></emph.end>co-ſinus anguli <emph type="italics"></emph>BVP<emph.end type="italics"></emph.end>ſeu 1/2 <emph type="italics"></emph>BEP,<emph.end type="italics"></emph.end>adeoque <emph type="italics"></emph>BV-VP<emph.end type="italics"></emph.end><lb></lb>ſinus verſus ejuſdem anguli; & propterea in hac Rota, cujus radius <lb></lb>eſt 1/2 <emph type="italics"></emph>BV,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>BV-VP<emph.end type="italics"></emph.end>duplus ſinus verſus arcus 1/2 <emph type="italics"></emph>BP.<emph.end type="italics"></emph.end>Ergo <lb></lb><emph type="italics"></emph>AP<emph.end type="italics"></emph.end>eſt ad duplum ſinum verſum arcus 1/2 <emph type="italics"></emph>BP<emph.end type="italics"></emph.end>ut 2 <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CB. <lb></lb><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note113"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Lineam autem <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>in Propoſitione priore Cycloidem extra <lb></lb>Globum, alteram in poſteriore Cycloidem intra Globum diſtincti<lb></lb>onis gratia nominabimus. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi deſcribatur Cyclois integra <emph type="italics"></emph>ASL<emph.end type="italics"></emph.end>& biſecetur <lb></lb>ea in <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>erit longitudo partis <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad longitudinem <emph type="italics"></emph>VP<emph.end type="italics"></emph.end>(quæ du<lb></lb>plus eſt ſinus anguli <emph type="italics"></emph>VBP,<emph.end type="italics"></emph.end>exiſtente <emph type="italics"></emph>EB<emph.end type="italics"></emph.end>radio) ut 2 <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CB,<emph.end type="italics"></emph.end><lb></lb>atque adeo in ratione data. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et longitudo ſemiperimetri Cycloidis <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>æquabitur <lb></lb>lineæ rectæ quæ eſt ad Rotæ diametrum <emph type="italics"></emph>BV,<emph.end type="italics"></emph.end>ut 2 <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CB.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO L. PROBLEMA XXXIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Facere ut Corpus pendulum oſcilletur in Cycloide data.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Intra Globum <emph type="italics"></emph>QVS,<emph.end type="italics"></emph.end>centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>deſcriptum, detur Cyclois <emph type="italics"></emph>QRS<emph.end type="italics"></emph.end><lb></lb>biſecta in <emph type="italics"></emph>R<emph.end type="italics"></emph.end>& punctis ſuis extremis <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ſuperficiei Globi hinc <lb></lb>inde occurrens. </s> <s>Agatur <emph type="italics"></emph>CR<emph.end type="italics"></emph.end>biſecans arcum <emph type="italics"></emph>QS<emph.end type="italics"></emph.end>in <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>& produca<lb></lb>tur ea ad <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CO<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CR.<emph.end type="italics"></emph.end>Centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>in-<pb xlink:href="039/01/166.jpg" pagenum="138"></pb><arrow.to.target n="note114"></arrow.to.target>tervallo <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>deſeribatur Globus exterior <emph type="italics"></emph>ABD,<emph.end type="italics"></emph.end>& intra hunc Glo<lb></lb>bum a Rota, cujus diameter ſit <emph type="italics"></emph>AO,<emph.end type="italics"></emph.end>deſcribantur duæ Semicycloides <lb></lb><emph type="italics"></emph>AQ, AS,<emph.end type="italics"></emph.end>quæ Globum interiorem tangant in <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end>& Globo ex<lb></lb>teriori occurrant in <emph type="italics"></emph>A.<emph.end type="italics"></emph.end>A puncto illo <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>Filo <emph type="italics"></emph>APT<emph.end type="italics"></emph.end>longitudinem <lb></lb><emph type="italics"></emph>AR<emph.end type="italics"></emph.end>æquante, pendeat corpus <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>& ita intra Semicycloides <emph type="italics"></emph>AQ, <lb></lb>AS<emph.end type="italics"></emph.end>oſcilletur, ut quoties pendulum digreditur a perpendiculo <emph type="italics"></emph>AR,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.166.1.jpg" xlink:href="039/01/166/1.jpg"></figure><lb></lb>Filum parte ſui ſuperiore <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>applicetur ad Semicycloidem illam <lb></lb><emph type="italics"></emph>APS<emph.end type="italics"></emph.end>verſus quam peragitur motus, & circum eam ceu obſtacu<lb></lb>lum flectatur, parteque reliqua <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>cui Semicyclois nondum obji<lb></lb>citur, protendatur in lineam rectam; & pondus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>oſcillabitur in <lb></lb>Cycloide data <emph type="italics"></emph>QRS. <expan abbr="q.">que</expan> E. F.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note114"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Occurrat enim Filum <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>tum Cycloidi <emph type="italics"></emph>QRS<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>tum circulo <lb></lb><emph type="italics"></emph>QOS<emph.end type="italics"></emph.end>in <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>agaturque <emph type="italics"></emph>CV;<emph.end type="italics"></emph.end>& ad Fili partem rectam <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>e punctis <lb></lb>extremis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>erigantur perpendicula <emph type="italics"></emph>PB, TW,<emph.end type="italics"></emph.end>occurrentia re<lb></lb>ctæ <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>in <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>W.<emph.end type="italics"></emph.end>Patet, ex conſtructione & geneſi ſimilium Fi<lb></lb>gurarum <emph type="italics"></emph>AS, SR,<emph.end type="italics"></emph.end>perpendicula illa <emph type="italics"></emph>PB, TW<emph.end type="italics"></emph.end>abſcindere de <emph type="italics"></emph>CV<emph.end type="italics"></emph.end>lon<lb></lb>gitudines <emph type="italics"></emph>VB, VW<emph.end type="italics"></emph.end>Rotarum diametris <emph type="italics"></emph>OA, OR<emph.end type="italics"></emph.end>æquales. </s> <s>Eſt igi<lb></lb>tur <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>VP<emph.end type="italics"></emph.end>(duplum ſinum anguli <emph type="italics"></emph>VBP<emph.end type="italics"></emph.end>exiſtente 1/2 <emph type="italics"></emph>BV<emph.end type="italics"></emph.end>ra-<pb xlink:href="039/01/167.jpg" pagenum="139"></pb>dio) ut <emph type="italics"></emph>BW<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BV,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>AO+OR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AO,<emph.end type="italics"></emph.end>id eſt (cum ſint <emph type="italics"></emph>CA<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note115"></arrow.to.target>ad <emph type="italics"></emph>CO, CO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CR<emph.end type="italics"></emph.end>& diviſim <emph type="italics"></emph>AO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OR<emph.end type="italics"></emph.end>proportionales,) ut <lb></lb><emph type="italics"></emph>CA+CO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>vel, ſi biſecetur <emph type="italics"></emph>BV<emph.end type="italics"></emph.end>in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>ut 2 <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CB.<emph.end type="italics"></emph.end><lb></lb>Proinde, per Corol. </s> <s>1. Prop. </s> <s>XLIX, longitudo partis rectæ Fili <emph type="italics"></emph>PT<emph.end type="italics"></emph.end><lb></lb>æquatur ſemper Cycloidis arcui <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end>& Filum totum <emph type="italics"></emph>APT<emph.end type="italics"></emph.end>æquatur <lb></lb>ſemper Cycloidis arcui dimidio <emph type="italics"></emph>APS,<emph.end type="italics"></emph.end>hoc eſt (per Corol. </s> <s>2. Prop. </s> <s><lb></lb>XLIX) longitudini <emph type="italics"></emph>AR.<emph.end type="italics"></emph.end>Et propterea viciſſim ſi Filum manet ſem<lb></lb>per æquale longitudini <emph type="italics"></emph>AR<emph.end type="italics"></emph.end>movebitur punctum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>in Cycloide <lb></lb>data <emph type="italics"></emph>QRS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note115"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Filum <emph type="italics"></emph>AR<emph.end type="italics"></emph.end>æquatur Semicycloidi <emph type="italics"></emph>AS,<emph.end type="italics"></emph.end>adeoque ad ſemi<lb></lb>diametrum <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>eandem habet rationem quam ſimilis illi Semicy<lb></lb>clois <emph type="italics"></emph>SR<emph.end type="italics"></emph.end>habet ad ſemidiametrum <emph type="italics"></emph>CO.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LI. THEOREMA XVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Vis centripeta tendens undique ad Globi centrum<emph.end type="italics"></emph.end>C <emph type="italics"></emph>ſit in locis <lb></lb>ſingulis ut diſtantia loci cujuſque a centro, & hac ſola Vi a<lb></lb>gente corpus<emph.end type="italics"></emph.end>T <emph type="italics"></emph>oſcilletur (modo jam deſcripto) in perimetro Cy<lb></lb>cloidis<emph.end type="italics"></emph.end>QRS: <emph type="italics"></emph>dico quod oſcillationum utcunQ.E.I.æqualium <lb></lb>æqualia erunt Tempora.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam in Cycloidis tangentem <emph type="italics"></emph>TW<emph.end type="italics"></emph.end>infinite productam cadat per<lb></lb>pendiculum <emph type="italics"></emph>CX<emph.end type="italics"></emph.end>& jungatur <emph type="italics"></emph>CT.<emph.end type="italics"></emph.end>Quoniam vis centripeta qua cor<lb></lb>pus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>impellitur verſus <emph type="italics"></emph>C<emph.end type="italics"></emph.end>eſt ut diſtantia <emph type="italics"></emph>CT,<emph.end type="italics"></emph.end>atque hæc (per Legum <lb></lb>Corol. </s> <s>2.) reſolvitur in partes <emph type="italics"></emph>CX, TX,<emph.end type="italics"></emph.end>quarum <emph type="italics"></emph>CX<emph.end type="italics"></emph.end>impellen<lb></lb>do corpus directe a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>diſtendit filum <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>& per ejus reſiſtentiam <lb></lb>tota ceſſat, nullum alium edens effectum; pars autem altera <emph type="italics"></emph>TX,<emph.end type="italics"></emph.end><lb></lb>urgendo corpus tranſverſim ſeu verſus <emph type="italics"></emph>X,<emph.end type="italics"></emph.end>directe accelerat motum <lb></lb>ejus in Cycloide; manifeſtum eſt quod corporis acceleratio, huic <lb></lb>vi acceleratrici proportionalis, ſit ſingulis momentis ut longitudo <lb></lb><emph type="italics"></emph>TX,<emph.end type="italics"></emph.end>id eſt, (ob datas <emph type="italics"></emph>CV, WV<emph.end type="italics"></emph.end>iiſque proportionales <emph type="italics"></emph>TX, TW,<emph.end type="italics"></emph.end>) <lb></lb>ut longitudo <emph type="italics"></emph>TW,<emph.end type="italics"></emph.end>hoc eſt (per Corol. </s> <s>1. Prop. </s> <s>XLIX,) ut longitudo <lb></lb>arcus Cycloidis <emph type="italics"></emph>TR.<emph.end type="italics"></emph.end>Pendulis igitur duobus <emph type="italics"></emph>APT, Apt<emph.end type="italics"></emph.end>de per<lb></lb>pendiculo <emph type="italics"></emph>AR<emph.end type="italics"></emph.end>inæqualiter deductis & ſimul dimiſſis, acceleratio<lb></lb>nes eorum ſemper erunt ut arcus deſcribendi <emph type="italics"></emph>TR, tR.<emph.end type="italics"></emph.end>Sunt au<lb></lb>tem partes ſub initio deſcriptæ ut accelerationes, hoc eſt, ut totæ <lb></lb>ſub initio deſcribendæ, & propterea partes quæ manent deſcriben-<pb xlink:href="039/01/168.jpg" pagenum="140"></pb><arrow.to.target n="note116"></arrow.to.target>dæ & accelerationes ſubſequentes, his partibus proportionales, ſunt <lb></lb>etiam ut totæ; & ſic deinceps. </s> <s>Sunt igitur accelerationes atque <lb></lb>adeo velocitates genitæ & partes his velocitatibus deſcriptæ par<lb></lb>teſQ.E.D.ſcribendæ, ſemper ut totæ; & propterea partes deſcriben<lb></lb>dæ datam ſervantes rationem ad invicem ſimul evaneſcent, id eſt, <lb></lb>corpora duo oſcillantia ſimul pervenient ad perpendiculum <emph type="italics"></emph>AR.<emph.end type="italics"></emph.end><lb></lb>Cumque viciſſim aſcenſus perpendiculorum de loco inſimo <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>per <lb></lb>eoſdem arcus Cycloidales motu retrogrado facti, retardentur in <lb></lb>locis ſingulis a viribus iiſdem a quibus deſcenſus accelerabantur, <lb></lb>patet velocitates aſcenſuum ac deſcenſuum per eoſdem arcus fa<lb></lb>ctorum æquales eſſe, atque adeo temporibus æqualibus fieri; & <lb></lb>propterea, cum Cycloidis partes duæ <emph type="italics"></emph>RS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>RQ<emph.end type="italics"></emph.end>ad utrumque per<lb></lb>pendiculi latus jacentes ſint ſimiles & æquales, pendula duo oſcil<lb></lb>lationes ſuas tam totas quam dimidias iiſdem temporibus ſemper <lb></lb>peragent. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note116"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Vis qua corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>in loco quovis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>acceleratur vel retar<lb></lb>tur in Cycloide, eſt ad totum corporis ejuſdem Pondus in loco <lb></lb>altiſſimo <emph type="italics"></emph>S<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end>ut Cycloidis arcus <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>ad ejuſdem arcum <emph type="italics"></emph>SR<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>QR.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LII. PROBLEMA XXXIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Definire & Velocitates Pendulorum in locis ſingulis, & Tempora <lb></lb>quibus tum oſcillationes totæ, tum ſingulæ oſcillationum partes <lb></lb>peraguntur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Centro quovis <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>Cycloidis arcum <emph type="italics"></emph>RS<emph.end type="italics"></emph.end>æquante, <lb></lb>deſcribe ſemicirculum <emph type="italics"></emph>HKMG<emph.end type="italics"></emph.end>ſemidiametro <emph type="italics"></emph>GK<emph.end type="italics"></emph.end>biſectum. </s> <s>Et <lb></lb>ſi vis centripeta, diſtantiis loeorum a centro proportionalis, tendat <lb></lb>ad centrum <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>ſitque ea in perimetro <emph type="italics"></emph>HIK<emph.end type="italics"></emph.end>æqualis vi centripetæ <lb></lb>in perimetro Globi <emph type="italics"></emph>QOS (Vide Fig. </s> <s>Prop.<emph.end type="italics"></emph.end>L.) ad ipſius cen<lb></lb>trum tendenti; & eodem tempore quo pendulum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>dimittitur e <lb></lb>loco ſupremo <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>cadat corpus aliquod <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ab <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>G:<emph.end type="italics"></emph.end>quoniam <lb></lb>vires quibus corpora urgentur ſunt æquales ſub initio & ſpatiis <lb></lb>deſcribendis <emph type="italics"></emph>TR, LG<emph.end type="italics"></emph.end>ſemper proportionales, atque adeo, ſi æ<lb></lb>quantur <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>LG,<emph.end type="italics"></emph.end>æquales in locis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end>; patet corpora illa <lb></lb>deſcribere ſpatia <emph type="italics"></emph>ST, HL<emph.end type="italics"></emph.end>æqualia ſub initio, adeoque ſubinde per<lb></lb>gere æqualiter urgeri, & æqualia ſpatia deſcribere. </s> <s>Quare, per Prop. </s> <s><lb></lb>XXXVIII, tempus quo corpus deſcribit arcum <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>eſt ad tempus <pb xlink:href="039/01/169.jpg" pagenum="141"></pb>oſcillationis unius, ut arcus <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>(tempus quo corpus <emph type="italics"></emph>H<emph.end type="italics"></emph.end>perveniet <lb></lb><arrow.to.target n="note117"></arrow.to.target>ad <emph type="italics"></emph>L<emph.end type="italics"></emph.end>) ad ſemiperipheriam <emph type="italics"></emph>HKM<emph.end type="italics"></emph.end>(tempus quo corpus <emph type="italics"></emph>H<emph.end type="italics"></emph.end>per<lb></lb>veniet ad <emph type="italics"></emph>M.<emph.end type="italics"></emph.end>) Et velocitas corporis penduli in loco <emph type="italics"></emph>T<emph.end type="italics"></emph.end>eſt ad ve<lb></lb>locitatem ipſius in loco infimo <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>(hoc eſt, velocitas corporis <emph type="italics"></emph>H<emph.end type="italics"></emph.end>in <lb></lb>loco <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ad velocitatem ejus in loco <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>ſeu incrementum momenta<lb></lb>neum lineæ <emph type="italics"></emph>HL<emph.end type="italics"></emph.end>ad incrementum momentaneum lineæ <emph type="italics"></emph>HG,<emph.end type="italics"></emph.end>arcu<lb></lb>bus <emph type="italics"></emph>HI, HK<emph.end type="italics"></emph.end>æquabili fluxu creſcentibus) ut ordinatim applicata <lb></lb><emph type="italics"></emph>LI<emph.end type="italics"></emph.end>ad radium <emph type="italics"></emph>GK,<emph.end type="italics"></emph.end>ſive ut √<emph type="italics"></emph><expan abbr="SRq.-TRq.">SRq.-TRque</expan><emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SR.<emph.end type="italics"></emph.end>Unde cum, <lb></lb>in oſcillationibus inæqualibus, deſcribantur æqualibus temporibus <lb></lb>arcus totis oſcillationum arcubus proportionales; habentur, ex da<lb></lb>tis temporibus, & velocitates & arcus deſcripti in oſcillationibus <lb></lb>univerſis. </s> <s>Quæ erant primo invenienda. </s></p> <p type="margin"> <s><margin.target id="note117"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Oſcillentur jam Funipendula <lb></lb><figure id="id.039.01.169.1.jpg" xlink:href="039/01/169/1.jpg"></figure><lb></lb>corpora in Cycloidibus diverſis <lb></lb>intra Globos diverſos, quorum <lb></lb>diverſæ ſunt etiam Vires abſolu<lb></lb>tæ, deſcriptis: &, ſi Vis abſolu<lb></lb>ta Globi cujuſvis <emph type="italics"></emph>QOS<emph.end type="italics"></emph.end>dicatur V, <lb></lb>Vis acceleratrix qua <expan abbr="Pendulũ">Pendulum</expan> urge<lb></lb>tur in circumferentia hujus Globi, <lb></lb>ubi incipit directe verſus centrum <lb></lb>ejus moveri, erit ut diſtantia Cor<lb></lb>poris penduli a centro illo & Vis abſoluta Globi conjunctim, hoc <lb></lb>eſt, ut <emph type="italics"></emph>CO<emph.end type="italics"></emph.end>XV. </s> <s>Itaque lineola <emph type="italics"></emph>HY,<emph.end type="italics"></emph.end>quæ ſit ut hæc Vis accelera<lb></lb>trix <emph type="italics"></emph>CO<emph.end type="italics"></emph.end>XV, deſcribetur dato tempore; &, ſi erigatur normalis <emph type="italics"></emph>YZ<emph.end type="italics"></emph.end><lb></lb>circumferentiæ occurrens in <emph type="italics"></emph>Z,<emph.end type="italics"></emph.end>arcus naſcens <emph type="italics"></emph>HZ<emph.end type="italics"></emph.end>denotabit datum <lb></lb>illud tempus. </s> <s>Eſt autem arcus hic naſcens <emph type="italics"></emph>HZ<emph.end type="italics"></emph.end>in ſubduplicata ra<lb></lb>tione rectanguli <emph type="italics"></emph>GHY,<emph.end type="italics"></emph.end>adeoque ut √<emph type="italics"></emph>GHXCO<emph.end type="italics"></emph.end>XV. </s> <s>Unde Tem<lb></lb>pus oſcillationis integræ in Cycloide <emph type="italics"></emph>QRS<emph.end type="italics"></emph.end>(cum ſit ut ſemiperi<lb></lb>pheria <emph type="italics"></emph>HKM,<emph.end type="italics"></emph.end>quæ oſcillationem illam integram denotat, directe, <lb></lb>utque arcus <emph type="italics"></emph>HZ,<emph.end type="italics"></emph.end>qui datum tempus ſimiliter denotat, inverſe) fiet <lb></lb>ut <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>directe & √<emph type="italics"></emph>GHXCO<emph.end type="italics"></emph.end>XV inverſe, hoc eſt, ob æquales <emph type="italics"></emph>GH<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>SR,<emph.end type="italics"></emph.end>ut √(<emph type="italics"></emph>SR/CO<emph.end type="italics"></emph.end>XV), ſive (per Corol. </s> <s>Prop. </s> <s>L) ut √(<emph type="italics"></emph>AR/AC<emph.end type="italics"></emph.end>XV). <lb></lb>Itaque Oſcillationes in Globis & Cycloidibus omnibus, quibuſ<lb></lb>cunque cum Viribus abſolutis factæ, ſunt in ratione quæ compo<lb></lb>nitur ex ſubduplicata ratione longitudinis Fili directe, & ſubdu<lb></lb>plicata ratione diſtantiæ inter punctum ſuſpenſionis & centrum <pb xlink:href="039/01/170.jpg" pagenum="142"></pb><arrow.to.target n="note118"></arrow.to.target>Globi inverſe, & ſubduplicata ratione Vis abſolutæ Globi etiam <lb></lb>inverſe. <emph type="italics"></emph><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note118"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc etiam Oſcillantium, Cadentium & Revolventium <lb></lb>corporum tempora poſſunt inter ſe conferri. </s> <s>Nam ſi Rotæ, qua Cy<lb></lb>clois intra globum deſcribitur, diameter conſtituatur æqualis ſemi<lb></lb>diametro globi, Cyclois evadet Linea recta per centrum globi tran<lb></lb>ſiens, & Oſcillatio jam erit deſcenſus & ſubſequens aſcenſus in hac <lb></lb>recta. </s> <s>Unde datur tum tempus deſcenſus de loco quovis ad <lb></lb>centrum, tum tempus huic æquale quo corpus uniformiter cir<lb></lb>ca centrum globi ad diſtantiam quamvis revolvendo arcum qua<lb></lb>drantalem deſcribit. </s> <s>Eſt enim hoc tempus (per Caſum ſecun<lb></lb>dum) ad tempus ſemioſcillationis in Cycloide quavis <emph type="italics"></emph>QRS<emph.end type="italics"></emph.end>ut <lb></lb>1 ad √(<emph type="italics"></emph>AR/AC<emph.end type="italics"></emph.end>). </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Hinc etiam conſectantur quæ <emph type="italics"></emph>Wrennus<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Hugenius<emph.end type="italics"></emph.end>de <lb></lb>Cycloide vulgari adinvenerunt. </s> <s>Nam ſi Globi diameter augeatur <lb></lb>in infinitum: mutabitur ejus ſuperficies ſphærica in planum, Viſque <lb></lb>centripeta aget uniformiter ſecundum lineas huic plano perpendi<lb></lb>culares, & Cyclois noſtra abibit in Cycloidem vulgi. </s> <s>Iſto autem <lb></lb>in caſu longitudo arcus Cycloidis, inter planum illud & punctum <lb></lb>deſcribens, æqualis evadet quadruplicato ſinui verſo dimidii arcus <lb></lb>Rotæ inter idem planum & punctum deſcribens; ut invenit <emph type="italics"></emph>Wren<lb></lb>nus:<emph.end type="italics"></emph.end>Et Pendulum inter duas ejuſmodi Cycloides in ſimili & æ<lb></lb>quali Cycloide temporibus æqualibus Oſcillabitur, ut demonſtravit <lb></lb><emph type="italics"></emph>Hugenius.<emph.end type="italics"></emph.end>Sed & Deſcenſus gravium, tempore Oſcillationis unius, <lb></lb>is erit quem <emph type="italics"></emph>Hugenius<emph.end type="italics"></emph.end>indicavit. </s></p> <p type="main"> <s>Aptantur autem Propoſitiones a nobis demonſtratæ ad veram <lb></lb>conſtitutionem Terræ, quatenus Rotæ eundo in ejus circulis maxi<lb></lb>mis deſcribunt motu Clavorum, perimetris ſuis infixorum, Cycloi<lb></lb>des extra globum; & Pendula inferius in fodinis & cavernis Terra <lb></lb>ſuſpenſa, in Cycloidibus intra globos Oſcillari debent, ut Oſcilla<lb></lb>tiones omnes evadant Iſochronæ. </s> <s>Nam Gravitas (ut in Libro <lb></lb>tertio docebitur) decreſcit in progreſſu a ſuperficie Terræ, ſur<lb></lb>ſum quidem in duplicata ratione diſtantiarum a centro ejus, de <lb></lb>orſum vero in ratione ſimplici. <pb xlink:href="039/01/171.jpg" pagenum="143"></pb><arrow.to.target n="note119"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note119"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LIII. PROBLEMA XXXV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Conceſſis Figurarum curvilinearum quadraturis, invenire Vires qui<lb></lb>bus corpora in datis curvis lineis Oſcillationes ſemper Iſochro<lb></lb>nas peragent.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Oſcilletur corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>in curva quavis linea <emph type="italics"></emph>STRQ,<emph.end type="italics"></emph.end>cujus axis ſit <lb></lb><emph type="italics"></emph>OR<emph.end type="italics"></emph.end>tranſiens per virium centrum <emph type="italics"></emph>C.<emph.end type="italics"></emph.end>Agatur <emph type="italics"></emph>TX<emph.end type="italics"></emph.end>quæ curvam il<lb></lb>lam in corporis loco quovis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>contingat, inque hac tangente <emph type="italics"></emph>TX<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.171.1.jpg" xlink:href="039/01/171/1.jpg"></figure><lb></lb>capiatur <emph type="italics"></emph>TY<emph.end type="italics"></emph.end>æqualis arcui <emph type="italics"></emph>TR.<emph.end type="italics"></emph.end>Nam longitudo arcus illius ex Fi<lb></lb>gurarum quadraturis (per Methodos vulgares) innoteſcit. </s> <s>De pun<lb></lb>cto <emph type="italics"></emph>Y<emph.end type="italics"></emph.end>educatur recta <emph type="italics"></emph>YZ<emph.end type="italics"></emph.end>tangenti perpendicularis. </s> <s>Agatur <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>per<lb></lb>pendiculari illi occurrens in <emph type="italics"></emph>Z,<emph.end type="italics"></emph.end>& erit Vis centripeta proportiona<lb></lb>lis rectæ <emph type="italics"></emph>TZ. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end><pb xlink:href="039/01/172.jpg" pagenum="144"></pb><arrow.to.target n="note120"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note120"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Nam ſi vis, qua corpus trahitur de <emph type="italics"></emph>T<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>exponatur per <lb></lb>rectam <emph type="italics"></emph>TZ<emph.end type="italics"></emph.end>captam ipſi proportionalem, reſolvetur hæc in vires <lb></lb><emph type="italics"></emph>TY, YZ<emph.end type="italics"></emph.end>; quarum <emph type="italics"></emph>YZ<emph.end type="italics"></emph.end>trahendo corpus ſecundum longitudinem <lb></lb>Fili <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>motum ejus nil mutat, vis autem altera <emph type="italics"></emph>TY<emph.end type="italics"></emph.end>motum ejus <lb></lb>in curva <emph type="italics"></emph>STRQ<emph.end type="italics"></emph.end>directe accelerat vel directe retardat. </s> <s>Proinde <lb></lb>cum hæc ſit ut via deſcribenda <emph type="italics"></emph>TR,<emph.end type="italics"></emph.end>accelerationes corporis vel re<lb></lb>tardationes in Oſcillationum duarum (majoris & minoris) parti<lb></lb>bus proportionalibus deſcribendis, erunt ſemper ut partes illæ, & <lb></lb>propterea facient ut partes illæ ſimul deſcribantur. </s> <s>Corpora autem <lb></lb>quæ partes totis ſemper proportionales ſimul deſcribunt, ſimul de<lb></lb>ſcribent totas. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>Filo rectilineo <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>a centro <emph type="italics"></emph>A<emph.end type="italics"></emph.end>pen<lb></lb>dens, deſcribat arcum circularem <emph type="italics"></emph>STRQ,<emph.end type="italics"></emph.end>& interea urgeatur ſe<lb></lb>cundum lineas parallelas deorſum a vi aliqua, quæ ſit ad vim uNI<lb></lb>formem Gravitatis, ut arcus <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>ad ejus ſinum <emph type="italics"></emph>TN:<emph.end type="italics"></emph.end>æqualia e<lb></lb>runt Oſcillationum ſingularum tempora. </s> <s>Etenim ob parallelas <lb></lb><emph type="italics"></emph>TZ, AR,<emph.end type="italics"></emph.end>ſimilia erunt triangula <emph type="italics"></emph>ATN, ZTY<emph.end type="italics"></emph.end>; & propterea <lb></lb><emph type="italics"></emph>TZ<emph.end type="italics"></emph.end>erit ad <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>TY<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TN<emph.end type="italics"></emph.end>; hoc eſt, (ſi Gravitatis vis unifor<lb></lb>mis exponatur per longitudinem datam <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>) vis <emph type="italics"></emph>TZ,<emph.end type="italics"></emph.end>qua Oſcil<lb></lb>lationes evadent Iſochronæ, erit ad vim Gravitatis <emph type="italics"></emph>AT,<emph.end type="italics"></emph.end>ut arcus <lb></lb><emph type="italics"></emph>TR<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>TY<emph.end type="italics"></emph.end>æqualis ad arcus illius ſinum <emph type="italics"></emph>TN.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Igitur in Horologiis, ſi vires a Machina in Pendulum <lb></lb>ad motum conſervandum impreſſæ ita cum vi Gravitatis componi <lb></lb>poſſint, ut vis tota deorſum ſemper ſit ut linea quæ oritur appli<lb></lb>cando rectangulum ſub arcu <emph type="italics"></emph>TR<emph.end type="italics"></emph.end>& radio <emph type="italics"></emph>AR<emph.end type="italics"></emph.end>ad ſinum <emph type="italics"></emph>TN,<emph.end type="italics"></emph.end><lb></lb>Oſcillationes omnes erunt Iſochronæ. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LIV. PROBLEMA XXXVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Conceſſis Figurarum curvilinearum quadraturis, invenire Tempora <lb></lb>quibus corpora Vi qualibet centripeta in lineis quibuſcunque cur<lb></lb>vis, in plano per centrum Virium tranſeunte deſcriptis, deſcen<lb></lb>dent & aſcendent.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Deſcendat corpus de loco quovis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>per lineam quamvis curvam <lb></lb><emph type="italics"></emph>STtR,<emph.end type="italics"></emph.end>in plano per virium centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>tranſeunte datam. </s> <s>Junga<lb></lb>tur <emph type="italics"></emph>CS<emph.end type="italics"></emph.end>& dividatur eadem in partes innumeras æquales, ſitque <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end><pb xlink:href="039/01/173.jpg" pagenum="145"></pb>partium illarum aliqua. </s> <s>Centro <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>intervallis <emph type="italics"></emph>CD, Cd<emph.end type="italics"></emph.end>deſcriban</s></p> <p type="main"> <s><arrow.to.target n="note121"></arrow.to.target>tur circuli <emph type="italics"></emph>DT, dt,<emph.end type="italics"></emph.end>lineæ curvæ <emph type="italics"></emph>STtR<emph.end type="italics"></emph.end>occurrentes in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>t.<emph.end type="italics"></emph.end>Et <lb></lb>ex data tum lege vis centripetæ, tum <lb></lb><figure id="id.039.01.173.1.jpg" xlink:href="039/01/173/1.jpg"></figure><lb></lb>altitudine <emph type="italics"></emph>CS<emph.end type="italics"></emph.end>de qua corpus cecidit; <lb></lb>dabitur velocitas corporis in alia qua<lb></lb>vis altitudine <emph type="italics"></emph>CT,<emph.end type="italics"></emph.end>per Prop. </s> <s>XXXIX. </s> <s><lb></lb>Tempus autem, quo corpus deſcribit <lb></lb>lineolam <emph type="italics"></emph>Tt,<emph.end type="italics"></emph.end>eſt ut lineolæ hujus lon<lb></lb>gitudo (id eſt ut ſecans anguli <emph type="italics"></emph>tTC<emph.end type="italics"></emph.end>) <lb></lb>directe, & velocitas inverſe. </s> <s>Tempori <lb></lb>huic proportionalis ſit ordinatim appli<lb></lb>cata <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ad rectam <emph type="italics"></emph>CS<emph.end type="italics"></emph.end>per punctum <lb></lb><emph type="italics"></emph>D<emph.end type="italics"></emph.end>perpendicularis, & ob datam <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end><lb></lb>erit rectangulum <emph type="italics"></emph>DdXDN,<emph.end type="italics"></emph.end>hoc eſt <lb></lb>area <emph type="italics"></emph>DNnd,<emph.end type="italics"></emph.end>eidem tempori propor<lb></lb>tionale. </s> <s>Ergo ſi <emph type="italics"></emph>SNn<emph.end type="italics"></emph.end>ſit curva illa li<lb></lb>nea quam punctum <emph type="italics"></emph>N<emph.end type="italics"></emph.end>perpetuo tangit, <lb></lb>erit area <emph type="italics"></emph>SNDS<emph.end type="italics"></emph.end>proportionalis tem<lb></lb>pori quo corpus deſcendendo deſcrip<lb></lb>ſit lineam <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>; proindeque ex inventa illa area dabitur Tempus. <lb></lb><emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note121"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LV. THEOREMA XIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpus movetur in ſuperficie quacunque curva, cujus axis per <lb></lb>centrum Virium tranſit, & a corpore in axem demittatur per<lb></lb>pendicularis, eique parallela & æqualis ab axis puncto quovis <lb></lb>dato ducatur: dico quod parallela illa aream tempori proportio<lb></lb>nalem deſcribet.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>BSKL<emph.end type="italics"></emph.end>ſuperficies curva, <emph type="italics"></emph>T<emph.end type="italics"></emph.end>corpus in ea revolvens, <emph type="italics"></emph>STtR<emph.end type="italics"></emph.end><lb></lb>Trajectoria quam corpus in eadem deſcribit, <emph type="italics"></emph>S<emph.end type="italics"></emph.end>initium Trajecto<lb></lb>riæ, <emph type="italics"></emph>OMNK<emph.end type="italics"></emph.end>axis ſuperficiei curvæ, <emph type="italics"></emph>TN<emph.end type="italics"></emph.end>recta a corpore in axem <lb></lb>perpendicularis, <emph type="italics"></emph>OP<emph.end type="italics"></emph.end>huic parallela & æqualis a puncto <emph type="italics"></emph>O<emph.end type="italics"></emph.end>quod in <lb></lb>axe datur educta, <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>veſtigium Trajectoriæ a puncto <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in lineæ <lb></lb>volubilis <emph type="italics"></emph>OP<emph.end type="italics"></emph.end>plano <emph type="italics"></emph>AOP<emph.end type="italics"></emph.end>deſcriptum, <emph type="italics"></emph>A<emph.end type="italics"></emph.end>veſtigii initium puncto <emph type="italics"></emph>S<emph.end type="italics"></emph.end><lb></lb>reſpondens, <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>recta a corpore ad centrum ducta; <emph type="italics"></emph>TG<emph.end type="italics"></emph.end>pars ejus <lb></lb>vi centripetæ qua corpus urgetur in centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>proportionalis; <lb></lb><emph type="italics"></emph>TM<emph.end type="italics"></emph.end>recta ad ſuperficiem curvam perpendicularis, <emph type="italics"></emph>TI<emph.end type="italics"></emph.end>pars ejus vi <lb></lb>preſſionis, qua corpus urget ſuperficiem viciſſimque urgetur verſus <emph type="italics"></emph>M<emph.end type="italics"></emph.end><pb xlink:href="039/01/174.jpg" pagenum="146"></pb><arrow.to.target n="note122"></arrow.to.target>a ſuperficie, proportiona<lb></lb><figure id="id.039.01.174.1.jpg" xlink:href="039/01/174/1.jpg"></figure><lb></lb>lis; <emph type="italics"></emph>PHTF<emph.end type="italics"></emph.end>recta axi <lb></lb>parallela per corpus tran<lb></lb>ſiens, & <emph type="italics"></emph>GF, IH<emph.end type="italics"></emph.end>rectæ <lb></lb>a punctis <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I<emph.end type="italics"></emph.end>in pa<lb></lb>rallelam illam <emph type="italics"></emph>PHTF<emph.end type="italics"></emph.end><lb></lb>perpendiculariter demiſ<lb></lb>ſæ. </s> <s>Dico jam quod area <lb></lb><emph type="italics"></emph>AOP,<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>OP<emph.end type="italics"></emph.end>ab iNI<lb></lb>tio motus deſcripta, ſit <lb></lb>tempori proportionalis. </s> <s><lb></lb>Nam vis <emph type="italics"></emph>TG<emph.end type="italics"></emph.end>(per Le<lb></lb>gum Corol. </s> <s>2.) reſolvitur <lb></lb>in vires <emph type="italics"></emph>TF, FG<emph.end type="italics"></emph.end>; & vis <lb></lb><emph type="italics"></emph>TI<emph.end type="italics"></emph.end>in vires <emph type="italics"></emph>TH, HI:<emph.end type="italics"></emph.end><lb></lb>Vires autem <emph type="italics"></emph>TF, TH<emph.end type="italics"></emph.end><lb></lb>agendo ſecundum lineam <lb></lb><emph type="italics"></emph>PF<emph.end type="italics"></emph.end>plano <emph type="italics"></emph>AOP<emph.end type="italics"></emph.end>per<lb></lb>pendicularem mutant ſo<lb></lb>lummodo motum cor<lb></lb>poris quatenus huic plano perpendicularem. </s> <s>Ideoque motus ejus <lb></lb>quatenus ſecundum poſitionem plani factus, hoc eſt, motus pun<lb></lb>cti <emph type="italics"></emph>P<emph.end type="italics"></emph.end>quo Trajectoriæ veſtigium <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>in hoc plano deſcri<lb></lb>bitur, idem eſt ac ſi vires <emph type="italics"></emph>TF, TH<emph.end type="italics"></emph.end>tollerentur, & corpus ſolis vi<lb></lb>ribus <emph type="italics"></emph>FG, HI<emph.end type="italics"></emph.end>agitaretur; hoc eſt, idem ac ſi corpus in plano <lb></lb><emph type="italics"></emph>AOP,<emph.end type="italics"></emph.end>vi centripeta ad centrum <emph type="italics"></emph>O<emph.end type="italics"></emph.end>tendente & ſummam virium <lb></lb><emph type="italics"></emph>FG<emph.end type="italics"></emph.end>& <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>æquante, deſcriberet curvam <emph type="italics"></emph>AP.<emph.end type="italics"></emph.end>Sed vi tali deſcribi<lb></lb>tur area <emph type="italics"></emph>AOP<emph.end type="italics"></emph.end>(per Prop. </s> <s>1.) tempori proportionalis. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note122"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Eodem argumento ſi corpus a viribus agitatum ad centra <lb></lb>duo vel plura in eadem quavis recta <emph type="italics"></emph>CO<emph.end type="italics"></emph.end>data tendentibus, deſcri<lb></lb>beret in ſpatio libero lineam quamcunque curvam <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>; foret area <lb></lb><emph type="italics"></emph>AOP<emph.end type="italics"></emph.end>tempori ſemper proportionalis. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LVI. PROBLEMA XXXVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Conceſſis Figurarum curvilinearum quadraturis, datiſque tum lege <lb></lb>Vis centripetæ ad centrum datum tendentis, tum ſuperficie cur<lb></lb>va cujus axis per centrum illud trænſit; invenieuda est Traje<lb></lb>ctoria quam corpus in eadem ſuperficie deſcribet, de loco dato, data <lb></lb>cum Velocitate, verſus plagam in ſuperficie illa datam egreſſum.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/175.jpg" pagenum="147"></pb> <p type="main"> <s>Stantibus quæ in ſuperiore Propoſitione conſtructa ſunt, exeat <lb></lb><arrow.to.target n="note123"></arrow.to.target>corpus de loco <emph type="italics"></emph>S<emph.end type="italics"></emph.end>in Trajectoriam inveniendam <emph type="italics"></emph>STtR<emph.end type="italics"></emph.end>; &, ex da<lb></lb>ta ejus velocitate in altitudine <emph type="italics"></emph>SC,<emph.end type="italics"></emph.end>dabitur ejus velocitas in alia <lb></lb>quavis altitudine <emph type="italics"></emph>TC.<emph.end type="italics"></emph.end>Ea cum velocitate, dato tempore quam <lb></lb>minimo, deſcribat corpus Trajectoriæ ſuæ particulam <emph type="italics"></emph>Tt,<emph.end type="italics"></emph.end>ſitque <lb></lb><emph type="italics"></emph>Pp<emph.end type="italics"></emph.end>veſtigium ejus in plano <emph type="italics"></emph>AOP<emph.end type="italics"></emph.end>deſcriptum. </s> <s>Jungatur <emph type="italics"></emph>Op,<emph.end type="italics"></emph.end>& <lb></lb>Circelli centro <emph type="italics"></emph>T<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>Tt<emph.end type="italics"></emph.end>in ſuperficie curva deſcripti ſit <emph type="italics"></emph>PpQ<emph.end type="italics"></emph.end><lb></lb>veſtigium Ellipticum in eodem plano <emph type="italics"></emph>OAPp<emph.end type="italics"></emph.end>deſcriptum. </s> <s>Et ob <lb></lb>datum magnitudine & poſitione Circellum, dabitur Ellipſis illa <lb></lb><emph type="italics"></emph><expan abbr="Ppq.">Ppque</expan><emph.end type="italics"></emph.end>Cumque area <emph type="italics"></emph>POp<emph.end type="italics"></emph.end>ſit tempori proportionalis, atque ad<lb></lb>eo ex dato tempore detur, dabitur <emph type="italics"></emph>Op<emph.end type="italics"></emph.end>poſitione, & inde dabitur <lb></lb>communis ejus & Ellipſeos interſectio <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>una cum angulo <emph type="italics"></emph>OPp,<emph.end type="italics"></emph.end><lb></lb>in quo Trajectoriæ veſtigium <emph type="italics"></emph>APp<emph.end type="italics"></emph.end>ſecat lineam <emph type="italics"></emph>OP.<emph.end type="italics"></emph.end>Inde au<lb></lb>tem invenietur Trajectoriæ veſtigium illud <emph type="italics"></emph>APp,<emph.end type="italics"></emph.end>eadem methodo <lb></lb>qua curva linea <emph type="italics"></emph>VIKk,<emph.end type="italics"></emph.end>in Propoſitione XLI, ex ſimilibus datis <lb></lb>inventa fuit. </s> <s>Tum ex ſingulis veſtigii punctis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erigendo ad pla<lb></lb>num <emph type="italics"></emph>AOP<emph.end type="italics"></emph.end>perpendicula <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ſuperficiei curvæ occurrentia in <emph type="italics"></emph>T,<emph.end type="italics"></emph.end><lb></lb>dabuntur ſingula Trajectoriæ puncta <emph type="italics"></emph>T. Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note123"></margin.target>LIBER <lb></lb>PRIMUS.</s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"></emph>SECTIO XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Motu Corporum Viribus centripetis ſe mutuo petentium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hactenus expoſui Motus corporum attractorum ad centrum Im<lb></lb>mobile, quale tamen vix extat in rerum natura. </s> <s>Attractiones enim <lb></lb>fieri ſolent ad corpora; & corporum trahentium & attractorum <lb></lb>actiones ſemper mutuæ ſunt & æquales, per Legem tertiam: ad<lb></lb>eo ut neque attrahens poſſit quieſcere neque attractum, ſi duo ſint <lb></lb>corpora, ſed ambo (per Legum Corollarium quartum) quaſi at<lb></lb>tractione mutua, circum gravitatis centrum commune revolvantur: <lb></lb>& ſi plura ſint corpora (quæ vel ab unico attrahantur vel omnia <lb></lb>ſe mutuo attrahant) hæc ita inter ſe moveri debeant, ut gravitatis <lb></lb>centrum commune vel quieſcat vel uniformiter moveatur in direc<lb></lb>tum. </s> <s>Qua de cauſa jam pergo Motum exponere corporum ſe mu<lb></lb>tuo trahentium, conſiderando Vires centripetas tanquam Attractio<lb></lb>nes, quamvis fortaſſe, ſi phyſice loquamur, verius dicantur Im<lb></lb>pulſus. </s> <s>In Mathematicis enim jam verſamur, & propterea miſſis <lb></lb>diſputationibus Phyſicis, familiari utimur ſermone, quo poſſimus <lb></lb>a Lectoribus Mathematicis facilius intelligi. <pb xlink:href="039/01/176.jpg" pagenum="148"></pb><arrow.to.target n="note124"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note124"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LVII. THEOREMA XX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Corpora duo ſe invicem trahentia deſcribunt, & circum commune <lb></lb>centrum gravitatis, & circum ſe mutuo, Figuras ſimiles.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sunt enim diſtantiæ a communi gravitatis centro reciproce pro<lb></lb>portionales corporibus, atque adeo in data ratione ad invicem, & <lb></lb>componendo, in data ratione ad diſtantiam totam inter corpora. </s> <s><lb></lb>Feruntur autem hæ diſtantiæ circum terminos ſuos communi motu <lb></lb>angulari, propterea quod in directum ſemper jacentes non mutant <lb></lb>inclinationem ad ſe mutuo. </s> <s>Lineæ autem rectæ, quæ ſunt in data <lb></lb>ratione ad invicem, & æquali motu angulari circum terminos ſuos <lb></lb>feruntur, Figuras circum eoſdem terminos (in planis quæ una cum <lb></lb>his terminis vel quieſcunt vel motu quovis non angulari moven<lb></lb>tur) deſcribunt omnino ſimiles. </s> <s>Proinde ſimiles ſunt Figuræ quæ <lb></lb>his diſtantiis circumactis deſcribuntur. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LVIII. THEOREMA XXI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpora duo Viribus quibuſvis ſe mutuo trahunt, & interea re<lb></lb>volvuntur circa gravitatis centrum commune: dico quod Fi<lb></lb>guris, quas corpora ſic mota deſcribunt circum ſe mutuo, potest <lb></lb>Figura ſimilis & æqualis, circum corpus alterutrum immotum, <lb></lb>Viribus iiſdem deſcribi.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Revolvantur corpora <emph type="italics"></emph>S, P<emph.end type="italics"></emph.end>circa commune gravitatis centrum <lb></lb><emph type="italics"></emph>C,<emph.end type="italics"></emph.end>pergendo de <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>T<emph.end type="italics"></emph.end>deque <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end>A dato puncto <emph type="italics"></emph>s<emph.end type="italics"></emph.end>ipſis <lb></lb><figure id="id.039.01.176.1.jpg" xlink:href="039/01/176/1.jpg"></figure><lb></lb><emph type="italics"></emph>SP, TQ<emph.end type="italics"></emph.end>æquales & parallelæ ducantur ſemper <emph type="italics"></emph>sp, sq<emph.end type="italics"></emph.end>; & Curva <lb></lb><emph type="italics"></emph>pqv<emph.end type="italics"></emph.end>quam punctum <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>revolvendo circum punctum immotum <emph type="italics"></emph>s,<emph.end type="italics"></emph.end><pb xlink:href="039/01/177.jpg" pagenum="149"></pb>deſcribit, erit ſimilis & æqualis Curvis quas corpora <emph type="italics"></emph>S, P<emph.end type="italics"></emph.end>deſcri<lb></lb><arrow.to.target n="note125"></arrow.to.target>bunt circum ſe mutuo: proindeque (per Theor. </s> <s>XX) ſimilis Curvis <lb></lb><emph type="italics"></emph>ST<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PQV,<emph.end type="italics"></emph.end>quas eadem corpora deſcribunt circum commune <lb></lb>gravitatis centrum <emph type="italics"></emph>C:<emph.end type="italics"></emph.end>id adeo quia proportiones linearum <emph type="italics"></emph>SC, CP<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>vel. <emph type="italics"></emph>sp<emph.end type="italics"></emph.end>ad invicem dantur. </s></p> <p type="margin"> <s><margin.target id="note125"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Commune illud gravitatis centrum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>per Legum Co<lb></lb>rollarium quartum, vel quieſcit vel movetur uniformiter in direc<lb></lb>tum. </s> <s>Ponamus primo quod id quieſcit, inque <emph type="italics"></emph>s<emph.end type="italics"></emph.end>& <emph type="italics"></emph>p<emph.end type="italics"></emph.end>locentur cor<lb></lb>pora duo, immobile in <emph type="italics"></emph>s,<emph.end type="italics"></emph.end>mobile in <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>corporibus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ſimilia <lb></lb>& æqualia. </s> <s>Dein tangant rectæ <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pr<emph.end type="italics"></emph.end>Curvas <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pq<emph.end type="italics"></emph.end>in <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>& producantur <emph type="italics"></emph>CQ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>sq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>R<emph.end type="italics"></emph.end>& <emph type="italics"></emph>r.<emph.end type="italics"></emph.end>Et, ob ſimilitudi<lb></lb>nem Figurarum <emph type="italics"></emph>CPRQ, sprq,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>RQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>rq<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>sp,<emph.end type="italics"></emph.end>ad<lb></lb>eoQ.E.I. data ratione. </s> <s>Proinde ſi vis qua corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>verſus cor<lb></lb>pus <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>atque adeo verſus centrum intermedium <emph type="italics"></emph>C<emph.end type="italics"></emph.end>attrahitur, eſſet <lb></lb>ad vim qua corpus <emph type="italics"></emph>p<emph.end type="italics"></emph.end>verſus centrum <emph type="italics"></emph>s<emph.end type="italics"></emph.end>attrahitur in eadem illa ra<lb></lb>tione data; hæ vires æqualibus temporibus attraherent ſemper cor<lb></lb>pora de tangentibus <emph type="italics"></emph>PR, pr<emph.end type="italics"></emph.end>ad arcus <emph type="italics"></emph>PQ, pq,<emph.end type="italics"></emph.end>per intervalla ipſis <lb></lb>proportionalia <emph type="italics"></emph>RQ, rq;<emph.end type="italics"></emph.end>adeoque vis poſterior efficeret ut corpus <lb></lb><emph type="italics"></emph>p<emph.end type="italics"></emph.end>gyraretur in Curva <emph type="italics"></emph>pqv,<emph.end type="italics"></emph.end>quæ ſimilis eſſet Curvæ <emph type="italics"></emph>PQV,<emph.end type="italics"></emph.end>in qua <lb></lb>vis prior efficit ut corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>gyretur, & revolutiones iiſdem tem<lb></lb>poribus complerentur. </s> <s>At quoniam vires illæ non ſunt ad invi<lb></lb>cem in ratione <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>sp,<emph.end type="italics"></emph.end>ſed (ob ſimilitudinem & æqualitatem <lb></lb>corporum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>& <emph type="italics"></emph>s, P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>æqualitatem diſtantiarum <emph type="italics"></emph>SP, sp<emph.end type="italics"></emph.end>) <lb></lb>ſibi mutuo æquales; corpora æqualibus temporibus æqualiter tra<lb></lb>hentur de tangentibus: & propterea, ut corpus poſterius <emph type="italics"></emph>p<emph.end type="italics"></emph.end>trahatur <lb></lb>per intervallum majus <emph type="italics"></emph>rq,<emph.end type="italics"></emph.end>requiritur tempus majus, idQ.E.I. ſub<lb></lb>duplicata ratione intervallorum; propterea quod (per Lemma de<lb></lb>cimum) ſpatia, ipſo motus initio deſcripta, ſunt in duplicata ratione <lb></lb>temporum. </s> <s>Ponatur igitur velocitas corporis <emph type="italics"></emph>p<emph.end type="italics"></emph.end>eſſe ad velocita<lb></lb>tem corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in ſubduplicata ratione diſtantiæ <emph type="italics"></emph>sp<emph.end type="italics"></emph.end>ad diſtantiam <lb></lb><emph type="italics"></emph>CP,<emph.end type="italics"></emph.end>eo ut temporibus quæ ſint in eadem ſubduplicata ratione de<lb></lb>ſcribantur arcus <emph type="italics"></emph>pq, PQ,<emph.end type="italics"></emph.end>qui ſunt in ratione integra: Et corpora <lb></lb><emph type="italics"></emph>P, p<emph.end type="italics"></emph.end>viribus æqualibus ſemper attracta deſcribent circum centra <lb></lb>quieſcentia <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& <emph type="italics"></emph>s<emph.end type="italics"></emph.end>Figuras ſimiles <emph type="italics"></emph>PQV, pqv,<emph.end type="italics"></emph.end>quarum poſterior <lb></lb><emph type="italics"></emph>pqv<emph.end type="italics"></emph.end>ſimilis eſt & æqualis Figuræ quam corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>circum corpus <lb></lb>mobile <emph type="italics"></emph>S<emph.end type="italics"></emph.end>deſcribit. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Ponamus jam quod commune gravitatis centrum, una <lb></lb>cum ſpatio in quo corpora moventur inter ſe, progreditur unifor<lb></lb>miter in directum; &, per Legum Corollarium ſextum, motus <lb></lb>omnes in hoc ſpatio peragentur ut prius, adeoque corpora deſcri-<pb xlink:href="039/01/178.jpg" pagenum="150"></pb><arrow.to.target n="note126"></arrow.to.target>bent circum ſe mutuo Figuras eaſdem ac prius, & propterea Figuræ <lb></lb><emph type="italics"></emph>pqv<emph.end type="italics"></emph.end>ſimiles & æquales. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note126"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc corpora duo Viribus diſtantiæ ſuæ proportionali<lb></lb>bus ſe mutuo trahentia, deſcribunt (per Prop. </s> <s>X,) & circum com<lb></lb>mune gravitatis centrum, & circum ſe mutuo, Ellipſes concentri<lb></lb>cas: & vice verſa, ſi tales Figuræ deſcribuntur, ſunt Vires diſtan<lb></lb>tiæ proportionales. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et corpora duo Viribus quadrato diſtantiæ ſuæ recipro<lb></lb>ce proportionalibus deſcribunt (per Prop. </s> <s>XI, XII, XIII) & circum <lb></lb>commune gravitatis centrum, & circum ſe mutuo, Sectiones conicas <lb></lb>umbilicum habentes in centro circum quod Figuræ deſcribuntur. </s> <s>Et <lb></lb>vice verſa, ſi tales Figuræ deſcribuntur, Vires centripetæ ſunt qua<lb></lb>drato diſtantiæ reciproce proportionales. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Corpora duo quævis cirum gravitatis centrum com<lb></lb>mune gyrantia, radiis & ad centrum illud & ad ſe mutuo ductis, <lb></lb>deſcribunt areas temporibus proportionales. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LIX. THEOREMA XXII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corporum duorum<emph.end type="italics"></emph.end>S <emph type="italics"></emph>&<emph.end type="italics"></emph.end>P <emph type="italics"></emph>circa commune gravitatis centrum<emph.end type="italics"></emph.end>C <lb></lb><emph type="italics"></emph>revolventium Tempus periodicum eſſe ad Tempus periodicum cor<lb></lb>poris alterutrius<emph.end type="italics"></emph.end>P, <emph type="italics"></emph>circa alterum immotum<emph.end type="italics"></emph.end>S <emph type="italics"></emph>gyrantis & Figu<lb></lb>ris quæ corpora circum ſe mutuo deſcribunt Figuram ſimilem & <lb></lb>æqualem deſcribentis, in ſubduplicata ratione corporis alterins<emph.end type="italics"></emph.end>S, <lb></lb><emph type="italics"></emph>ad ſummam corporum<emph.end type="italics"></emph.end>S+P. </s></p> <p type="main"> <s>Namque, ex demonſtratione ſuperioris Propoſitionis, tempora <lb></lb>quibus arcus quivis ſimiles <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pq<emph.end type="italics"></emph.end>deſcribuntur, ſunt in ſub<lb></lb>duplicata ratione diſtantiarum <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>sp,<emph.end type="italics"></emph.end>hoc eſt, in ſub<lb></lb>duplicata ratione corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad ſummam corporum <emph type="italics"></emph>S+P.<emph.end type="italics"></emph.end>Et com<lb></lb>ponendo, ſummæ temporum quibus arcus omnes ſimiles <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pq<emph.end type="italics"></emph.end><lb></lb>deſcribuntur, hoc eſt, tempora tota quibus Figuræ totæ ſimiles de<lb></lb>ſcribuntur, ſunt in eadem ſubduplicata ratione. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/179.jpg" pagenum="151"></pb> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LX. THEOREMA XXIII.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note127"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note127"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>St corpora duo<emph.end type="italics"></emph.end>S <emph type="italics"></emph>&<emph.end type="italics"></emph.end>P, <emph type="italics"></emph>Viribus quadrato diſtantiæ ſuæ reciproee <lb></lb>proportionalibus ſe mutuo trahentia, revalvuntur circa gravi<lb></lb>tatis centrum commune: dico quod Ellipſeos, quam corpus al<lb></lb>terutrum<emph.end type="italics"></emph.end>P <emph type="italics"></emph>hoc motu circa alterum<emph.end type="italics"></emph.end>S <emph type="italics"></emph>deſcribit, Axis principa<lb></lb>lis erit ad Axem principalem Ellipſeos, quam corpus idem<emph.end type="italics"></emph.end>P <lb></lb><emph type="italics"></emph>circa alterum quieſcens<emph.end type="italics"></emph.end>S <emph type="italics"></emph>eodem tempore periodico deſcribere <lb></lb>poſſet, ut ſumma corporum duorum<emph.end type="italics"></emph.end>S+P <emph type="italics"></emph>ad primam duarum <lb></lb>medie proportionalium inter hanc ſummam & corpus illud al<lb></lb>terum<emph.end type="italics"></emph.end>S. </s></p> <p type="main"> <s>Nam ſi deſcriptæ Ellipſes eſſent ſibi invicem æquales, tempora <lb></lb>periodica (per Theorema ſuperius) forent in ſubduplicata ratione <lb></lb>corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad ſummam corporum <emph type="italics"></emph>S+P.<emph.end type="italics"></emph.end>Minuatur in hac ratione <lb></lb>tempus periodicum in Ellipſi poſteriore, & tempora periodica eva<lb></lb>dent æqualia; Ellipſeos autem axis principalis (per Prop. </s> <s>XV.) minu<lb></lb>etur in ratione cujus hæc eſt ſeſquiplicata, id eſt in ratione, cujus <lb></lb>ratio <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>S+P<emph.end type="italics"></emph.end>eſt triplicata; adeoque erit ad axem principalem <lb></lb>Ellipſeos alterius, ut prima duarum medie proportionalium inter <lb></lb><emph type="italics"></emph>S+P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>S+P.<emph.end type="italics"></emph.end>Et inverſe, axis principalis Ellipſeos circa <lb></lb>corpus mobile deſcriptæ erit ad axem principalem deſcriptæ circa <lb></lb>immobile, ut <emph type="italics"></emph>S+P<emph.end type="italics"></emph.end>ad primam duarum medie proportionalium in<lb></lb>ter <emph type="italics"></emph>S+P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXI. THEOREMA XXIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpora duo Viribus quibuſvis ſe mutuo trahentia, neque alias <lb></lb>agitata vel impedita, quomodocunque moveantur; motus eo<lb></lb>rum perinde ſe habebunt ac ſi non traherent ſe mutuo, ſed u<lb></lb>trumque a corpore tertio in communi gravitatis centro conſtituto <lb></lb>Viribus iiſdem traberetur: Et Virium trahentium eadem erit Lex <lb></lb>reſpectu diſtantiæ corporum a centro illo communi atque reſpe<lb></lb>ctu diſtantiæ totius inter corpora.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam vires illæ, quibus corpora ſe mutuo trahunt, tendendo <lb></lb>ad corpora, tendunt ad commune gravitatis centrum interme-</s></p><pb xlink:href="039/01/180.jpg" pagenum="152"></pb> <p type="main"> <s><arrow.to.target n="note128"></arrow.to.target>dium, adeoque eædem ſunt ac ſi a corpore intermedio mana<lb></lb>rent. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note128"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Et quoniam data eſt ratio diſtantiæ corporis utriuſvis a centro <lb></lb>illo communi ad diſtantiam corporis ejuſdem a corpore altero, da<lb></lb>bitur ratio cujuſvis poteſtatis diſtantiæ unius ad eandem poteſta<lb></lb>tem diſtantiæ alterius; ut & ratio quantitatis cujuſvis, quæ ex una <lb></lb>diſtantia & quantitatibus datis utcunQ.E.D.rivatur, ad quantitatem <lb></lb>aliam, quæ ex altera diſtantia & quantitatibus totidem datis da<lb></lb>tamQ.E.I.lam diſtantiarum rationem ad priores habentibus ſimiliter <lb></lb>derivatur. </s> <s>Proinde ſi vis, qua corpus unum ab altero trahitur, ſit <lb></lb>directe vel inverſe ut diſtantia corporum ab invicem; vel ut quæ<lb></lb>libet hujus diſtantiæ poteſtas; vel denique ut quantitas quævis ex <lb></lb>hac diſtantia & quantitatibus datis quomodocunQ.E.D.rivata: erit <lb></lb>eadem vis, qua corpus idem ad commune gravitatis centrum tra<lb></lb>hitur, directe itidem vel inverſe ut corporis attracti diſtantia a cen<lb></lb>tro illo communi, vel ut eadem diſtantiæ hujus poteſtas, vel de<lb></lb>nique ut quantitas ex hac diſtantia & analogis quantitatibus da<lb></lb>tis ſimiliter derivata. </s> <s>Hoc eſt, Vis trahentis eadem erit Lex reſpe<lb></lb>ctu diſtantiæ utriuſque. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXII. PROBLEMA XXXVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corporum duorum quæ Viribus quadrato diſtantiæ ſuæ reciproce <lb></lb>proportionalibus ſe mutuo trahunt, ac de locis datis demittun<lb></lb>tur, determinare Motus.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Corpora (per Theorema noviſſimum) perinde movebuntur ac <lb></lb>ſi a corpore tertio, in communi gravitatis centro conſtituto, trahe<lb></lb>rentur; & centrum illud ipſo motus initio quieſcet per Hypothe<lb></lb>ſin; & propterea (per Legum Corol. </s> <s>4.) ſemper quieſcet. </s> <s>Deter<lb></lb>minandi ſunt igitur motus corporum (per Prob. </s> <s>XXV,) perinde <lb></lb>ac ſi a viribus ad centrum illud tendentibus urgerentur, & habe<lb></lb>buntur motus corporum ſe mutuo trahentium. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXIII. PROBLEMA XXXIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corporum duorum quæ Viribus quadrato diſtantiæ ſuæ reciproce pro<lb></lb>portionalibus ſe mutuo trahunt, deque locis datis, ſecundum datas <lb></lb>rectas, datis cum Velocitatibus exeunt, determinare Motus.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/181.jpg" pagenum="153"></pb> <p type="main"> <s>Ex datis corporum motibus ſub initio, datur uniformis motus <lb></lb><arrow.to.target n="note129"></arrow.to.target>centri communis gravitatis, ut & motus ſpatii quod una cum hoc <lb></lb>centro movetur uniformiter in directum, nec non corporum mo<lb></lb>tus initiales reſpectu hujus ſpatii. </s> <s>Motus autem ſubſequentes <lb></lb>(per Legum Corollarium quintum, & Theorema noviſſimum) <lb></lb>perinde fiunt in hoc ſpatio, ac ſi ſpatium ipſum una cum commu<lb></lb>ni illo gravitatis centro quieſceret, & corpora non traherent ſe <lb></lb>mutuo, ſed a corpore tertio ſito in centro illo traherentur. </s> <s>Cor<lb></lb>poris igitur alterutrius in hoc ſpatio mobili, de loco dato, ſecun<lb></lb>dum datam rectam, data cum velocitate exeuntis, & vi centripeta <lb></lb>ad centrum illud tendente correpti, determinandus eſt motus per <lb></lb>Problema nonum & viceſimum ſextum: & habebitur ſimul mo<lb></lb>tus corporis alterius e regione. </s> <s>Cum hoc motu componendus <lb></lb>eſt uniformis ille Syſtematis ſpatii & corporum in eo gyrantium <lb></lb>motus progreſſivus ſupra inventus, & habebitur motus abſolutus <lb></lb>corporum in ſpatio immobili. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note129"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXIV. PROBLEMA XL.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Viribus quibus Corpora ſe mutuo trahunt creſcentibus in ſimplici ra<lb></lb>tione diſtantiarum a centris: requiruntur Motus plurium Cor<lb></lb>porum inter ſe.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Ponantur primo corpora duo <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end>commune habentia gravi<lb></lb>tatis centrum <emph type="italics"></emph>D.<emph.end type="italics"></emph.end>Deſcribent hæc (per Corollarium primum Theo<lb></lb>rematis XXI) Ellipſes centra habentes in <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>quarum magnitudo ex <lb></lb>Problemate V, innoteſcit. </s></p> <p type="main"> <s>Trahat jam corpus tertium <lb></lb><figure id="id.039.01.181.1.jpg" xlink:href="039/01/181/1.jpg"></figure><lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end>priora duo <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end>viri<lb></lb>bus acceleratricibus <emph type="italics"></emph>ST, SL,<emph.end type="italics"></emph.end><lb></lb>& ab ipſis viciſſim trahatur. </s> <s><lb></lb>Vis <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>(per Legum Cor. </s> <s>2.) <lb></lb>reſolvitur in vires <emph type="italics"></emph>SD, DT<emph.end type="italics"></emph.end>; <lb></lb>& vis <emph type="italics"></emph>SL<emph.end type="italics"></emph.end>in vires <emph type="italics"></emph>SD, DL.<emph.end type="italics"></emph.end><lb></lb>Vires autem <emph type="italics"></emph>DT, DL,<emph.end type="italics"></emph.end>quæ <lb></lb>ſunt ut ipſarum ſumma <emph type="italics"></emph>TL,<emph.end type="italics"></emph.end><lb></lb>atque adeo ut vires accelera<lb></lb>trices quibus corpora <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ſe mutuo trahunt, additæ his viri<lb></lb>bus corporum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>prior priori & poſterior poſteriori, com<lb></lb>ponunt vires diſtantiis <emph type="italics"></emph>DT<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>DL<emph.end type="italics"></emph.end>proportionales, ut prius, ſed <pb xlink:href="039/01/182.jpg" pagenum="154"></pb><arrow.to.target n="note130"></arrow.to.target>viribus prioribus majores; adeoque (per Corol. </s> <s>1. Prop. </s> <s>X. & Corol. </s> <s><lb></lb>1 & 8. Prop, IV) efficiunt ut corpora illa deſcribant Ellipſes ut prius, <lb></lb>ſed motu celeriore. </s> <s>Vires reliquæ acceleratrices <emph type="italics"></emph>SD<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SD,<emph.end type="italics"></emph.end>actio<lb></lb>nibus motricibus <emph type="italics"></emph>SDXT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SDXL,<emph.end type="italics"></emph.end>quæ ſunt ut corpora, tra<lb></lb>hendo corpora illa æqualiter & ſecundum lineas <emph type="italics"></emph>TI, LK,<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>DS<emph.end type="italics"></emph.end><lb></lb>parallelas, nil mutant ſitus eorum ad invicem, ſed faciunt ut ipſa <lb></lb>æqualiter accedant ad lineam <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>; quam ductam concipe per me<lb></lb>dium corporis <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>& lineæ <emph type="italics"></emph>DS<emph.end type="italics"></emph.end>perpendicularem. </s> <s>Impedietur au<lb></lb>tem iſte ad lineam <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>acceſſus faciendo ut Syſtema corporum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end><lb></lb>ex una parte, & corpus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ex altera, juſtis cum velocitatibus, gyren<lb></lb>tur circa commune gravitatis centrum <emph type="italics"></emph>C.<emph.end type="italics"></emph.end>Tali motu corpus <emph type="italics"></emph>S<emph.end type="italics"></emph.end><lb></lb>(eo quod ſumma virium motricium <emph type="italics"></emph>SDXT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SDXL,<emph.end type="italics"></emph.end>diſtan<lb></lb>tiæ <emph type="italics"></emph>CS<emph.end type="italics"></emph.end>proportionalium, tendit verſus centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>) deſcribit El<lb></lb>lipſin circa idem <emph type="italics"></emph>C;<emph.end type="italics"></emph.end>& punctum <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ob proportionales <emph type="italics"></emph>CS, CD,<emph.end type="italics"></emph.end><lb></lb>deſcribet Ellipſin conſimilem e regione. </s> <s>Corpora autem <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end><lb></lb>viribus motricibus <emph type="italics"></emph>SDXT<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.182.1.jpg" xlink:href="039/01/182/1.jpg"></figure><lb></lb>& <emph type="italics"></emph>SDXL,<emph.end type="italics"></emph.end>(prius priore, <lb></lb>poſterius poſteriore) æqua<lb></lb>liter & ſecundum lineas pa<lb></lb>rallelas <emph type="italics"></emph>TI<emph.end type="italics"></emph.end>& <emph type="italics"></emph>LK<emph.end type="italics"></emph.end>(ut dic<lb></lb>tum eſt) attracta, pergent <lb></lb>(per Legum Corollarium <lb></lb>quintum & ſextum) circa cen<lb></lb>trum mobile <emph type="italics"></emph>D<emph.end type="italics"></emph.end>Ellipſes ſuas <lb></lb>deſcribere, ut prius. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note130"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Addatur jam corpus quartum <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>& ſimili argumento conclude<lb></lb>tur hoc & punctum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>Ellipſes circa omnium commune centrum <lb></lb>gravitatis <emph type="italics"></emph>B<emph.end type="italics"></emph.end>deſcribere; manentibus motibus priorum corporum <lb></lb><emph type="italics"></emph>T, L<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end>circa centra <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>ſed paulo acceleratis. </s> <s>Et eadem <lb></lb>methodo corpora plura adjungere licebit. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Hæc ita ſe habent ubi corpora <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L<emph.end type="italics"></emph.end>trahunt ſe mutuo viribus <lb></lb>acceleratricibus majoribus vel minoribus quam quibus trahunt cor<lb></lb>pora reliqua pro ratione diſtantiarum. </s> <s>Sunto mutuæ omnium at<lb></lb>tractiones acceleratrices ad invicem ut diſtantiæ ductæ in corpo<lb></lb>ra trahentia, & ex præcedentibus facile deducetur quod corpora <lb></lb>omnia æqualibus temporibus periodicis Ellipſes varias, circa om<lb></lb>nium commune gravitatis centrum <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>in plano immobili deſcri<lb></lb>bunt. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/183.jpg" pagenum="155"></pb> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXV. THEOREMA XXV.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note131"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note131"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corpora plura, quorum Vires decreſcunt in duplicata ratione di<lb></lb>ſtantiarum ab eorundem centris, moveri poſſe inter ſe in El<lb></lb>lipſibus; & radiis ad umbilicos ductis areas deſcribere tempo<lb></lb>ribus proportionales quam proxime.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>In Propoſitione ſuperiore demonſtratus eſt caſus ubi motus plu<lb></lb>res peraguntur in Ellipſibus accurate. </s> <s>Quo magis recedit Lex vi<lb></lb>rium a Lege ibi poſita, eo magis corpora perturbabunt mutuos <lb></lb>motus; neque fieri poteſt ut corpora, ſecundum Legem hic poſitam <lb></lb>ſe mutuo trahentia, moveantur in Ellipſibus accurate, niſi ſervando <lb></lb>certam proportionem diſtantiarum ab invicem. </s> <s>In ſequentibus au<lb></lb>tem caſibus non multum ab Ellipſibus errabitur. </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Pone corpora plura minora circa maximum aliquod ad <lb></lb>varias ab eo diſtantias revolvi, tendantque ad ſingula vires abſolu<lb></lb>tæ proportionales iiſdem corporibus. </s> <s>Et quoniam omnium com<lb></lb>mune gravitatis centrum (per Legum Corol. </s> <s>quartum) vel quie<lb></lb>ſcit vel movetur uniformiter in directum, fingamus corpora mi<lb></lb>nora tam parva eſſe, ut corpus maximum nunquam diſtet ſenſibi<lb></lb>liter ab hoc centro: & maximum illud vel quieſcet vel movebitur <lb></lb>uniformiter in directum, abſque errore ſenſibili; minora autem re<lb></lb>volventur circa hoc maximum in Ellipſibus, atque radiis ad idem <lb></lb>ductis deſcribent areas temporibus proportionales; niſi quatenus <lb></lb>errores inducuntur, vel per errorem maximi a communi illo gravi<lb></lb>tatis centro, vel per actiones minorum corporum in ſe mutuo. </s> <s>Di<lb></lb>minui autem poſſunt corpora minora uſQ.E.D.nec error iſte & ac<lb></lb>tiones mutuæ ſint datis quibuſvis minores, atque adeo donec Orbes <lb></lb>cum Ellipſibus quadrent, & areæ reſpondeant temporibus, abſque <lb></lb>errore qui non ſit minor quovis dato. <emph type="italics"></emph>q.E.O.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Fingamus jam Syſtema corporum minorum modo jam <lb></lb>deſcripto circa maximum revolventium, aliudve quodvis duorum <lb></lb>circum ſe mutuo revolventium corporum Syſtema progredi unifor<lb></lb>miter in directum, & interea vi corporis alterius longe maximi & <lb></lb>ad magnam diſtantiam ſiti urgeri ad latus. </s> <s>Et quoniam æquales <lb></lb>vires acceleratrices, quibus corpora ſecundum lineas parallelas ur<lb></lb>gentur, non mutant ſitus corporum ad invicem, ſed ut Syſtema <lb></lb>totum, ſervatis partium motibus inter ſe, ſimul transferatur effici<lb></lb>unt: manifeſtum eſt quod, ex attractionibus in corpus maximum, </s></p><pb xlink:href="039/01/184.jpg" pagenum="156"></pb> <p type="main"> <s><arrow.to.target n="note132"></arrow.to.target>nulla prorſus orietur mutatio motus attractorum inter ſe, niſi vel <lb></lb>ex attractionum acceleratricum inæqualitate, vel ex inclinatione li<lb></lb>nearum ad invicem, ſecundum quas attractiones fiunt. </s> <s>Pone ergo <lb></lb>attractiones omnes acceleratrices in corpus maximum eſſe inter ſe <lb></lb>reciproce ut quadrata diſtantiarum; &, augendo corporis maximi <lb></lb>diſtantiam, donec rectarum ab hoc ad reliqua ductarum differen<lb></lb>tiæ reſpectu earum longitudinis & inclinationes ad invicem mino<lb></lb>res ſint quam datæ quævis, perſeverabunt motus partium Syſtema<lb></lb>tis inter ſe abſque erroribus qui non ſint quibuſvis datis minores. </s> <s><lb></lb>Et quoniam, ob exiguam partium illarum ab invicem diſtantiam, <lb></lb>Syſtema totum ad modum corporis unius attrahitur; movebitur <lb></lb>idem hac attractione ad modum corporis unius; hoc eſt, centro <lb></lb>ſuo gravitatis deſcribet circa corpus maximum Sectionem aliquam <lb></lb>Conicam (<emph type="italics"></emph>viz.<emph.end type="italics"></emph.end>Hyperbolam vel Parabolam attractione languida, <lb></lb>Ellipſin fortiore,) & Radio ad maximum ducto deſcribet areas <lb></lb>temporibus proportionales, abſque ullis erroribus, niſi quas par<lb></lb>tium diſtantiæ (perexiguæ ſane & pro lubitu minuendæ) valeant <lb></lb>efficere. <emph type="italics"></emph>q.E.O.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note132"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Simili argumento pergere licet ad caſus magis compoſitos in in<lb></lb>finitum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. In caſu ſecundo; quo propius accedit corpus omnium <lb></lb>maximum ad Syſtema duorum vel plurium, eo magis turbabuntur <lb></lb>motus partium Syſtematis inter ſe; propterea quod linearum a cor<lb></lb>pore maximo ad has ductarum jam major eſt inclinatio ad invicem, <lb></lb>majorque proportionis inæqualitas. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Maxime autem turbabuntur, ponendo quod attractio<lb></lb>nes acceleratrices partium Syſtematis verſus corpus omnium maxi<lb></lb>mum, non ſint ad invicem reciproce ut quadrata diſtantiarum a <lb></lb>corpore illo maximo; præſertim ſi proportionis hujus inæqualitas <lb></lb>major ſit quam inæqualitas proportionis diſtantiarum a corpore <lb></lb>maximo: Nam ſi vis acceleratrix, æqualiter & ſecundum lineas pa<lb></lb>rallelas agendo, nil perturbat motus inter ſe, neceſſe eſt ut ex acti<lb></lb>onis inæqualitate perturbatio oriatur, majorque ſit vel minor pro <lb></lb>majore vel minore inæqualitate. </s> <s>Exceſſus impulſuum majorum, <lb></lb>agendo in aliqua corpora & non agendo in alia, neceſſario muta<lb></lb>bunt ſitum eorum inter ſe. </s> <s>Et hæc perturbatio, addita perturbatio<lb></lb>ni quæ ex linearum inclinatione & inæqualitate oritur, majorem <lb></lb>reddet perturbationem totam. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Unde ſi Syſtematis hujus partes in Ellipſibus vel Cir<lb></lb>culis ſine perturbatione inſigni moveantur; manifeſtum eſt, quod <pb xlink:href="039/01/185.jpg" pagenum="157"></pb>eædem a viribus acceleratricibus ad alia corpora tendentibus, aut <lb></lb><arrow.to.target n="note133"></arrow.to.target>non urgentur niſi leviſſime, aut urgentur æqualiter & ſecundum li<lb></lb>neas parallelas quamproxime. </s></p> <p type="margin"> <s><margin.target id="note133"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXVI. THEOREMA XXVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Corpora tria, quorum Vires decreſcunt in duplicata ratione di<lb></lb>ſtantiarum, ſe mutuo trahant, & attractiones acceleratrices bi<lb></lb>norum quorumcunQ.E.I. tertium ſint inter ſe reciproce ut qua<lb></lb>drata diſtantiarum; minora autem circa maximum revolvan<lb></lb>tur: Dico quod interius circa intimum & maximum, radiis <lb></lb>ad ipſum ductis, deſcribet areas temporibus magis proportio<lb></lb>nales, & Figuram ad formam Ellipſeos umbilicum in concur<lb></lb>ſu radiorum habentis magis accedentem, ſi corpus maximum <lb></lb>his attractionibus agitetur, quam ſi maximum illud vel a mi<lb></lb>noribus non attractum quieſcat, vel multo minus vel multo ma<lb></lb>gis attractum aut multo minus aut multo magis agitetur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Liquet fere ex demonſtratione Corollarii ſecundi Propoſitionis <lb></lb>præcedentis; ſed argumento magis diſtincto & latius cogente ſic <lb></lb>evincitur. </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Revolvantur <lb></lb><figure id="id.039.01.185.1.jpg" xlink:href="039/01/185/1.jpg"></figure><lb></lb>corpora minora <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end><lb></lb>in eodem plano circa <lb></lb>maximum <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>quorum <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>deſcribat Orbem in<lb></lb>teriorem <emph type="italics"></emph>PAB,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end><lb></lb>exteriorem <emph type="italics"></emph>SE.<emph.end type="italics"></emph.end>Sit <lb></lb><emph type="italics"></emph>SK<emph.end type="italics"></emph.end>mediocris diſtan<lb></lb>tia corporum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end>; <lb></lb>& corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>verſus <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end>attractio acceleratrix in mediocri illa diſtantia exponatur per e<lb></lb>andem. </s> <s>In duplicata ratione <emph type="italics"></emph>SK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>SL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SK,<emph.end type="italics"></emph.end>& e<lb></lb>rit <emph type="italics"></emph>SL<emph.end type="italics"></emph.end>attractio acceleratrix corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>in diſtantia quavis <lb></lb><emph type="italics"></emph>SP.<emph.end type="italics"></emph.end>Junge <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>eique parallelam age <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>occurrentem <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>in <emph type="italics"></emph>M,<emph.end type="italics"></emph.end><lb></lb>& attractio <emph type="italics"></emph>SL<emph.end type="italics"></emph.end>reſolvetur (per Legum Corol 2.) in attractiones <lb></lb><emph type="italics"></emph>SM, LM.<emph.end type="italics"></emph.end>Et ſic urgebitur corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>vi acceleratrice triplici: <pb xlink:href="039/01/186.jpg" pagenum="158"></pb><arrow.to.target n="note134"></arrow.to.target>una tendente ad <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& oriunda a mutua attractione corporum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P.<emph.end type="italics"></emph.end><lb></lb>Hac vi ſola corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>circum corpus <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>ſive immotum ſive hac <lb></lb>attractione agitatum, deſcribere deberet & areas, radio <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>tem<lb></lb>poribus proportionales, & Ellipſin cui umbilicus eſt in centro cor<lb></lb>poris <emph type="italics"></emph>T.<emph.end type="italics"></emph.end>Patet hoc per Prop. </s> <s>XI. & Corollaria 2 & 3 Theor. </s> <s>XXI. </s> <s>Vis <lb></lb>altera eſt attractionis <emph type="italics"></emph>LM,<emph.end type="italics"></emph.end>quæ quoniam tendit a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>ſuperad<lb></lb>dita vi priori coincidet cum ipſa, & ſic faciet ut areæ etiamnum tem<lb></lb>poribus proportionales deſcribantur per Corol. </s> <s>3. Theor. </s> <s>XXI. </s> <s>At <lb></lb>quoniam non eſt quadrato diſtantiæ <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>reciproce proportionalis, <lb></lb>componet ea cum vi priore vim ab hac proportione aberrantem, id<lb></lb>que eo magis quo major eſt proportio hujus vis ad vim priorem, <lb></lb>cæteris paribus. </s> <s>Proinde cum (per Prop. </s> <s>XI, & per Corol. </s> <s>2. <lb></lb>Theor. </s> <s>XXI) vis qua Ellipſis circa umbilicum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>deſcribitur tendere <lb></lb>debeat ad umbilicum illum, & eſſe quadrato diſtantiæ <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>reciproce <lb></lb>proportionalis; vis illa <lb></lb><figure id="id.039.01.186.1.jpg" xlink:href="039/01/186/1.jpg"></figure><lb></lb>compoſita, aberrando <lb></lb>ab hac proportione, fa<lb></lb>ciet ut Orbis <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end><lb></lb>aberret a forma Ellip<lb></lb>ſeos umbilicum haben<lb></lb>tis in <emph type="italics"></emph>S;<emph.end type="italics"></emph.end>idque eo ma<lb></lb>gis quo major eſt ab<lb></lb>erratio ab hac propor<lb></lb>tione; atque adeo eti<lb></lb>am quo major eſt proportio vis ſecundæ <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>ad vim primam, cæ<lb></lb>teris paribus. </s> <s>Jam vero vis tertia <emph type="italics"></emph>SM,<emph.end type="italics"></emph.end>trahendo corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ſecun<lb></lb>dum lineam ipſi <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>parallelam, componet cum viribus prioribus <lb></lb>vim quæ non amplius dirigitur a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>quæque ab hac determi<lb></lb>natione tanto magis aberrat, quanto major eſt proportio hujus ter<lb></lb>tiæ vis ad vires priores, cæteris paribus; atque adeo quæ faciet ut <lb></lb>corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>areas non amplius temporibus proportiona<lb></lb>les deſcribat, atque aberratio ab hac proportionalitate ut tanto ma<lb></lb>jor ſit, quanto major eſt proportio vis hujus tertiæ ad vires cæte<lb></lb>ras. </s> <s>Orbis vero <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end>aberrationem a forma Elliptica præfata hæc<lb></lb>vis tertia duplici de cauſa adaugebit, tum quod non dirigatur a <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>tum etiam quod non ſit proportionalis quadrato diſtantiæ <emph type="italics"></emph>PT.<emph.end type="italics"></emph.end><lb></lb>Quibus intellectis, manifeſtum eſt quod areæ temporibus tum max<lb></lb>ime fiunt proportionales, ubi vis tertia, manentibus viribus cæte<lb></lb>ris, fit minima; & quod Orbis <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end>tum maxime accedit ad præ<lb></lb>fatam formam Ellipticam, ubi vis tam ſecunda quam tertia, ſed præ<lb></lb>cipue vis tertia, fit minima, vi prima manente. </s></p><pb xlink:href="039/01/187.jpg" pagenum="159"></pb> <p type="margin"> <s><margin.target id="note134"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Exponatur corporis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>attractio acceleratrix verſus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>per lineam <lb></lb><arrow.to.target n="note135"></arrow.to.target><emph type="italics"></emph>SN;<emph.end type="italics"></emph.end>& ſi attractiones acceleratrices <emph type="italics"></emph>SM, SN<emph.end type="italics"></emph.end>æquales eſſent; hæ, <lb></lb>trahendo corpora <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>æqualiter & ſecundum lineas parallelas, <lb></lb>nil mutarent ſitum eorum ad invicem. </s> <s>Iidem jam forent corporum <lb></lb>illorum motus inter ſe (per Legum Corol. </s> <s>6.) ac ſi hæ attractio<lb></lb>nes tollerentur. </s> <s>Et pari ratione ſi attractio <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>minor eſſet at<lb></lb>tractione <emph type="italics"></emph>SM,<emph.end type="italics"></emph.end>tolleret ipſa attractionis <emph type="italics"></emph>SM<emph.end type="italics"></emph.end>partem <emph type="italics"></emph>SN,<emph.end type="italics"></emph.end>& ma<lb></lb>neret pars ſola <emph type="italics"></emph>MN,<emph.end type="italics"></emph.end>qua temporum & arearum proportionalitas <lb></lb>& Orbitæ forma illa Elliptica perturbaretur. </s> <s>Et ſimiliter ſi attra<lb></lb>ctio <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>major eſſet attractione <emph type="italics"></emph>SM,<emph.end type="italics"></emph.end>oriretur ex differentia ſola <lb></lb><emph type="italics"></emph>MN<emph.end type="italics"></emph.end>perturbatio proportionalitatis & Orbitæ. </s> <s>Sic per attractio<lb></lb>nem <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>reducitur ſemper attractio tertia ſuperior <emph type="italics"></emph>SM<emph.end type="italics"></emph.end>ad attra<lb></lb>ctionem <emph type="italics"></emph>MN,<emph.end type="italics"></emph.end>attractione prima & ſecunda manentibus prorſus im<lb></lb>mutatis: & propterea areæ ac tempora ad proportionalitatem, & <lb></lb>Orbita <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end>ad formam præfatam Ellipticam tum maxime acce<lb></lb>dunt, ubi attractio <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>vel nulla eſt, vel quam fieri poſſit miNI<lb></lb>ma; hoc eſt, ubi corporum <emph type="italics"></emph>P & T<emph.end type="italics"></emph.end>attractiones acceleratrices, fa<lb></lb>ctæ verſus corpus <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>accedunt quantum fieri poteſt ad æqualita<lb></lb>tem; id eſt, ubi attractio <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>non eſt nulla, neque minor minima <lb></lb>attractionum omnium <emph type="italics"></emph>SM,<emph.end type="italics"></emph.end>ſed inter attractionum omnium <emph type="italics"></emph>SM<emph.end type="italics"></emph.end><lb></lb>maximam & minimam quaſi mediocris, hoc eſt, non multo major <lb></lb>neque multo minor attractione <emph type="italics"></emph>SK. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note135"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Revolvantur jam corpora minora <emph type="italics"></emph>P, S<emph.end type="italics"></emph.end>circa maximum <emph type="italics"></emph>T<emph.end type="italics"></emph.end><lb></lb>in planis diverſis; & vis <emph type="italics"></emph>LM,<emph.end type="italics"></emph.end>agendo ſecundum lineam <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>in pla<lb></lb>no Orbitæ <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end>ſitam, eundem habebit effectum ac prius, neque <lb></lb>corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>de plano Orbitæ ſuæ deturbabit. </s> <s>At vis altera <emph type="italics"></emph>NM,<emph.end type="italics"></emph.end><lb></lb>agendo ſecundum lineam quæ ipſi <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>parallela eſt, (atque adco, <lb></lb>quando corpus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>verſatur extra lineam Nodorum, inclinatur ad <lb></lb>planum Orbitæ <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end>;) præter perturbationem motus in Longitu<lb></lb>dinem jam ante expoſitam, inducet perturbationem motus in Lati<lb></lb>tudinem, trahendo corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>de plano ſuæ Orbitæ. </s> <s>Et hæc per<lb></lb>turbatio, in dato quovis corporum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ad invicem ſitu, erit ut <lb></lb>vis illa generans <emph type="italics"></emph>MN,<emph.end type="italics"></emph.end>adeoque minima evadet ubi <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>eſt miNI<lb></lb>ma, hoc eſt (uti jam expoſui) ubi attractio <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>non eſt multo ma<lb></lb>jor, neque multo minor attractione <emph type="italics"></emph>SK. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Ex his facile colligitur quod, ſi corpora plura minora <lb></lb><emph type="italics"></emph>P, S, R,<emph.end type="italics"></emph.end>&c. </s> <s>revolvantur circa maximum <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>motus corporis inti<lb></lb>mi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>minime perturbabitur attractionibus exteriorum, ubi corpus <lb></lb>maximum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>pariter a cæteris, pro ratione virium acceleratricum, <lb></lb>attrahitur & agitatur atque cætera a ſe mutuo. <pb xlink:href="039/01/188.jpg" pagenum="160"></pb><arrow.to.target n="note136"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note136"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. In Syſtemate vero trium corporum <emph type="italics"></emph>T, P, S,<emph.end type="italics"></emph.end>ſi attracti<lb></lb>ones acceleratrices binorum quorumcunQ.E.I. tertium ſint ad invi<lb></lb>cem reciproce ut quadrata diſtantiarum; corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>are<lb></lb>am circa corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>velocius deſcribet prope Conjunctionem <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& Op<lb></lb>poſitionem <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>quam prope Quadraturas <emph type="italics"></emph>C, D.<emph.end type="italics"></emph.end>Namque vis omnis <lb></lb>qua corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>urgetur & corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>non urgetur, quæque non agit <lb></lb>ſecundum lineam <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>accelerat vel retardat deſcriptionem areæ, <lb></lb>perinde ut ipſa in conſequentia vel in antecedentia dirigitur. </s> <s>Talis <lb></lb>eſt vis <emph type="italics"></emph>NM.<emph.end type="italics"></emph.end>Hæc in tranſitu corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>a <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>A<emph.end type="italics"></emph.end>tendit in con<lb></lb>ſequentia, motumque accelerat; dein uſque ad <emph type="italics"></emph>D<emph.end type="italics"></emph.end>in antecedentia, <lb></lb>& motum retardat; tum in conſequentia uſque ad <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& ultimo in <lb></lb>antecedentia tranſeundo a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>C.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et eodem argumento patet quod corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>cæteris pa<lb></lb>ribus, velocius movetur in Conjunctione & Oppoſitione quam in <lb></lb>Quadraturis. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Orbita corporis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>cæteris paribus, curvior eſt in Qua<lb></lb>draturis quam in Conjunctione & Oppoſitione. </s> <s>Nam corpora ve<lb></lb>lociora minus deflec<lb></lb><figure id="id.039.01.188.1.jpg" xlink:href="039/01/188/1.jpg"></figure><lb></lb>tunt a recto tramite. </s> <s>Et <lb></lb>præterea vis <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>vel <lb></lb><emph type="italics"></emph>NM,<emph.end type="italics"></emph.end>in Conjunctione <lb></lb>& Oppoſitione, con<lb></lb>traria eſt vi qua cor<lb></lb>pus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>trahit corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end><lb></lb>adeoque vim illam mi<lb></lb>nuit; corpus autem <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>minus deflectet a recto <lb></lb>tramite, ubi minus urgetur in corpus <emph type="italics"></emph>T.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Unde corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>cæteris paribus, longius recedet a cor<lb></lb>pore <emph type="italics"></emph>T<emph.end type="italics"></emph.end>in Quadraturis, quam in Conjunctione & Oppoſitione. </s> <s>Hæc <lb></lb>ita ſe habent excluſo motu Excentricitatis. </s> <s>Nam ſi Orbita corpo<lb></lb>ris <emph type="italics"></emph>P<emph.end type="italics"></emph.end>excentrica ſit: Excentricitas ejus (ut mox in hujus Corol. </s> <s>9. <lb></lb>oſtendetur) evadet maxima ubi Apſides ſunt in Syzygiis; indeque <lb></lb>fieri poteſt ut corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ad Apſidem ſummam appellans, abſit lon<lb></lb>gius a corpore <emph type="italics"></emph>T<emph.end type="italics"></emph.end>in Syzygiis quam in Quadraturis. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Quoniam vis centripeta corporis centralis <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>qua cor<lb></lb>pus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>retinetur in Orbe ſuo, augetur in Quadraturis per additio<lb></lb>nem vis <emph type="italics"></emph>LM,<emph.end type="italics"></emph.end>ac diminuitur in Syzygiis per ablationem vis <emph type="italics"></emph>KL,<emph.end type="italics"></emph.end>& <lb></lb>ob magnitudinem vis <emph type="italics"></emph>KL,<emph.end type="italics"></emph.end>magis diminuitur quam augetur; eſt au<lb></lb>tem vis illa centripeta (per Corol. </s> <s>2, Prop. </s> <s>IV.) in ratione compo<lb></lb>ſita ex ratione ſimplici radii <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>directe & ratione duplicata tempo-<pb xlink:href="039/01/189.jpg" pagenum="161"></pb>ris periodici inverſe: patet hanc rationem compoſitam diminui per </s></p> <p type="main"> <s><arrow.to.target n="note137"></arrow.to.target>actionem vis <emph type="italics"></emph>KL,<emph.end type="italics"></emph.end>adeoque tempus periodicum, ſi maneat Orbis <lb></lb>radius <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>augeri, idQ.E.I. ſubduplicata ratione qua vis illa cen<lb></lb>tripeta diminuitur: auctoque adeo vel diminuto hoc Radio, tem<lb></lb>pus periodicum augeri magis, vel diminui minus quam in Radii hu<lb></lb>jus ratione ſeſquiplicata, per Corol. </s> <s>6. Prop. </s> <s>IV. </s> <s>Si vis illa corporis <lb></lb>centralis paulatim langueſceret, corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>minus ſemper & minus <lb></lb>attractum perpetuo recederet longius a centro <emph type="italics"></emph>T<emph.end type="italics"></emph.end>; & contra, ſi vis <lb></lb>illa augeretur, accederet propius. </s> <s>Ergo ſi actio corporis longin<lb></lb>qui <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>qua vis illa diminuitur, augeatur ac diminuatur per vices; <lb></lb>augebitur ſimul ac diminuetur Radius <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>per vices, & tempus pe<lb></lb>riodicum augebitur ac diminuetur in ratione compoſita ex ratione <lb></lb>ſeſquiplicata Radii & ratione ſubduplicata qua vis illa centripeta <lb></lb>corporis centralis <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>per incrementum vel decrementum actionis <lb></lb>corporis longinqui <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>diminuitur vel augetur. </s></p> <p type="margin"> <s><margin.target id="note137"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Ex præmiſſis conſequitur etiam quod Ellipſeos a cor<lb></lb>pore <emph type="italics"></emph>P<emph.end type="italics"></emph.end>deſcriptæ Axis, ſeu Apſidum linea, quoad motum angula<lb></lb>rem progreditur & regreditur per vices, ſed magis tamen progre<lb></lb>ditur, & in ſingulis corporis revolutionibus per exceſſum progreſ<lb></lb>ſionis fertur in conſequentia. </s> <s>Nam vis qua corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>urgetur in <lb></lb>corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>in Quadraturis, ubi vis <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>evanuit, componitur ex vi <lb></lb><emph type="italics"></emph>LM<emph.end type="italics"></emph.end>& vi centripeta qua corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>trahit corpus <emph type="italics"></emph>P.<emph.end type="italics"></emph.end>Vis prior <emph type="italics"></emph>LM,<emph.end type="italics"></emph.end><lb></lb>ſi augeatur diſtantia <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>augetur in eadem fere ratione cum hac <lb></lb>diſtantia, & vis poſterior decreſcit in duplicata illa ratione, adeo<lb></lb>que ſumma harum virium decreſcit in minore quam duplicata ra<lb></lb>tione diſtantiæ <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>& propterea (per Corol. </s> <s>1. Prop. </s> <s>XLV) efficit <lb></lb>ut Aux, ſeu Apſis ſumma, regrediatur. </s> <s>In Conjunctione vero & <lb></lb>Oppoſitione, vis qua corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>urgetur in corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>differentia eſt <lb></lb>inter vim qua corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>trahit corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& vim <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>; & differen<lb></lb>tia illa, propterea quod vis <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>augetur quamproxime in ratione <lb></lb>diſtantiæ <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>decreſcit in majore quam duplicata ratione diſtan<lb></lb>tiæ <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>adeoque (per Corol. </s> <s>1. Prop.XLV) efficit ut Aux progre<lb></lb>diatur. </s> <s>In locis inter Syzygias & Quadraturas pendet motus Au<lb></lb>gis ex cauſa utraque conjunctim, adeo ut pro hujus vel alterius <lb></lb>exceſſu progrediatur ipſa vel regrediatur. </s> <s>Unde cum vis <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>in <lb></lb>Syzygiis ſit quaſi duplo major quam vis <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>in Quadraturis, ex<lb></lb>ceſſus in tota revolutione erit penes vim <emph type="italics"></emph>KL,<emph.end type="italics"></emph.end>transferetque Au<lb></lb>gem ſingulis revolutionibus in conſequentia. </s> <s>Veritas autem hujus <lb></lb>& præcedentis Corollarii facilius intelligetur concipiendo Syſtema <lb></lb>corporum duorum <emph type="italics"></emph>T, P<emph.end type="italics"></emph.end>corporibus pluribus <emph type="italics"></emph>S, S, S,<emph.end type="italics"></emph.end>&c, in Or<lb></lb>be <emph type="italics"></emph>ESE<emph.end type="italics"></emph.end>conſiſtentibus, undique cingi. </s> <s>Namque horum actioni-<pb xlink:href="039/01/190.jpg" pagenum="162"></pb><arrow.to.target n="note138"></arrow.to.target>bus actio ipſius <emph type="italics"></emph>T<emph.end type="italics"></emph.end>minuetur undique, decreſcetQ.E.I. ratione pluſ<lb></lb>quam duplicata diſtantiæ. </s></p> <p type="margin"> <s><margin.target id="note138"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Cum autem pendeat Apſidum progreſſus vel regreſſus <lb></lb>a decremento vis centripetæ facto in majori vel minori quam du<lb></lb>plicata ratione diſtantiæ <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>in tranſitu corporis ab Apſide ima <lb></lb>ad Apſidem ſummam; ut & a ſimili incremento in reditu ad Ap<lb></lb>ſidem imam; atque adeo maximus ſit ubi proportio vis in Apſide <lb></lb>ſumma ad vim in Apſide ima maxime recedit a duplicata ratione <lb></lb>diſtantiarum inverſa: manifeſtum eſt quod Apſides in Syzygiis <lb></lb>ſuis, per vim ablatitiam <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>NM-LM,<emph.end type="italics"></emph.end>progredientur ve<lb></lb>locius, inque Quadraturis ſuis tardius recedent per vim addititiam <lb></lb><emph type="italics"></emph>LM.<emph.end type="italics"></emph.end>Ob diuturnitatem vero temporis quo velocitas progreſſus vel <lb></lb>tarditas regreſſus continuatur, fit hæc inæqualitas longe maxima. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>9. Si corpus aliquod vi reciproce proportionali quadrato <lb></lb>diſtantiæ ſuæ a centro, revolveretur circa hoc centrum in El<lb></lb>lipſi, & mox, in deſcenſu ab Apſide ſumma ſeu Auge ad Apſidem <lb></lb>imam, vis illa per acceſſum perpetuum vis novæ augeretur in ra<lb></lb>tione pluſquam dupli<lb></lb><figure id="id.039.01.190.1.jpg" xlink:href="039/01/190/1.jpg"></figure><lb></lb>cata diſtantiæ diminu<lb></lb>tæ: manifeſtum eſt <lb></lb>quod corpus, perpe<lb></lb>tuo acceſſu vis illius <lb></lb>novæ impulſum ſem<lb></lb>per in centrum, magis <lb></lb>vergeret in hoc cen<lb></lb>trum, quam ſi urge<lb></lb>retur vi ſola creſcente <lb></lb>in duplicata ratione diſtantiæ diminutæ, adeoque Orbem deſcri<lb></lb>beret Orbe Elliptico interiorem, & in Apſide ima propius acce<lb></lb>deret ad centrum quam prius. </s> <s>Orbis igitur, acceſſu hujus vis no<lb></lb>væ, fiet magis excentricus. </s> <s>Si jam vis, in receſſu corporis ab <lb></lb>Apſide ima ad Apſidem ſummam, decreſceret iiſdem gradibus qui<lb></lb>bus ante creverat, rediret corpus ad diſtantiam priorem, adeoque <lb></lb>ſi vis decreſcat in majori ratione, corpus jam minus attractum aſ<lb></lb>cendet ad diſtantiam majorem & ſic Orbis Excentricitas adhuc ma<lb></lb>gis augebitur. </s> <s>Igitur ſi ratio incrementi & decrementi vis centri<lb></lb>petæ ſingulis revolutionibus augeatur, augebitur ſemper Excentri<lb></lb>citas; & e contra, diminuetur eadem ſi ratio illa decreſcat. </s> <s>Jam <lb></lb>vero in Syſtemate corporum <emph type="italics"></emph>T, P, S,<emph.end type="italics"></emph.end>ubi Apſides Orbis <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end><lb></lb>ſunt in Quadraturis, ratio illa incrementi ac decrementi minima eſt, <pb xlink:href="039/01/191.jpg" pagenum="163"></pb>& maxima fit ubi Apſides ſunt in Syzygiis. </s> <s>Si Apſides conſtituan<lb></lb><arrow.to.target n="note139"></arrow.to.target>tur in Quadraturis, ratio prope Apſides minor eſt & prope Syzy<lb></lb>gias major quam duplicata diſtantiarum, & ex ratione illa majori <lb></lb>oritur Augis motus velociſſimus, uti jam dictum eſt. </s> <s>At ſi con<lb></lb>ſideretur ratio incrementi vel decrementi totius in progreſſu inter <lb></lb>Apſides, hæc minor eſt quam duplicata diſtantiarum. </s> <s>Vis in Ap<lb></lb>ſide ima eſt ad vim in Apſide ſumma in minore quam duplicata <lb></lb>ratione diſtantiæ Apſidis ſummæ ab umbilico Ellipſeos ad di<lb></lb>ſtantiam Apſidis imæ ab eodem umbilico: & e contra, ubi <lb></lb>Apſides conſtituuntur in Syzygiis, vis in Apſide ima eſt ad vim <lb></lb>in Apſide ſumma in majore quam duplicata ratione diſtantiarum. </s> <s><lb></lb>Nam vires <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>in Quadraturis additæ viribus corporis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>compo<lb></lb>nunt vires in ratione minore, & vires <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>in Syzygiis ſubductæ <lb></lb>viribus corporis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>relinquunt vires in ratione majore. </s> <s>Eſt igi<lb></lb>tur ratio decrementi & incrementi totius, in tranſitu inter Apſides, <lb></lb>minima in Quadraturis, maxima in Syzygiis: & propterea in tran<lb></lb>ſitu Apſidum a Quadraturis ad Syzygias perpetuo augetur, auget<lb></lb>que Excentricitatem Ellipſeos; inque tranſitu a Syzygiis ad <lb></lb>Quadraturas perpetuo diminuitur, & Excentricitatem diminuit. </s></p> <p type="margin"> <s><margin.target id="note139"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>10. Ut rationem ineamus errorum in Latitudinem, finga<lb></lb>mus planum Orbis <emph type="italics"></emph>EST<emph.end type="italics"></emph.end>immobile manere; & ex errorum expo<lb></lb>ſita cauſa manifeſtum eſt quod, ex viribus <emph type="italics"></emph>NM, ML,<emph.end type="italics"></emph.end>quæ ſunt <lb></lb>cauſa illa tota, vis <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>agendo ſemper ſecundum planum Orbis <lb></lb><emph type="italics"></emph>PAB,<emph.end type="italics"></emph.end>nunquam perturbat motus in Latitudinem; quodque vis <emph type="italics"></emph>NM,<emph.end type="italics"></emph.end><lb></lb>ubi Nodi ſunt in Syzygiis, agendo etiam ſecundum idem Orbis <lb></lb>planum, non perturbat hos motus; ubi vero ſunt in Quadraturis <lb></lb>eos maxime perturbat, corpuſque <emph type="italics"></emph>P<emph.end type="italics"></emph.end>de plano Orbis ſui perpetuo <lb></lb>trahendo, minuit inclinationem plani in tranſitu corporis a Qua<lb></lb>draturis ad Syzygias, augetque viciſſim eandem in tranſitu a Syzy<lb></lb>giis ad Quadraturas. </s> <s>Unde fit ut corpore in Syzygiis exiſtente in<lb></lb>clinatio evadat omnium minima, redeatque ad priorem magnitudi<lb></lb>nem circiter, ubi corpus ad Nodum proximum accedit. </s> <s>At ſi Nodi <lb></lb>conſtituantur in Octantibus poſt Quadraturas, id eſt, inter <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& <emph type="italics"></emph>A, <lb></lb>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>intelligetur ex modo expoſitis quod, in tranſitu corporis <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>a Nodo alterutro ad gradum inde nonageſimum, inclinatio pla<lb></lb>ni perpetuo minuitur; deinde in tranſitu per proximos 45 gradus, <lb></lb>uſque ad Quadraturam proximam, inclinatio augetur, & poſtea de<lb></lb>nuo in tranſitu per alios 45 gradus, uſque ad Nodum proximum, <lb></lb>diminuitur. </s> <s>Magis itaQ.E.D.minuitur inclinatio quam augetur, & <lb></lb>propterea minor eſt ſemper in Nodo ſubſequente quam in præce-<pb xlink:href="039/01/192.jpg" pagenum="164"></pb><arrow.to.target n="note140"></arrow.to.target>dente. </s> <s>Et ſimili ratiocinio, inclinatio magis augetur quam diminui<lb></lb>tur ubi Nodi ſunt in Octantibus alteris inter <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>D, B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>C.<emph.end type="italics"></emph.end>In<lb></lb>clinatio igitur ubi Nodi ſunt in Syzygiis eſt omnium maxima. </s> <s>In <lb></lb>tranſitu eorum a Syzygiis ad Quadraturas, in ſingulis corporis ad <lb></lb>Nodos appulſibus, diminuitur, fitque omnium minima ubi Nodi <lb></lb>ſunt in Quadraturis & corpus in Syzygiis: dein creſcit iiſdem gra<lb></lb>dibus quibus antea decreverat, Nodiſque ad Syzygias proximas ap<lb></lb>pulſis ad magnitudinem primam revertitur. </s></p> <p type="margin"> <s><margin.target id="note140"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>11. Quoniam corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ubi Nodi ſunt in Quadraturis per<lb></lb>petuo trahitur de plano Orbis ſui, idQ.E.I. partem verſus <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>in <lb></lb>tranſitu ſuo a Nodo <emph type="italics"></emph>C<emph.end type="italics"></emph.end>per Conjunctionem <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad Nodum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>; & in <lb></lb>contrariam partem in tranſitu a Nodo <emph type="italics"></emph>D<emph.end type="italics"></emph.end>per Oppoſitionem <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad <lb></lb>Nodum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>; manifeſtum eſt quod in motu ſuo a Nodo <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>corpus <lb></lb>perpetuo recedit ab Orbis ſui plano primo <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end>uſQ.E.D.m per<lb></lb>ventum eſt ad Nodum proximum; adeoQ.E.I. hoc Nodo, longiſſi<lb></lb>me diſtans a plano illo primo <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end>tranſit per planum Orbis <emph type="italics"></emph>EST<emph.end type="italics"></emph.end><lb></lb>non in plani illius Nodo altero <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ſed in puncto quod inde vergit <lb></lb>ad partes corporis <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>quodque proinde novus eſt Nodi locus in an<lb></lb>teriora vergens. </s> <s>Et ſimili argumento pergent Nodi recedere in <lb></lb>tranſitu corporis de hoc Nodo in Nodum proximum. </s> <s>Nodi igi<lb></lb>tur in Quadraturis conſtituti perpetuo recedunt; in Syzygiis (ubi <lb></lb>motus in Latitudinem nil perturbatur) quieſcunt; in locis inter<lb></lb>mediis, conditionis utriuſque participes, recedunt tardius; adeoque, <lb></lb>ſemper vel retrogradi vel ſtationarii, ſingulis revolutionibus ferun<lb></lb>tur in antecedentia. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>12. Omnes illi in his Corollariis deſcripti Errores ſunt pau<lb></lb>lo majores in Conjunctione corporum <emph type="italics"></emph>P, S<emph.end type="italics"></emph.end>quam in eorum Op<lb></lb>poſitione, idque ob majores vires generantes <emph type="italics"></emph>NM<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ML.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>13. Cumque rationes horum Corollariorum non pendeant <lb></lb>a magnitudine corporis <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>obtinent præcedentia omnia, ubi corporis <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end>tanta ſtatuitur magnitudo ut circa ipſum revolvatur corporum duo<lb></lb>rum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>Syſtema. </s> <s>Et ex aucto corpore <emph type="italics"></emph>S<emph.end type="italics"></emph.end>auctaque adeo ipſius <lb></lb>vi centripeta, a qua errores corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>oriuntur, evadent errores illi <lb></lb>omnes (paribus diſtantiis) majores in hoc caſu quam in altero, ubi <lb></lb>corpus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>circum Syſtema corporum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>T<emph.end type="italics"></emph.end>revolvitur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>14. Cum autem vires <emph type="italics"></emph>NM, ML,<emph.end type="italics"></emph.end>ubi corpus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>longin<lb></lb>quum eſt, ſint quamproxime ut vis <emph type="italics"></emph>SK<emph.end type="italics"></emph.end>& ratio <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>con<lb></lb>junctim, hoc eſt, ſi detur tum diſtantia <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>tum corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>vis <lb></lb>abſoluta, ut <emph type="italics"></emph>ST cub.<emph.end type="italics"></emph.end>reciproce; ſint autem vires illæ <emph type="italics"></emph>NM, ML<emph.end type="italics"></emph.end><lb></lb>cauſæ errorum & effectuum omnium de quibus actum eſt in præce-<pb xlink:href="039/01/193.jpg" pagenum="165"></pb>dentibus Corollariis: manifeſtum eſt quod effectus illi omnes, ſtan<lb></lb><arrow.to.target n="note141"></arrow.to.target>te corporum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>Syſtemate, & mutatis tantum diſtantia <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>& <lb></lb>vi abſoluta corporis <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>ſint quamproxime in ratione compoſita ex <lb></lb>ratione directa vis abſolutæ corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>& ratione triplicata inverſa <lb></lb>diſtantiæ <emph type="italics"></emph>ST.<emph.end type="italics"></emph.end>Unde ſi Syſtema corporum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>revolvatur cir<lb></lb>ca corpus longinquum <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>vires illæ <emph type="italics"></emph>NM, ML<emph.end type="italics"></emph.end>& earum effectus <lb></lb>erunt (per Corol. </s> <s>2. & 6. Prop. </s> <s>IV.) reciproce in duplicata ratione <lb></lb>temporis periodici. </s> <s>Et inde etiam, ſi magnitudo corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>propor<lb></lb>tionalis ſit ipſius vi abſolutæ, erunt vires illæ <emph type="italics"></emph>NM, ML<emph.end type="italics"></emph.end>& earum <lb></lb>effectus directe ut cubus diametri apparentis longinqui corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>e <lb></lb>corpore <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ſpectati, & vice verſa. </s> <s>Namque hæ rationes eædem ſunt <lb></lb>atque ratio ſuperior compoſita. </s></p> <p type="margin"> <s><margin.target id="note141"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>15. Et quoniam ſi, manentibus Orbium <emph type="italics"></emph>ESE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end><lb></lb>forma, proportionibus & inclinatione ad invicem, mutetur eorum <lb></lb>magnitudo, & ſi corporum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>& <emph type="italics"></emph>T<emph.end type="italics"></emph.end>vel maneant vel mutentur vires <lb></lb>in data quavis ratio<lb></lb><figure id="id.039.01.193.1.jpg" xlink:href="039/01/193/1.jpg"></figure><lb></lb>ne, hæ vires (hoc eſt, <lb></lb>vis corporis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>qua cor<lb></lb>pus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>de recto trami<lb></lb>te in Orbitam <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end><lb></lb>deflectere, & vis cor<lb></lb>poris <emph type="italics"></emph>S<emph.end type="italics"></emph.end>qua corpus <lb></lb>idem <emph type="italics"></emph>P<emph.end type="italics"></emph.end>de Orbita illa <lb></lb>deviare cogitur) agunt <lb></lb>ſemper eodem mo<lb></lb>do & eadem proportione: neceſſe eſt ut ſimiles & proportiona<lb></lb>les ſint effectus omnes & proportionalia effectuum tempora; hoc <lb></lb>eſt, ut errores omnes lineares ſint ut Orbium diametri, angulares <lb></lb>vero iidem qui prius, & errorum linearium ſimilium vel angularium <lb></lb>æqualium tempora ut Orbium tempora periodica. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>16. Unde, ſi dentur Orbium formæ & inclinatio ad invi<lb></lb>cem, & mutentur utcunque corporum magnitudines, vires & di<lb></lb>ſtantiæ; ex datis erroribus & errorum temporibus in uno Caſu, col<lb></lb>ligi poſſunt errores & errorum tempora in alio quovis, quam pro<lb></lb>xime: Sed brevius hac Methodo. </s> <s>Vires <emph type="italics"></emph>NM, ML,<emph.end type="italics"></emph.end>cæteris ſtan<lb></lb>tibus, ſunt ut Radius <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>& harum effectus periodici (per Corol.2, <lb></lb>Lem. </s> <s>X) ut vires & quadratum temporis periodici corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>con<lb></lb>junctim. </s> <s>Hi ſunt errores lineares corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>; & hinc errores an<lb></lb>gulares e centro <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ſpectati (id eſt, tam motus Augis & Nodorum, <lb></lb>quam omnes in Longitudinem & Latitudinem errores apparentes) <lb></lb>ſunt, in qualibet revolutione corporis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ut quadratum temporis <pb xlink:href="039/01/194.jpg" pagenum="166"></pb><arrow.to.target n="note142"></arrow.to.target>revolutionis quam proxime. </s> <s>Conjungantur hæ rationes cum ratio<lb></lb>nibus Corollarii 14, & in quolibet corporum <emph type="italics"></emph>T, P, S<emph.end type="italics"></emph.end>Syſtemate, <lb></lb>ubi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>circum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ſibi propinquum, & <emph type="italics"></emph>T<emph.end type="italics"></emph.end>circum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>longinquum re<lb></lb>volvitur, errores angulares corporis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>de centro <emph type="italics"></emph>T<emph.end type="italics"></emph.end>apparentes, <lb></lb>erunt, in ſingulis revolutionibus corporis illius <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ut quadratum <lb></lb>temporis periodici corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>directe & quadratum temporis pe<lb></lb>riodici corporis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>inverſe. </s> <s>Et inde motus medius Augis erit in da<lb></lb>ta ratione ad motum medium Nodorum; & motus uterque erit ut tempus periodicum corporis &c. </s> <s><lb></lb>quadratum temporis periodici corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>directe & quadratum <lb></lb>temporis periodici corporis <emph type="italics"></emph>T<emph.end type="italics"></emph.end>inverſe. </s> <s>Augendo vel minuendo <lb></lb>Excentricitatem & Inclinationem Orbis <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end>non mutantur mo<lb></lb>tus Augis & Nodorum ſenſibiliter, niſi ubi eædem ſunt nimis <lb></lb>magnæ. </s></p> <p type="margin"> <s><margin.target id="note142"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>17. Cum autem linea <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>nunc major ſit nunc minor <lb></lb>quam radius <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>exponatur vis mediocris <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>per radium il<lb></lb>lum <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>; & erit hæc ad <lb></lb><figure id="id.039.01.194.1.jpg" xlink:href="039/01/194/1.jpg"></figure><lb></lb>vim mediocrem <emph type="italics"></emph>SK<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>(quam expo<lb></lb>nere licet per <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>) ut <lb></lb>longitudo <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ad lon<lb></lb>gitudinem <emph type="italics"></emph>ST.<emph.end type="italics"></emph.end>Eſt au<lb></lb>tem vis mediocris <emph type="italics"></emph>SN<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>ST,<emph.end type="italics"></emph.end>qua corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end><lb></lb>retinetur in Orbe ſuo <lb></lb>circum <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>ad vim qua <lb></lb>corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>retinetur in Orbe ſuo circum <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>in ratione compoſita ex <lb></lb>ratione radii <emph type="italics"></emph>ST<emph.end type="italics"></emph.end>ad radium <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>& ratione duplicata temporis pe<lb></lb>riodici corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>circum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ad tempus periodicum corporis <emph type="italics"></emph>T<emph.end type="italics"></emph.end><lb></lb>circum <emph type="italics"></emph>S.<emph.end type="italics"></emph.end>Et ex æquo, vis mediocris <emph type="italics"></emph>LM,<emph.end type="italics"></emph.end>ad vim qua corpus <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>retinetur in Orbe ſuo circum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>(quave corpus idem <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>eo<lb></lb>dem tempore periodico, circum punctum quodvis immobile <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ad <lb></lb>diſtantiam <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>revolvi poſſet) eſt in ratione illa duplicata periodi<lb></lb>eorum temporum. </s> <s>Datis igitur temporibus periodicis una cum di<lb></lb>ſtantia <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>datur vis mediocris <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>; & ea data, datur etiam vis <lb></lb><emph type="italics"></emph>MN<emph.end type="italics"></emph.end>quamproxime per analogiam linearum <emph type="italics"></emph>PT, MN.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>18. Iiſdem legibus quibus corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>circum corpus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>re<lb></lb>volvitur, fingamus corpora plura fluida circum idem <emph type="italics"></emph>T<emph.end type="italics"></emph.end>ad æqua<lb></lb>les ab ipſo diſtantias moveri; deinde ex his contiguis factis confla<lb></lb>ri Annulum fluidum, rotundum ac corpori <emph type="italics"></emph>T<emph.end type="italics"></emph.end>concentricum; & <lb></lb>ſingulæ Annuli partes, motus ſuos omnes ad legem corporis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>per-<pb xlink:href="039/01/195.jpg" pagenum="167"></pb>agendo, propius accedent ad corpus <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>& celerius movebuntur <lb></lb><arrow.to.target n="note143"></arrow.to.target>in Conjunctione & Oppoſitione ipſarum & corporis <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>quam in <lb></lb>Quadraturis. </s> <s>Et Nodi Annuli hujus ſeu interſectiones ejus cum <lb></lb>plano Orbitæ corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>quieſcent in Syzygiis; extra Syzy<lb></lb>gias vero movebuntur in antecedentia, & velociſſime quidem in <lb></lb>Quadraturis, tardius aliis in locis. </s> <s>Annuli quoQ.E.I.clinatio varia<lb></lb>bitur, & axis ejus ſingulis revolutionibus oſcillabitur, completaque <lb></lb>revolutione ad priſtinum ſitum redibit, niſi quatenus per præceſſi<lb></lb>onem Nodorum circumfertur. </s></p> <p type="margin"> <s><margin.target id="note143"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>19. Fingas jam Globum corporis <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>ex materia non fluida <lb></lb>conſtantem, ampliari & extendi uſque ad hunc Annulum, & alveo <lb></lb>per circuitum excavato continere Aquam, motuque eodem perio<lb></lb>dico circa axem ſuum uniformiter revolvi. </s> <s>Hic liquor per vices <lb></lb>acceleratus & retardatus (ut in ſuperiore Corollario) in Syzygiis <lb></lb>velocior erit, in Quadraturis tardior quam ſuperficies Globi, & <lb></lb>ſic fluet in alveo refluet que ad modum Maris. </s> <s>Aqua revolvendo cir<lb></lb>ca Globi centrum quieſcens, ſi tollatur attractio corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>nullum <lb></lb>acquiret motum fluxus & refluxus. </s> <s>Par eſt ratio Globi uniformiter <lb></lb>progredientis in directum & interea revolventis circa centrum <lb></lb>ſuum (per Legum Corol. </s> <s>5.) ut & Globi de curſu rectilineo uNI<lb></lb>formiter tracti, per Legum Corol. </s> <s>6. Accedat autem corpus <emph type="italics"></emph>S,<emph.end type="italics"></emph.end><lb></lb>& ab ipſius inæquabili attractione mox turbabitur Aqua. </s> <s>Etenim <lb></lb>major erit attractio aquæ propioris, minor ea remotioris. </s> <s>Vis <lb></lb>autem <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>trahet aquam deorſum in Quadraturis, facietQ.E.I.<lb></lb>ſam deſcendere uſque ad Syzygias; & vis <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>trahet eandem ſur<lb></lb>ſum in Syzygiis, ſiſtetQ.E.D.ſcenſum ejus & faciet ipſam aſcendere <lb></lb>uſque ad Quadraturas. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>20. Si Annulus jam rigeat & minuatur Globus, ceſſa<lb></lb>bit motus fluendi & refluendi; ſed Oſcillatorius ille inclinationis <lb></lb>motus & præceſſio Nodorum manebunt. </s> <s>Habeat Globus eundem <lb></lb>axem cum Annulo, gyroſque compleat iiſdem temporibus, & ſuper<lb></lb>ficie ſua contingat ipſum interius, eiQ.E.I.hæreat; & participando <lb></lb>motum ejus, compages utriuſque Oſcillabitur & Nodi regredien<lb></lb>tur. </s> <s>Nam Globus, ut mox dicetur, ad ſuſcipiendas impreſſiones <lb></lb>omnes indifferens eſt. </s> <s>Annuli Globo orbati maximus inclinationis <lb></lb>angulus eſt ubi Nodi ſunt in Syzygiis. </s> <s>Inde in progreſſu Nodo<lb></lb>rum ad Quadraturas conatur is inclinationem ſuam minuere, & iſto <lb></lb>conatu motum imprimit Globo toti. </s> <s>Retinet Globus motum im<lb></lb>preſſum uſQ.E.D.m Annulus conatu contrario motum hunc tollat, <lb></lb>imprimatque motum novum in contrariam partem: Atque hac ra-<pb xlink:href="039/01/196.jpg" pagenum="168"></pb><arrow.to.target n="note144"></arrow.to.target>tione maximus decreſcentis inclinationis motus fit in Quadraturis <lb></lb>Nodorum, & minimus inclinationis angulus in Octantibus poſt <lb></lb>Quadraturas; dein maximus reclinationis motus in Syzygiis, & <lb></lb>maximus angulus in Octantibus proximis. </s> <s>Et eadem eſt ratio Glo<lb></lb>bi Annulo nudati, qui in regionibus æquatoris vel altior eſt paulo <lb></lb>quam juxta polos, vel conſtat ex nateria paulo denſiore. </s> <s>Sup<lb></lb>plet enim vicem Annuli iſte materiæ in æquatoris regionibus exceſ<lb></lb>ſus. </s> <s>Et quanquam, aucta utcunque Globi hujus vi centripeta, <lb></lb>tendere ſupponantur omnes ejus partes deorſum, ad modum gra<lb></lb>vitantium partium telluris, tamen Phænomena hujus & præceden<lb></lb>tis Corollarii vix inde mutabuntur. </s></p> <p type="margin"> <s><margin.target id="note144"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>21. Eadem ratione qua materia Globi juxta æquatorem <lb></lb>redundans efficit ut Nodi regrediantur, atque adeo per hujus in<lb></lb>crementum augetur iſte regreſſus, per diminutionem vero diminui<lb></lb>tur & per ablationem tollitur; ſi materia pluſquam redundans tol<lb></lb>latur, hoc eſt, ſi Globus juxta æquatorem vel depreſſior reddatur <lb></lb>vel rarior quam juxta polos, orietur motus Nodorum in con<lb></lb>ſequentia. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>22. Et inde viciſſim, ex motu Nodorum innoteſcit conſti<lb></lb>tutio Globi. </s> <s>Nimirum ſi Globus polos eoſdem conſtanter ſervat, <lb></lb>& motus fit in antecedentia, materia juxta æquatorem redundat; <lb></lb>ſi in conſequentia, deficit. </s> <s>Pone Globum uniformem & perfecte <lb></lb>circinatum in ſpatiis liberis primo quieſcere; dein impetu quocun<lb></lb>que obliQ.E.I. ſuperficiem ſuam facto propelli, & motum inde <lb></lb>concipere partim circularem, partim in directum. </s> <s>Quoniam Glo<lb></lb>bus iſte ad axes omnes per centrum ſuum tranſeuntes indifferenter <lb></lb>ſe habet, neque propenſior eſt in unum axem, unumve axis ſitum, <lb></lb>quam in alium quemvis; perſpicuum eſt quod is axem ſuum axiſ<lb></lb>Q.E.I.clinationem vi propria nunquam mutabit. </s> <s>Impellatur jam <lb></lb>Globus oblique, in eadem illa ſuperficiei parte qua prius, impulſu <lb></lb>quocunque novo; & cum citior vel ferior impulſus effectum nil <lb></lb>mutet, manifeſtum eſt quod hi duo impulſus ſucceſſive impreſſi <lb></lb>eundem producent motum ac ſi ſimul impreſſi fuiſſent, hoc eſt, <lb></lb>eundem ac ſi Globus vi ſimplici ex utroque (per Legum Corol. </s> <s>2.) <lb></lb>compoſita impulſus fuiſſet, atque adeo ſimplicem, circa axem in<lb></lb>clinatione datum. </s> <s>Et par eſt ratio impulſus ſecundi facti in lo<lb></lb>cum alium quemvis in æquatore motus primi; ut & impulſus pri<lb></lb>mi facti in locum quemvis in æquatore motus, quem impulſus ſe<lb></lb>cundus abſque primo generaret; atque adeo impulſuum amborum <lb></lb>factorum in loca quæcunque: Generabunt hi eundem motum cir-<pb xlink:href="039/01/197.jpg" pagenum="169"></pb>cularem ac ſi ſimul & ſemel in locum interſectionis æquatorum <lb></lb><arrow.to.target n="note145"></arrow.to.target>motuum illorum, quos feorſim generarent, fuiſſent impreſſi. </s> <s><lb></lb>Globus igitur homogeneus & perfectus non retinet motus plures <lb></lb>diſtinctos, ſed impreſſos omnes componit & ad unum reducit, & <lb></lb>quatenus in ſe eſt, gyratur ſemper motu ſimplici & uniformi circa <lb></lb>axem unicum, inclinatione ſemper invariabili datum. </s> <s>Sed nec vis <lb></lb>centripeta inclinationem axis, aut rotationis velocitatem mutare <lb></lb>poteſt. </s> <s>Si Globus plano quocunque, per centrum ſuum & cen<lb></lb>trum in quod vis dirigitur tranſeunte, dividi intelligatur in duo he<lb></lb>miſphæria; urgebit ſemper vis illa utrumque hemiſphærium æqua<lb></lb>liter, & propterea Globum, quoad motum rotationis, nullam in <lb></lb>partem inclinabit. </s> <s>Addatur vero alicubi inter polum & æquato<lb></lb>rem materia nova in formam montis cumulata, & hæc, perpetuo <lb></lb>conatu recedendi a centro ſui motus, turbabit motum Globi, fa<lb></lb>cietque polos ejus errare per ipſius ſuperficiem, & circulos circum <lb></lb>ſe punctumque ſibi oppoſitum perpetuo deſcribere. </s> <s>Neque corrige<lb></lb>tur iſta vagationis enormitas, niſi locando montem illum vel in polo <lb></lb>alterutro, quo in Caſu (per Corol. </s> <s>21) Nodi æquatoris progredien<lb></lb>tur; vel in æquatore, qua ratione (per Corol. </s> <s>20) Nodi regredi<lb></lb>entur; vel denique ex altera axis parte addendo materiam novam, <lb></lb>qua mons inter movendum libretur, & hoc pacto Nodi vel pro<lb></lb>gredientur, vel recedent, perinde ut mons & hæcce nova materia <lb></lb>ſunt vel polo vel æquatori propiores. </s></p> <p type="margin"> <s><margin.target id="note145"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXVII. THEOREMA XXVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſitis iiſdem attractionum legibus, dico quod corpus exterius<emph.end type="italics"></emph.end>S, <lb></lb><emph type="italics"></emph>circa interiorum<emph.end type="italics"></emph.end>P, T <emph type="italics"></emph>commune gravitatis centrum<emph.end type="italics"></emph.end>C, <emph type="italics"></emph>radiis <lb></lb>ad centrum illud ductis, deſcribit areas temporibus magis pro<lb></lb>portionales & Orbem ad formam Ellipſeos umbilicum in centro <lb></lb>eodem habentis magis accedentem, quam circa corpus intimum <lb></lb>& maximum<emph.end type="italics"></emph.end>T, <emph type="italics"></emph>radiis ad ipſum ductis, deſcribere potest.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam corporis <emph type="italics"></emph>S<emph.end type="italics"></emph.end>attractiones verſus <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>componunt ipſius at<lb></lb>tractionem abſolutam, quæ magis dirigitur in corporum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>com<lb></lb>mune gravitatis centrum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>quam in corpus maximum <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>quæque <lb></lb>quadrato diſtantiæ <emph type="italics"></emph>SC<emph.end type="italics"></emph.end>magis eſt proportionalis reciproce, quam <lb></lb>quadrato diſtantiæ <emph type="italics"></emph>ST:<emph.end type="italics"></emph.end>ut rem perpendenti facile conſtabit. <pb xlink:href="039/01/198.jpg" pagenum="170"></pb><arrow.to.target n="note146"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note146"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXVIII. THEOREMA XXVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſitis iiſdem attractionum legibus, dico quod corpus exterius<emph.end type="italics"></emph.end>S, <lb></lb><emph type="italics"></emph>circa interiorum<emph.end type="italics"></emph.end>P & T <emph type="italics"></emph>commune gravitatis centrum<emph.end type="italics"></emph.end>C, <emph type="italics"></emph>ra<lb></lb>diis ad centrum illud ductis, deſcribit areas temporibus magis <lb></lb>proportionales, & Orbem ad formam Ellipſeos umbilicum in <lb></lb>centro eodem habentis magis accedentem, ſi corpus intimum & <lb></lb>maximum his attractionibus perinde atque cætera agitetur, quam <lb></lb>ſi id vel non attractum quieſcat, vel multo magis aut multo <lb></lb>minus attractum aut multo magis aut multo minus agitetur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Demonſtratur eo<lb></lb><figure id="id.039.01.198.1.jpg" xlink:href="039/01/198/1.jpg"></figure><lb></lb>dem fere modo cum <lb></lb>Prop. </s> <s>LXVI, ſed ar<lb></lb>gumento prolixiore, <lb></lb>quod ideo prætereo. </s> <s><lb></lb>Suffecerit rem ſic æſti<lb></lb>mare. </s> <s>Ex demonſtra<lb></lb>tione Propoſitionis <lb></lb>noviſſimæ liquet cen<lb></lb>trum in quod corpus <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end>conjunctis viribus urgetur, proximum eſſe communi centro gra<lb></lb>vitatis duorum illorum. </s> <s>Si coincideret hoc centrum cum centro <lb></lb>illo communi, & quieſceret commune centrum gravitatis corporum <lb></lb>trium; deſcriberent corpus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ex una parte, & commune centrum <lb></lb>aliorum duorum ex altera parte, circa commune omnium centrum <lb></lb>quieſcens, Ellipſes accuratas. </s> <s>Liquet hoc per Corollarium ſecun<lb></lb>dum Propoſitionis LVIII collatum cum demonſtratis in Propoſ. </s> <s><lb></lb>LXIV & LXV. </s> <s>Perturbatur iſte motus Ellipticus aliquantulum per <lb></lb>diſtantiam centri duorum a centro in quod tertium <emph type="italics"></emph>S<emph.end type="italics"></emph.end>attrahitur. </s> <s><lb></lb>Detur præterea motus communi trium centro, & augebitur per<lb></lb>turbatio. </s> <s>Proinde minima eſt perturbatio ubi commune trium <lb></lb>centrum quieſcit, hoc eſt, ubi corpus intimum & maximum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>lege <lb></lb>cæterorum attrahitur: fitque major ſemper ubi trium commune il<lb></lb>lud centrum, minuendo motum corporis <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>moveri incipit & ma<lb></lb>gis deinceps magiſque agitatur. </s></p><pb xlink:href="039/01/199.jpg" pagenum="171"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Et hinc, ſi corpora plura minora revolvantur circa maxi<lb></lb><arrow.to.target n="note147"></arrow.to.target>mum, colligere licet quod Orbitæ deſcriptæ propius accedent ad <lb></lb>Ellipticas, & arearum deſcriptiones fient magis æquabiles, ſi cor<lb></lb>pora omnia viribus acceleratricibus, quæ ſunt ut eorum vires ab<lb></lb>ſolutæ directe & quadrata diſtantiarum inverſe, ſe mutuo trahant <lb></lb>agitentque, & Orbitæ cujuſque umbilicus collocetur in communi <lb></lb>centro gravitatis corporum omnium interiorum (nimirum umbi<lb></lb>licus Orbitæ primæ & intimæ in centro gravitatis corporis maxi<lb></lb>mi & intimi; ille Orbitæ ſecundæ, in communi centro gravi<lb></lb>tatis corporum duorum intimorum; iſte tertiæ, in communi cen<lb></lb>tro gravitatis trium interiorum; & ſic deinceps) quam ſi corpus <lb></lb>intimum quieſcat & ſtatuatur communis umbilicus Orbitarum <lb></lb>omnium. </s></p> <p type="margin"> <s><margin.target id="note147"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXIX. THEOREMA XXIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>In Syſtemate corporum plurium<emph.end type="italics"></emph.end>A, B, C, D, <emph type="italics"></emph>&c. </s> <s>ſi corpus aliquod<emph.end type="italics"></emph.end><lb></lb>A <emph type="italics"></emph>trahit cætera omnia<emph.end type="italics"></emph.end>B, C, D, <emph type="italics"></emph>&c. </s> <s>viribus acceler atricibus <lb></lb>quæ ſunt reciproce ut quadrata diſtantiarum a trahente; & <lb></lb>corpus aliud<emph.end type="italics"></emph.end>B <emph type="italics"></emph>trahit etiam cætera<emph.end type="italics"></emph.end>A, C, D, <emph type="italics"></emph>&c. </s> <s>viribus quæ <lb></lb>ſunt reciproce ut quadrata diſtantiarum a trahente: erunt Ab<lb></lb>ſolutæ corporum trahentium<emph.end type="italics"></emph.end>A, B <emph type="italics"></emph>vires ad invicem, ut ſunt <lb></lb>ipſa corpora<emph.end type="italics"></emph.end>A, B, <emph type="italics"></emph>quorum ſunt vires.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam attractiones acceleratrices corporum omnium <emph type="italics"></emph>B, C, D<emph.end type="italics"></emph.end>ver<lb></lb>ſus <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>paribus diſtantiis, ſibi invicem æquantur ex Hypotheſi; & <lb></lb>ſimiliter attractiones acceleratrices corporum omnium verſus <emph type="italics"></emph>B,<emph.end type="italics"></emph.end><lb></lb>paribus diſtantiis, ſibi invicem æquantur. </s> <s>Eſt autem abſoluta vis <lb></lb>attractiva corporis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad vim abſolutam attractivam corporis <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>ut <lb></lb>attractio acceleratrix corporum omnium verſus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad attractionem <lb></lb>acceleratricem corporum omnium verſus <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>paribus diſtantiis; & <lb></lb>ita eſt attractio acceleratrix corporis <emph type="italics"></emph>B<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ad attractionem <lb></lb>acceleratricem corporis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>B.<emph.end type="italics"></emph.end>Sed attractio acceleratrix cor<lb></lb>poris <emph type="italics"></emph>B<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>eſt ad attractionem acceleratricem corporis <emph type="italics"></emph>A<emph.end type="italics"></emph.end><lb></lb>verſus <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>ut maſſa corporis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad maſſam corporis <emph type="italics"></emph>B<emph.end type="italics"></emph.end>; propterea <lb></lb>quod vires motrices, quæ (per Definitionem ſecundam, ſepti<lb></lb>mam & octavam) ex viribus acceleratricibus in corpora attracta <lb></lb>ductis oriuntur, ſunt (per motus Legem tertiam) ſibi invicem æqua-<pb xlink:href="039/01/200.jpg" pagenum="172"></pb><arrow.to.target n="note148"></arrow.to.target>les. </s> <s>Ergo abſoluta vis attractiva corporis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>eſt ad abſolutam vim <lb></lb>attractivam corporis <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>ut maſſa corporis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad maſſam corpo<lb></lb>ris <emph type="italics"></emph>B. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note148"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi ſingula Syſtematis corpora <emph type="italics"></emph>A, B, C, D,<emph.end type="italics"></emph.end>&c. <lb></lb></s> <s>ſeorſim ſpectata trahant cætera omnia viribus acceleratricibus quæ <lb></lb>ſunt reciproce ut quadrata diſtantiarum a trahente; erunt corpo<lb></lb>rum illorum omnium vires abſolutæ ad invicem ut ſunt ipſa cor<lb></lb>pora. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Eodem argumento, ſi ſingula Syſtematis corpora <lb></lb><emph type="italics"></emph>A, B, C, D,<emph.end type="italics"></emph.end>&c. </s> <s>ſeorſim ſpectata trahant cætera omnia viribus <lb></lb>acceleratricibus quæ ſunt vel reciproce vel directe in ratione dig<lb></lb>nitatis cujuſcunQ.E.D.ſtantiarum a trahente, quæve ſecundum Le<lb></lb>gem quamcunque communem ex diſtantiis ab unoquoque trahente <lb></lb>definiuntur; conſtat quod corporum illorum vires abſolutæ ſunt <lb></lb>ut corpora. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. In Syſtemate corporum, quorum vires decreſcunt in <lb></lb>ratione duplicata diſtantiarum, ſi minora circa maximum in Ellipſi<lb></lb>bus umbilicum communem in maximi illius centro habentibus quam <lb></lb>fieri poteſt accuratiſſimis revolvantur, & radiis ad maximum illud <lb></lb>ductis deſcribant areas temporibus quam maxime proportionales: <lb></lb>erunt corporum illorum vires abſolutæ ad invicem, aut accurate aut <lb></lb>quamproxime in ratione corporum; & contra. </s> <s>Patet per Corol. </s> <s><lb></lb>Prop. </s> <s>LXVIII collatum cum hujus Corol. </s> <s>1. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>His Propoſitionibus manuducimur ad analogiam inter vires cen<lb></lb>tripetas & corpora centralia, ad quæ vires illæ dirigi ſolent. </s> <s>Ra<lb></lb>tioni enim conſentaneum eſt, ut vires quæ ad corpora diriguntur <lb></lb>pendeant ab eorundem natura & quantitate, ut fit in Magneticis. </s> <s><lb></lb>Et quoties hujuſmodi caſus incidunt, æſtimandæ erunt corporum <lb></lb>attractiones, aſſignando ſingulis eorum particulis vires proprias, <lb></lb>& colligendo ſummas virium. </s> <s>Vocem Attractionis hic generaliter <lb></lb>uſurpo pro corporum conatu quocunque accedendi ad invicem; <lb></lb>ſive conatus iſte fiat ab actione corporum, vel ſe mutuo petentium, <lb></lb>vel per Spiritus emiſſos ſe invicem agitantium, ſive is ab actione <lb></lb>Ætheris, aut Aeris, Mediive cujuſcunque ſeu corporei ſeu incorpo<lb></lb>rei oriatur corpora innatantia in ſe invicem utcunQ.E.I.pellentis. </s> <s><lb></lb>Eodem ſenſu generali uſurpo vocem Impulſus, non ſpecies virium <pb xlink:href="039/01/201.jpg" pagenum="173"></pb>& qualitates Phyſicas, ſed quantitates & proportiones Mathema<lb></lb><arrow.to.target n="note149"></arrow.to.target>ticas in hoc Tractatu expendens, ut in Definitionibus explicui. </s> <s>In <lb></lb>Matheſi inveſtigandæ ſunt virium quantitates & rationes illæ, quæ <lb></lb>ex conditionibus quibuſcunque poſitis conſequentur: deinde, ubi <lb></lb>in Phyſicam deſcenditur, conferendæ ſunt hæ rationes cum Phæ<lb></lb>nomenis, ut innoteſcat quænam virium conditiones ſingulis cor<lb></lb>porum attractivorum generibus competant. </s> <s>Et tum demum de vi<lb></lb>rium ſpeciebus, cauſis & rationibus Phyſicis tutius diſputare lice<lb></lb>bit. </s> <s>Videamus igitur quibus viribus corpora Sphærica, ex particu<lb></lb>lis modo jam expoſito attractivis conſtantia, debeant in ſe mutuo<lb></lb>agere, & quales motus inde conſequantur. </s></p> <p type="margin"> <s><margin.target id="note149"></margin.target>LIBER <lb></lb>PRIMUS.</s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"></emph>SECTIO XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Corporum Sphæriccrum Viribus attractivis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXX. THEOREMA XXX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ad Sphæricæ ſuperficiei puncta ſingula tendant vires æquales cen<lb></lb>tripetæ decreſcentes in duplicata ratione diſtantiarum a punctis: <lb></lb>dico quod corpuſculum intra ſuperficiem conſtitutum his viri<lb></lb>bus nullam in partem attrahitur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>HIKL<emph.end type="italics"></emph.end>ſuperficies illa Sphæri<lb></lb><figure id="id.039.01.201.1.jpg" xlink:href="039/01/201/1.jpg"></figure><lb></lb>ca, & <emph type="italics"></emph>P<emph.end type="italics"></emph.end>corpuſculum intus conſtitu<lb></lb>tum. </s> <s>Per <emph type="italics"></emph>P<emph.end type="italics"></emph.end>agantur ad hanc ſuper<lb></lb>ficiem lineæ duæ <emph type="italics"></emph>HK, IL,<emph.end type="italics"></emph.end>arcus <lb></lb>quam minimos <emph type="italics"></emph>HI, KL<emph.end type="italics"></emph.end>intercipi<lb></lb>entes; &, ob triangula <emph type="italics"></emph>HPI, LPK<emph.end type="italics"></emph.end><lb></lb>(per Corol. </s> <s>3. Lem. </s> <s>VII) ſimilia, arcus <lb></lb>illi erunt diſtantiis <emph type="italics"></emph>HP, LP<emph.end type="italics"></emph.end>pro<lb></lb>portionales; & ſuperficiei Sphæricæ <lb></lb>particulæ quævis ad <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>& <emph type="italics"></emph>KL,<emph.end type="italics"></emph.end>rec<lb></lb>tis per punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>tranſeuntibus un<lb></lb>dique terminatæ, erunt in duplicata <lb></lb>illa ratione. </s> <s>Ergo vires harum particularum in corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>exercitæ <lb></lb>ſunt inter ſe æquales. </s> <s>Sunt enim ut particulæ directe & quadrata <lb></lb>diſtantiarum inverſe. </s> <s>Et hæ duæ rationes componunt rationem <pb xlink:href="039/01/202.jpg" pagenum="174"></pb><arrow.to.target n="note150"></arrow.to.target>æqualitatis. </s> <s>Attractiones igitur, in contrarias partes æqualiter fac<lb></lb>tæ, ſe mutuo deſtruunt. </s> <s>Et ſimili argumento, attractiones omnes <lb></lb>per totam Sphæricam ſuperficiem a contrariis attractionibus de<lb></lb>ſtruuntur. </s> <s>Proinde corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>nullam in partem his attractionibus <lb></lb>impellitur. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note150"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXI. THEOREMA XXXI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, dico quod corpuſculum extra Sphæricam ſuperficiem <lb></lb>conſtitutum attrahitur ad centrum Sphæræ, vi reciproce propor<lb></lb>tionali quadrato diſtantiæ ſuæ ab eodem centro.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sint <emph type="italics"></emph>AHKB, ahkb<emph.end type="italics"></emph.end>æquales duæ ſuperficies Sphæricæ, centris <lb></lb><emph type="italics"></emph>S, s,<emph.end type="italics"></emph.end>diametris <emph type="italics"></emph>AB, ab<emph.end type="italics"></emph.end>deſcriptæ, & <emph type="italics"></emph>P, p<emph.end type="italics"></emph.end>corpuſcula ſita extrin<lb></lb>ſecus in diametris illis productis. </s> <s>Agantur a corpuſculis lineæ <lb></lb><figure id="id.039.01.202.1.jpg" xlink:href="039/01/202/1.jpg"></figure><lb></lb><emph type="italics"></emph>PHK, PIL, phk, pil,<emph.end type="italics"></emph.end>auferentes a circulis maximis <emph type="italics"></emph>AHB, <lb></lb>ahb,<emph.end type="italics"></emph.end>æquales arcus <emph type="italics"></emph>HK, hk<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IL, il:<emph.end type="italics"></emph.end>Et ad eas de<lb></lb>mittantur perpendicula <emph type="italics"></emph>SD, sd; SE, se; IR, ir;<emph.end type="italics"></emph.end>quorum <lb></lb><emph type="italics"></emph>SD, sd<emph.end type="italics"></emph.end>ſecent <emph type="italics"></emph>PL, pl<emph.end type="italics"></emph.end>in <emph type="italics"></emph>F<emph.end type="italics"></emph.end>& <emph type="italics"></emph>f:<emph.end type="italics"></emph.end>Demittantur etiam ad diame<lb></lb>tros perpendicula <emph type="italics"></emph>IQ, <expan abbr="iq.">ique</expan><emph.end type="italics"></emph.end>Evaneſcant anguli <emph type="italics"></emph>DPE, dpe:<emph.end type="italics"></emph.end>& <lb></lb>(ob æquales <emph type="italics"></emph>DS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ds, ES<emph.end type="italics"></emph.end>& <emph type="italics"></emph>es,<emph.end type="italics"></emph.end>) lineæ <emph type="italics"></emph>PE, PF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pe, pf<emph.end type="italics"></emph.end><lb></lb>& lineolæ <emph type="italics"></emph>DF, df<emph.end type="italics"></emph.end>pro æqualibus habeantur; quippe quarum ra<lb></lb>tio ultima, angulis illis <emph type="italics"></emph>DPE, dpe<emph.end type="italics"></emph.end>ſimul evaneſcentibus, eſt æ<lb></lb>qualitatis. </s> <s>His itaque conſtitutis, erit <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PF<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>RI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DF,<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>pf<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>pi<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>df<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ri<emph.end type="italics"></emph.end>; & ex æquo <emph type="italics"></emph>PIXpf<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PFXpi<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>RI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ri,<emph.end type="italics"></emph.end>hoc eſt (per Corol. </s> <s>3. Lem. </s> <s>VII,) ut arcus <emph type="italics"></emph>IH<emph.end type="italics"></emph.end>ad <lb></lb>arcum <emph type="italics"></emph>ih.<emph.end type="italics"></emph.end>Rurſus <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>IQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SE,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ps<emph.end type="italics"></emph.end>and <emph type="italics"></emph>pi<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>se<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>SE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="iq;">ique</expan><emph.end type="italics"></emph.end>& ex æquo <emph type="italics"></emph>PIXps<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PSXpi<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>IQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="iq.">ique</expan><emph.end type="italics"></emph.end>ET <lb></lb>conjunctis rationibus <emph type="italics"></emph>PI quad.XpfXps<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>pi quad.XPFXPS,<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>IHXIQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="ihXiq;">ihXique</expan><emph.end type="italics"></emph.end>hoc eſt, ut ſuperficies circularis, quam <pb xlink:href="039/01/203.jpg" pagenum="175"></pb>arcus <emph type="italics"></emph>IH<emph.end type="italics"></emph.end>convolutione ſemicirculi <emph type="italics"></emph>AKB<emph.end type="italics"></emph.end>circa diametrum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note151"></arrow.to.target>deſcribet, ad ſuperficiem circularem, quam arcus <emph type="italics"></emph>ih<emph.end type="italics"></emph.end>convolutione <lb></lb>ſemicirculi <emph type="italics"></emph>akb<emph.end type="italics"></emph.end>circa diametrum <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>deſcribet. </s> <s>Et vires, quibus <lb></lb>hæ ſuperficies ſecundum lineas ad ſe tendentes attrahunt corpuſcu<lb></lb>la <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>p,<emph.end type="italics"></emph.end>ſunt (per Hypotheſin) ut ipſæ ſuperficies applicatæ <lb></lb>ad quadrata diſtantiarum ſuarum a corporibus, hoc eſt, ut <emph type="italics"></emph>pfXps<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>PFXPS.<emph.end type="italics"></emph.end>Suntque hæ vires ad ipſarum partes obliquas <lb></lb>quæ (facta per Legum Corol. </s> <s>2. reſolutione virium) ſecundum <lb></lb>lineas <emph type="italics"></emph>PS, ps<emph.end type="italics"></emph.end>ad centra tendunt, ut <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pi<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="pq;">pque</expan><emph.end type="italics"></emph.end>id <lb></lb>eſt (ob ſimilia triangula <emph type="italics"></emph>PIQ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PSF, piq<emph.end type="italics"></emph.end>& <emph type="italics"></emph>psf<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>PF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ps<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>pf.<emph.end type="italics"></emph.end>Unde, ex æquo, fit attractio corpuſculi hujus <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>verſus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ad attractionem corpuſculi <emph type="italics"></emph>p<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>s,<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>PFXpfXps/PS<emph.end type="italics"></emph.end>) ad <lb></lb>(<emph type="italics"></emph>pfXPFXPS/ps<emph.end type="italics"></emph.end>), hoc eſt, ut <emph type="italics"></emph>ps quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PS quad.<emph.end type="italics"></emph.end>Et ſimili argu<lb></lb>mento vires, quibus ſuperficies convolutione arcuum <emph type="italics"></emph>KL, kl<emph.end type="italics"></emph.end>de<lb></lb>ſcriptæ trahunt corpuſcula, erunt ut <emph type="italics"></emph>ps quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PS quad.<emph.end type="italics"></emph.end>; inque <lb></lb>eadem ratione erunt vires ſuperficierum omnium circularium in quas <lb></lb>utraque ſuperficies Sphærica, capiendo ſemper <emph type="italics"></emph>sd<emph.end type="italics"></emph.end>æqualem <emph type="italics"></emph>SD<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>se<emph.end type="italics"></emph.end>æqualem <emph type="italics"></emph>SE,<emph.end type="italics"></emph.end>diſtingui poteſt. </s> <s>Et, per compoſitionem, vires <lb></lb>totarum ſuperficierum Sphæricarum in corpuſcula exercitæ erunt <lb></lb>in eadem ratione. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note151"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXII. THEOREMA XXXII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ad Sphæræ cujuſvis puncta ſingula tendant vires æquales cen<lb></lb>tripetæ decreſcentes in duplicata ratione diſtantiarum a punctis, <lb></lb>ac detur tum Sphæræ denſitas, tum ratio diametri Sphæræ ad <lb></lb>diſtantiam corpuſculi a centro ejus; dico quod vis qua corpuſ<lb></lb>culum attrahitur proportionalis erit ſemidiametro Sphæræ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam concipe corpuſcula duo ſeorſim a Sphæris duabus attrahi, <lb></lb>unum ab una & alterum ab altera, & diſtantias eorum a Sphæra<lb></lb>rum centris proportionales eſſe diametris Sphærarum reſpective, <lb></lb>Sphæras autem reſolvi in particulas ſimiles & ſimiliter poſitas ad <lb></lb>corpuſcula. </s> <s>Et attractiones corpuſculi unius, factæ verſus ſingulas <lb></lb>particulas Sphæræ unius, erunt ad attractiones alterius verſus ana<lb></lb>logas totidem particulas Sphæræ alterius, in ratione compoſita ex <lb></lb>ratione particularum directe & ratione duplicata diſtantiarum in-<pb xlink:href="039/01/204.jpg" pagenum="176"></pb><arrow.to.target n="note152"></arrow.to.target>verſe. </s> <s>Sed particulæ ſunt ut Sphæræ, hoc eſt, in ratione triplicata <lb></lb>diametrorum, & diſtantiæ ſunt ut diametri, & ratio prior directe <lb></lb>una cum ratione poſteriore bis inverſe eſt ratio diametri ad diame<lb></lb>trum. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note152"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi corpuſcula in Circulis, circa Sphæras ex materia <lb></lb>æqualiter attractiva conſtantes, revolvantur; ſintQ.E.D.ſtantiæ a cen<lb></lb>tris Sphærarum proportionales earundem diametris: Tempora peri<lb></lb>odica erunt æqualia. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et vice verſa, ſi Tempora periodica ſunt æqualia; <lb></lb>diſtantiæ erunt proportionales diametris. </s> <s>Conſtant hæc duo per <lb></lb>Corol. </s> <s>3. Prop. </s> <s>IV. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si ad Solidorum durorum quorumvis ſimilium & æquali<lb></lb>ter denſorum puncta ſingula tendant vires æquales centripetæ de<lb></lb>creſcentes in duplicata ratione diſtantiarum a punctis: vires qui<lb></lb>bus corpuſcula, ad Solida illa duo ſimiliter ſita, attrahentur ab iiſ<lb></lb>dem, erunt ad invicem ut diametri Solidorum. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXIII. THEOREMA XXXIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ad Sphæræ alicujus datæ puncta ſingula tendant æquales vires <lb></lb>centripetæ decreſcentes in duplicata ratione diſtantiarum a pun<lb></lb>ctis: dico quod corpuſculum intra Sphæram conſtitutum attra<lb></lb>bitur vi proportionali diſtantiæ ſuæ ab ipſius centro.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>In Sphæra <emph type="italics"></emph>ABCD,<emph.end type="italics"></emph.end>centro <emph type="italics"></emph>S<emph.end type="italics"></emph.end>deſcripta, <lb></lb><figure id="id.039.01.204.1.jpg" xlink:href="039/01/204/1.jpg"></figure><lb></lb>locetur corpuſculum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>; & centro eodem <emph type="italics"></emph>S,<emph.end type="italics"></emph.end><lb></lb>intervallo <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>concipe Sphæram interiorem <lb></lb><emph type="italics"></emph>PEQF<emph.end type="italics"></emph.end>deſcribi. </s> <s>Manifeſtum eſt, per Prop. </s> <s><lb></lb>LXX, quod Sphæricæ ſuperficies concentri<lb></lb>cæ ex quibus Sphærarum differentia <emph type="italics"></emph>AEBF<emph.end type="italics"></emph.end><lb></lb>componitur, attractionibus per attractiones <lb></lb>contrarias deſtructis, nil agunt in corpus <lb></lb><emph type="italics"></emph>P.<emph.end type="italics"></emph.end>Reſtat ſola attractio Sphæræ interioris <lb></lb><emph type="italics"></emph>PEQF.<emph.end type="italics"></emph.end>Et per Prop. </s> <s>LXXII, hæc eſt ut <lb></lb>diſtantia <emph type="italics"></emph>PS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Superficies ex quibus ſolida componuntur, hic non ſunt pure <lb></lb>Mathematicæ, ſed Orbes adeo tenues ut eorum craſſitudo inſtar <pb xlink:href="039/01/205.jpg" pagenum="177"></pb>nihili ſit; nimirum Orbes evaneſcentes ex quibus Sphæra ultimo <lb></lb><arrow.to.target n="note153"></arrow.to.target>conſtat, ubi Orbium illorum numerus augetur & craſſitudo minui<lb></lb>tur in infinitum. </s> <s>Similiter per Puncta, ex quibus lineæ, ſuperficies <lb></lb>& ſolida componi dicuntur, intelligendæ ſunt particulæ æquales <lb></lb>magnitudinis contemnendæ. </s></p> <p type="margin"> <s><margin.target id="note153"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXIV. THEOREMA XXXIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, dico quod corpuſculum extra Sphæram conſtitutum <lb></lb>attrabitur vi reciproce proportionali quadrato diſtantiæ ſuæ ab <lb></lb>ipſius centro.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam diſtinguatur Sphæra in ſuperficies Sphæricas innumeras <lb></lb>concentricas, & attractiones corpuſculi a ſingulis ſuperficiebus <lb></lb>oriundæ erunt reciproce proportionales quadrato diſtantiæ cor<lb></lb>puſculi a centro, per Prop. </s> <s>LXXI. </s> <s>Et componendo, fiet ſum<lb></lb>ma attractionum, hoc eſt attractio corpuſculi in Sphæram totam, in <lb></lb>eadem ratione. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc in æqualibus diſtantiis a centris homogenearum <lb></lb>Sphærarum, attractiones ſunt ut Sphæræ. </s> <s>Nam per Prop. </s> <s>LXXII, <lb></lb>ſi diſtantiæ ſunt proportionales diametris Sphærarum, vires erunt <lb></lb>ut diametri. </s> <s>Minuatur diſtantia major in illa ratione; &, diſtan<lb></lb>tiis jam factis æqualibus, augebitur attractio in duplicata illa ratio<lb></lb>ne, adeoque erit ad attractionem alteram in triplicata illa ratione, <lb></lb>hoc eſt, in ratione Sphærarum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. In diſtantiis quibuſvis attractiones ſunt ut Sphæræ ap<lb></lb>plicatæ ad quadrata diſtantiarum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si corpuſculum, extra Sphæram homogeneam poſitum, <lb></lb>trahitur vi reciproce proportionali quadrato diſtantiæ ſuæ ab ipſius <lb></lb>centro, conſtet autem Sphæra ex particulis attractivis; decreſcet vis <lb></lb>particulæ cujuſQ.E.I. duplicata ratione diſtantiæ a particula. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXV. THEOREMA XXXV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ad Sphæræ datæ puncta ſingula tendant vires æquales centripe<lb></lb>tæ, decreſcentes in duplicata ratione diſtantiarum a punctis; dico <lb></lb>quod Sphæra quævis alia ſimilaris ab eadem attrahitur vi reci<lb></lb>proce proportionali quadrato diſtantiæ centrorum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam particulæ cujuſvis attractio eſt reciproce ut quadratum di<lb></lb>ſtantiæ ſuæ a centro Sphæræ trahentis, (per Prop. </s> <s>LXXIV) & prop-<pb xlink:href="039/01/206.jpg" pagenum="178"></pb><arrow.to.target n="note154"></arrow.to.target>terea eadem eſt ac ſi vis tota attrahens manaret de corpuſculo uNI<lb></lb>co ſito in centro hujus Sphæræ. </s> <s>Hæc autem attractio tanta eſt <lb></lb>quanta foret viciſſim attractio corpuſculi ejuſdem, ſi modo illud a <lb></lb>ſingulis Sphæræ attractæ particulis eadem vi traheretur qua ipſas <lb></lb>attrahit. </s> <s>Foret autem illa corpuſculi attractio (per Prop. </s> <s>LXXIV) <lb></lb>reciproce proportionalis quadrato diſtantiæ ſuæ a centro Sphæ<lb></lb>ræ; adeoque huic æqualis attractio Sphæræ eſt in eadem ratio<lb></lb>ne. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note154"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Attractiones Sphærarum, verſus alias Sphæras homoge<lb></lb>neas, ſunt ut Sphæræ trahentes applicatæ ad quadrata diſtantiarum <lb></lb>centrorum ſuorum a centris earum quas attrahunt. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Idem valet ubi Sphæra attracta etiam attrahit. </s> <s>Nam<lb></lb>que hujus puncta ſingula trahent ſingula alterius, eadem vi qua ab <lb></lb>ipſis viciſſim trahuntur, adeoque cum in omni attractione urgea<lb></lb>tur (per Legem III) tam punctum attrahens, quam punctum at<lb></lb>tractum, geminabitur vis attractionis mutuæ, conſervatis propor<lb></lb>tionibus. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Eadem omnia, quæ ſuperius de motu corporum circa <lb></lb>umbilicum Conicarum Sectionum demonſtrata ſunt, obtinent ubi <lb></lb>Sphæra attrahens locatur in umbilico & corpora moventur extra <lb></lb>Sphæram. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Ea vero quæ de motu corporum circa centrum Co<lb></lb>nicarum Sectionum demonſtrantur, obtinent ubi motus peraguntur <lb></lb>intra Sphæram. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXVI. THEOREMA XXXVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Sphæræ in progreſſu a centro ad circumferentiam (quoad mate<lb></lb>riæ denſitatem & vim attractivam) utcunQ.E.D.ſſimilares, in <lb></lb>progreſſu vero per circuitum ad datam omnem a centro diſtan<lb></lb>tiam ſunt undique ſimilares, & vis attractiva puncti cujuſque <lb></lb>decreſcit in duplicata ratione diſtantiæ corporis attracti: dico <lb></lb>quod vis tota qua hujuſmodi Sphæra una attrahit aliam ſit reci<lb></lb>proce proportionalis quadrato diſtantiæ centrorum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sunto Sphæræ quotcunque concentricæ ſimilares <emph type="italics"></emph>AB, CD, EF,<emph.end type="italics"></emph.end><lb></lb>&c. </s> <s>quarum interiores additæ exterioribus componant materiam <pb xlink:href="039/01/207.jpg" pagenum="179"></pb>denſiorem verſus centrum, vel ſubductæ relinquant tenuiorem; & <lb></lb><arrow.to.target n="note155"></arrow.to.target>hæ (per Prop. </s> <s>LXXV) trahent Sphæras alias quotcunque concentri<lb></lb>cas ſimilares <emph type="italics"></emph>GH, IK, LM,<emph.end type="italics"></emph.end>&c. </s> <s>ſingulæ ſingulas, viribus reci<lb></lb>proce proportionalibus quadrato diſtantiæ <emph type="italics"></emph>SP.<emph.end type="italics"></emph.end>Et componendo <lb></lb>vel dividendo, ſumma virium illarum omnium, vel exceſſus ali<lb></lb>quarum ſupra alias, hoc eſt, vis quas Sphæra tota ex concen<lb></lb>tricis quibuſcunque vel concentricarum differentiis compoſita <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end><lb></lb>trahit totam ex concentricis quibuſcunque vel concentricarum dif<lb></lb>ferentiis compoſitam <emph type="italics"></emph>GH,<emph.end type="italics"></emph.end>erit in eadem ratione. </s> <s>Augeatur nu<lb></lb>merus Sphærarum concentricarum in infinitum ſic, ut materiæ den<lb></lb>ſitas una cum vi attractiva, in progreſſu a circumferentia ad cen<lb></lb>trum, ſecundum Legem quamcunque creſcat vel decreſcat: &, ad<lb></lb><figure id="id.039.01.207.1.jpg" xlink:href="039/01/207/1.jpg"></figure><lb></lb>dita materia non attractiva, compleatur ubivis denſitas deficiens, eo <lb></lb>ut Sphæræ acquirant formam quamvis optatam; & vis qua harum <lb></lb>una attrahet alteram erit etiamnum (per argumentum ſuperius) in <lb></lb>eadem illa diſtantiæ quadratæ ratione inverſa. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note155"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi ejuſmodi Sphæræ complures, ſibi invicem per <lb></lb>omnia ſimiles, ſe mutuo trahant; attractiones acceleratrices ſingula<lb></lb>rum in ſingulas erunt, in æqualibus quibuſvis centrorum diſtantiis, <lb></lb>ut Sphæræ attrahentes. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. InQ.E.D.ſtantiis quibuſvis inæqualibus, ut Sphæræ attra<lb></lb>hentes applicatæ ad quadrata diſtantiarum inter centra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Attractiones vero motrices, ſeu pondera Sphærarum in <lb></lb>Sphæras erunt, in æqualibus centrorum diſtantiis, ut Sphæræ attra<lb></lb>hentes & attractæ conjunctim, id eſt, ut contenta ſub Sphæris per <lb></lb>multiplicationem producta. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. InQ.E.D.ſtantiis inæqualibus, ut contenta illa applicata <lb></lb>ad quadrata diſtantiarum inter centra. <pb xlink:href="039/01/208.jpg" pagenum="180"></pb><arrow.to.target n="note156"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note156"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Eadem valent ubi attractio oritur a Sphæræ utriuſque <lb></lb>virtute attractiva, mutuo exercita in Sphæram alteram. </s> <s>Nam viri<lb></lb>bus ambabus geminatur attractio, proportione ſervata. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Si hujuſmodi Sphæræ aliquæ circa alias quieſcentes re<lb></lb>volvantur, ſingulæ circa ſingulas, ſintQ.E.D.ſtantiæ inter centra re<lb></lb>volventium & quieſcentium proportionales quieſcentium diame<lb></lb>tris; æqualia erunt Tempora periodica. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Et viciſſim, ſi Tempora periodica ſunt æqualia; diſtan<lb></lb>tiæ erunt proportionales diametris. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Eadem omnia, quæ ſuperius de motu corporum circa <lb></lb>umbilicos Conicarum Sectionum demonſtrata ſunt, obtinent ubi <lb></lb>Sphæra attrahens, formæ & conditionis cujuſvis jam deſcriptæ, lo<lb></lb>catur in umbilico. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>9. Ut & ubi gyrantia ſunt etiam Sphæræ attrahentes, con<lb></lb>ditionis cujuſvis jam deſcriptæ. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXVII. THEOREMA XXXVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ad ſingula Sphærarum puncta tendant vires centripetæ, proper<lb></lb>tionales diſtantiis punctorum a corporibus attractis: dico quod <lb></lb>vis compoſita, qua Sphæræ duæ ſe mutuo trahent, est ut di<lb></lb>ſtantia inter centra Sphærarum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Sit <emph type="italics"></emph>AEBF<emph.end type="italics"></emph.end>Sphæra, <emph type="italics"></emph>S<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.208.1.jpg" xlink:href="039/01/208/1.jpg"></figure><lb></lb>centrum ejus, <emph type="italics"></emph>P<emph.end type="italics"></emph.end>corpuſculum at<lb></lb>tractum, <emph type="italics"></emph>PASB<emph.end type="italics"></emph.end>axis Sphæræ per <lb></lb>centrum corpuſculi tranſiens, <emph type="italics"></emph>EF, <lb></lb>ef<emph.end type="italics"></emph.end>plana duo quibus Sphæra ſe<lb></lb>catur, huic axi perpendicularia & <lb></lb>hinc inde æqualiter diſtantia a <lb></lb>centro Sphæræ; <emph type="italics"></emph>G, g<emph.end type="italics"></emph.end>interſectio<lb></lb>nes planorum & axis, & <emph type="italics"></emph>H<emph.end type="italics"></emph.end>pun<lb></lb>ctum quodvis in plano <emph type="italics"></emph>EF.<emph.end type="italics"></emph.end>Pun<lb></lb>cti <emph type="italics"></emph>H<emph.end type="italics"></emph.end>vis centripeta in corpuſculum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſecundum lineam <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>exer<lb></lb>cita, eſt ut diſtantia <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>; & (per Legum Corol. </s> <s>2.) ſecundum li<lb></lb>neam <emph type="italics"></emph>PG,<emph.end type="italics"></emph.end>ſeu verſus centrum <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>ut longitudo <emph type="italics"></emph>PG.<emph.end type="italics"></emph.end>Igitur pun<lb></lb>ctorum omnium in plano <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>hoc eſt plani totius vis, qua corpuſ<lb></lb>culum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>trahitur verſus centrum <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>eſt ut numerus punctorum <lb></lb>ductus in diſtantiam <emph type="italics"></emph>PG:<emph.end type="italics"></emph.end>id eſt, ut contentum ſub plano ipſo <emph type="italics"></emph>EF<emph.end type="italics"></emph.end><lb></lb>& diſtantia illa <emph type="italics"></emph>PG.<emph.end type="italics"></emph.end>Et ſimiliter vis plani <emph type="italics"></emph>ef,<emph.end type="italics"></emph.end>qua corpuſculum <emph type="italics"></emph>P<emph.end type="italics"></emph.end><pb xlink:href="039/01/209.jpg" pagenum="181"></pb>trahitur verſus centrum <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>eſt ut planum illud ductum in diſtantiam </s></p> <p type="main"> <s><arrow.to.target n="note157"></arrow.to.target>ſuam <emph type="italics"></emph>Pg,<emph.end type="italics"></emph.end>ſive ut huic æquale planum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ductum in diſtantiam <lb></lb>illam <emph type="italics"></emph>Pg<emph.end type="italics"></emph.end>; & ſumma virium plani utriuſque ut planum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>duc<lb></lb>tum in ſummam diſtantiarum <emph type="italics"></emph>PG+Pg,<emph.end type="italics"></emph.end>id eſt, ut planum illud <lb></lb>ductum in duplam centri & corpuſculi diſtantiam <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end>hoc eſt, ut <lb></lb>duplum planum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ductum in diſtantiam <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end>vel ut ſumma æ<lb></lb>qualium planorum <emph type="italics"></emph>EF+ef<emph.end type="italics"></emph.end>ducta in diſtantiam eandem. </s> <s>Et ſi<lb></lb>mili argumento, vires omnium planorum in Sphæra tota, hinc in<lb></lb>de æqualiter a centro Sphæræ diſtantium, ſunt ut ſumma planorum <lb></lb>ducta in diſtantiam <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end>hoc eſt, ut Sphæra tota ducta in diſtan<lb></lb>tiam centri ſui <emph type="italics"></emph>S<emph.end type="italics"></emph.end>a corpuſculo <emph type="italics"></emph>P. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note157"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Trahat jam corpuſculum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>Sphæram <emph type="italics"></emph>AEBF.<emph.end type="italics"></emph.end>Et eo<lb></lb>dem argumento probabitur quod vis, qua Sphæra illa trahitur, erit: <lb></lb>ut diſtantia <emph type="italics"></emph>PS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Componatur jam Sphæra altera ex corpuſculis innume<lb></lb>ris <emph type="italics"></emph>P<emph.end type="italics"></emph.end>; & quoniam vis; qua corpuſculum unumquodque trahitur, <lb></lb>eſt ut diſtantia corpuſculi a centro Sphæræ primæ ducta in Sphæ<lb></lb>ram eandem, atque adeo eadem eſt ac ſi prodiret tota de corpuſ<lb></lb>culo unico in centro Sphæræ; vis tota qua corpuſcula omnia in <lb></lb>Sphæra ſecunda trahuntur, hoc eſt, qua Sphæra illa tota trahitur, <lb></lb>eadem erit ac ſi Sphæra illa traheretur vi prodeunte de corpuſculo <lb></lb>unico in centro Sphæræ primæ, & propterea proportionalis eſt di<lb></lb>ſtantiæ inter centra Sphærarum. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>4. Trahant Sphæræ ſe mutuo, & vis geminata proportio<lb></lb>nem priorem ſervabit. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>5. Locetur jam corpuſculum <emph type="italics"></emph>p<emph.end type="italics"></emph.end>intra Sphæram <emph type="italics"></emph>AEBF<emph.end type="italics"></emph.end>; & <lb></lb>quoniam vis plani <emph type="italics"></emph>ef<emph.end type="italics"></emph.end>in corpuſculum eſt ut contentum ſub plano <lb></lb>illo & diſtantia <emph type="italics"></emph>pg<emph.end type="italics"></emph.end>; & vis contraria plani <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ut contentum ſub <lb></lb>plano illo & diſtantia <emph type="italics"></emph>pG<emph.end type="italics"></emph.end>; erit vis ex utraque compoſita ut diffe<lb></lb>rentia contentorum, hoc eſt, ut ſumma æqualium planorum ducta <lb></lb>in ſemiſſem differentiæ diſtantiarum, id eſt, ut ſumma illa ducta in <lb></lb><emph type="italics"></emph>pS<emph.end type="italics"></emph.end>diſtantiam corpuſculi a centro Sphæræ. </s> <s>Et ſimili argumento, <lb></lb>attractio planorum omnium <emph type="italics"></emph>EF, ef<emph.end type="italics"></emph.end>in Sphæra tota, hoc eſt, at<lb></lb>tractio Sphæræ totius, eſt ut ſumma planorum omnium, ſeu Sphæra <lb></lb>tota, ducta in <emph type="italics"></emph>pS<emph.end type="italics"></emph.end>diſtantiam corpuſculi a centro Sphæræ. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>6. Et ſi ex corpuſculis innumeris <emph type="italics"></emph>p<emph.end type="italics"></emph.end>componatur Sphæra <lb></lb>nova, intra Sphæram priorem <emph type="italics"></emph>AEBF<emph.end type="italics"></emph.end>ſita; probabitur ut prius <lb></lb>quod attractio, ſive ſimplex Sphæræ unius in alteram, ſive mutua <lb></lb>utriuſQ.E.I. ſe invicem, erit ut diſtantia centrorum <emph type="italics"></emph>pS. Q.E.D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/210.jpg" pagenum="182"></pb><arrow.to.target n="note158"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note158"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXVIII. THEOREMA XXXVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Sphæræ in progreſſu a centro ad circumferentiam ſint utcunque <lb></lb>diſſimilares & inæquabiles, in progreſſu vero per circuitum ad <lb></lb>datam omnem a centro diſtantiam ſint undique ſimilares; & <lb></lb>vis attractiva puncti cujuſque ſit ut diſtantia corporis attracti: <lb></lb>dico quod vis tota qua hujuſmodi Sphæræ duæ ſe mutuo trahunt <lb></lb>ſit proportionalis diſtantiæ inter centra Sphærarum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Demonſtratur ex Propoſitione præcedente, eodem modo quo <lb></lb>Propoſitio LXXVI ex Propoſitione LXXV demonſtrata fuit. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Quæ ſuperius in Propoſitionibus X & LXIV de motu <lb></lb>corporum circa centra Conicarum Sectionum demonſtrata ſunt, <lb></lb>valent ubi attractiones omnes fiunt vi Corporum Sphærieorum <lb></lb>conditionis jam deſcriptæ, ſuntque corpora attracta Sphæræ con<lb></lb>ditionis ejuſdem. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Attractionum Caſus duos inſigniores jam dedi expoſitos; nimi<lb></lb>rum ubi Vires centripetæ decreſcunt in duplicata diſtantiarum ra<lb></lb>tione, vel creſcunt in diſtantiarum ratione ſimplici; efficientes <lb></lb>in utroque Caſu ut corpora gyrentur in Conicis Sectionibus, & <lb></lb>componentes corporum Sphærieorum Vires centripetas eadem Lege, <lb></lb>in receſſu a centro, decreſcentes vel creſcentes cum ſeipſis: Quod <lb></lb>eſt notatu dignum. </s> <s>Caſus cæteros, qui concluſiones minus ele<lb></lb>gantes exhibent, ſigillatim percurrere longum eſſet. </s> <s>Malim <lb></lb>cunctos methodo generali ſimul comprehendere ac determinare, <lb></lb>ut ſequitur. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XXIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si deſcribantur centro<emph.end type="italics"></emph.end>S <emph type="italics"></emph>circulus quilibet<emph.end type="italics"></emph.end>AEB, <emph type="italics"></emph>& centro<emph.end type="italics"></emph.end>P <emph type="italics"></emph>cir<lb></lb>culi duo<emph.end type="italics"></emph.end>EF, ef, <emph type="italics"></emph>ſecantes priorem in<emph.end type="italics"></emph.end>E, e, <emph type="italics"></emph>lineamque<emph.end type="italics"></emph.end>PS <emph type="italics"></emph>in<emph.end type="italics"></emph.end><lb></lb>F, f; <emph type="italics"></emph>& ad<emph.end type="italics"></emph.end>PS <emph type="italics"></emph>demittantur perpendicula<emph.end type="italics"></emph.end>ED, ed: <emph type="italics"></emph>dico quod, <lb></lb>fi diſtantia arcuum<emph.end type="italics"></emph.end>EF, ef <emph type="italics"></emph>in infinitum minui intelligatur, ra<lb></lb>tio ultima lineæ evaneſcentis<emph.end type="italics"></emph.end>Dd <emph type="italics"></emph>ad lineam evaneſcentem<emph.end type="italics"></emph.end>Ff <lb></lb><emph type="italics"></emph>ea ſit, quæ lineæ<emph.end type="italics"></emph.end>PE <emph type="italics"></emph>ad lineam<emph.end type="italics"></emph.end>PS. </s></p><pb xlink:href="039/01/211.jpg" pagenum="183"></pb> <p type="main"> <s>Nam ſi linea <emph type="italics"></emph>Pe<emph.end type="italics"></emph.end>ſecet arcum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>in <emph type="italics"></emph>q<emph.end type="italics"></emph.end>; & recta <emph type="italics"></emph>Ee,<emph.end type="italics"></emph.end>quæ cum <lb></lb><arrow.to.target n="note159"></arrow.to.target>arcu evaneſcente <emph type="italics"></emph>Ee<emph.end type="italics"></emph.end>coincidit, producta occurrat rectæ <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>; <lb></lb>& ab <emph type="italics"></emph>S<emph.end type="italics"></emph.end>demittatur in <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>normalis <emph type="italics"></emph>SG:<emph.end type="italics"></emph.end>ob ſimilia triangula <lb></lb><emph type="italics"></emph>DTE, dTe, DES<emph.end type="italics"></emph.end>; erit <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ee,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TE,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>ad <lb></lb><figure id="id.039.01.211.1.jpg" xlink:href="039/01/211/1.jpg"></figure><lb></lb><emph type="italics"></emph>ES<emph.end type="italics"></emph.end>; & ob triangula <emph type="italics"></emph>Eeq, ESG<emph.end type="italics"></emph.end>(per Lem. </s> <s>VIII, & Corol. </s> <s>3. <lb></lb>Lem. </s> <s>VII) ſimilia, erit <emph type="italics"></emph>Ee<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>eq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>Ff,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>ES<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SG<emph.end type="italics"></emph.end>; & ex <lb></lb>æquo, <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SG<emph.end type="italics"></emph.end>; hoc eſt (ob ſimilia triangula <lb></lb><emph type="italics"></emph>PDE, PGS<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note159"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXIX. THEOREMA XXXIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ſuperficies ob latitudinem infinite diminutam jamjam evaneſcens<emph.end type="italics"></emph.end><lb></lb>EF fe, <emph type="italics"></emph>convolutione ſui circa axem<emph.end type="italics"></emph.end>PS, <emph type="italics"></emph>deſcribat ſolidum <lb></lb>Sphæricum concavo convexum, ad cujus particulas ſingulas æqua<lb></lb>les tendant æquales vires centripetæ: dico quod Vis, qua ſoli<lb></lb>dum illud trahit corpuſculum ſitum in<emph.end type="italics"></emph.end>P, <emph type="italics"></emph>est in ratione compo<lb></lb>ta ex ratione ſolidi<emph.end type="italics"></emph.end>DE<emph type="italics"></emph>q<emph.end type="italics"></emph.end>XFf <emph type="italics"></emph>& ratione vis qua particula <lb></lb>data in loco<emph.end type="italics"></emph.end>Ff <emph type="italics"></emph>traheret idem corpuſculum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi primo conſideremus vim ſuperficiei Sphæricæ <emph type="italics"></emph>FE,<emph.end type="italics"></emph.end>quæ <lb></lb>convolutione arcus <emph type="italics"></emph>FE<emph.end type="italics"></emph.end>generatur, & a linea <emph type="italics"></emph>de<emph.end type="italics"></emph.end>ubivis ſecatur in <emph type="italics"></emph>r<emph.end type="italics"></emph.end>; <lb></lb>erit ſuperficiei pars annularis, convolutione arcus <emph type="italics"></emph>rE<emph.end type="italics"></emph.end>genita, ut <lb></lb>lineola <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>manente Sphæræ radio <emph type="italics"></emph>PE,<emph.end type="italics"></emph.end>(uti demonſtravit <emph type="italics"></emph>Ar<lb></lb>chimedes<emph.end type="italics"></emph.end>in Lib. </s> <s>de <emph type="italics"></emph>Sphæra<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Cylindro.<emph.end type="italics"></emph.end>) Et hujus vis ſecundum li<lb></lb>neas <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Pr<emph.end type="italics"></emph.end>undiQ.E.I. ſuperficie conica ſitas exercita, ut <lb></lb>hæc ipſa ſuperficiei pars annularis; hoc eſt, ut lineola <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>vel, <lb></lb>quod perinde eſt, ut rectangulum ſub dato Sphæræ radio <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>& <pb xlink:href="039/01/212.jpg" pagenum="184"></pb><arrow.to.target n="note160"></arrow.to.target>lineola illa <emph type="italics"></emph>Dd:<emph.end type="italics"></emph.end>at ſecundum lineam <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>tendentem <lb></lb>minor, in ratione <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PE,<emph.end type="italics"></emph.end>adeoque ut <emph type="italics"></emph>PDXDd.<emph.end type="italics"></emph.end>Dividi <lb></lb>jam intelligatur linea <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>in particulas innumeras æquales, quæ <lb></lb>ſingulæ nominentur <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>; & ſuperficies <emph type="italics"></emph>FE<emph.end type="italics"></emph.end>dividetur in totidem <lb></lb>æquales annulos, quorum vires erunt ut ſumma omnium <emph type="italics"></emph>PDXDd,<emph.end type="italics"></emph.end><lb></lb>hoc eſt, ut 1/2 <emph type="italics"></emph>PFq<emph.end type="italics"></emph.end>-1/2<emph type="italics"></emph>PDq,<emph.end type="italics"></emph.end>adeoque ut <emph type="italics"></emph>DE quad.<emph.end type="italics"></emph.end>Ducatur <lb></lb><figure id="id.039.01.212.1.jpg" xlink:href="039/01/212/1.jpg"></figure><lb></lb>jam ſuperficies <emph type="italics"></emph>FE<emph.end type="italics"></emph.end>in altitudinem <emph type="italics"></emph>Ef<emph.end type="italics"></emph.end>; & fiet ſolidi <emph type="italics"></emph>EFfe<emph.end type="italics"></emph.end>vis ex<lb></lb>ercita in corpuſculum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DEqXFf:<emph.end type="italics"></emph.end>puta ſi detur vis quam <lb></lb>particula aliqua data <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end>in diſtantia <emph type="italics"></emph>PF<emph.end type="italics"></emph.end>exercet in corpuſculum <lb></lb><emph type="italics"></emph>P.<emph.end type="italics"></emph.end>At ſi vis illa non detur, fiet vis ſolidi <emph type="italics"></emph>EFfe<emph.end type="italics"></emph.end>ut ſolidum <lb></lb><emph type="italics"></emph>DEqXFf<emph.end type="italics"></emph.end>& vis illa non data conjunctim. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note160"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXX. THEOREMA XL.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ad Sphæræ alicujus<emph.end type="italics"></emph.end>ABE, <emph type="italics"></emph>centro<emph.end type="italics"></emph.end>S <emph type="italics"></emph>deſcriptæ, particulas ſingu<lb></lb>las æquales tendant æquales vires centripetæ, & ad Sphæræ <lb></lb>axem<emph.end type="italics"></emph.end>AB, <emph type="italics"></emph>in quo corpuſculum aliquod<emph.end type="italics"></emph.end>P <emph type="italics"></emph>locatur, erigantur de <lb></lb>punctis ſingulis<emph.end type="italics"></emph.end>D <emph type="italics"></emph>perpendicula<emph.end type="italics"></emph.end>DE, <emph type="italics"></emph>Sphæræ occurrentia in<emph.end type="italics"></emph.end>E, <lb></lb><emph type="italics"></emph>& in ipſis capiantur longitudines<emph.end type="italics"></emph.end>DN, <emph type="italics"></emph>quæ ſint ut quantitas<emph.end type="italics"></emph.end><lb></lb>(DE<emph type="italics"></emph>q<emph.end type="italics"></emph.end>XPS/PE) <emph type="italics"></emph>& vis quam Sphæræ particula ſita in axe ad di<lb></lb>ſtantiam<emph.end type="italics"></emph.end>PE <emph type="italics"></emph>exercet in corpuſculum<emph.end type="italics"></emph.end>P <emph type="italics"></emph>conjunctim: dico quod <lb></lb>Vis tota, qua corpuſculum<emph.end type="italics"></emph.end>P <emph type="italics"></emph>trahitur verſus Sphæram, est ut <lb></lb>area comprehenſa ſub axe Sphæræ<emph.end type="italics"></emph.end>AB <emph type="italics"></emph>& linea curva<emph.end type="italics"></emph.end>ANB, <lb></lb><emph type="italics"></emph>quam punctum<emph.end type="italics"></emph.end>N <emph type="italics"></emph>perpetuo tangit.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/213.jpg" pagenum="185"></pb> <p type="main"> <s>Etenim ſtantibus quæ in Lemmate & Theoremate noviſſimo <lb></lb><arrow.to.target n="note161"></arrow.to.target>conſtructa ſunt, concipe axem Sphæræ <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>dividi in particulas <lb></lb>innumeras æquales <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>& Sphæram totam dividi in totidem <lb></lb>laminas Sphæricas concavo-convexas <emph type="italics"></emph>EFfe<emph.end type="italics"></emph.end>; & erigatur perpen<lb></lb>diculum <emph type="italics"></emph>dn.<emph.end type="italics"></emph.end>Per Theorema ſuperius, vis qua lamina <emph type="italics"></emph>EFfe<emph.end type="italics"></emph.end><lb></lb>trahit corpuſculum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>eſt ut <emph type="italics"></emph>DEqXFf<emph.end type="italics"></emph.end>& vis particulæ unius ad <lb></lb>diſtantiam <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>PF<emph.end type="italics"></emph.end>exercita conjunctim. </s> <s>Eſt autem per Lem<lb></lb>ma noviſſimum, <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end>& inde <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end>æqualis <lb></lb>(<emph type="italics"></emph>PSXDd/PE<emph.end type="italics"></emph.end>); & <emph type="italics"></emph>DEqXFf<emph.end type="italics"></emph.end>æquale <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>in (<emph type="italics"></emph>DEqXPS/PE<emph.end type="italics"></emph.end>), & propter<lb></lb>ea vis laminæ <emph type="italics"></emph>EFfe<emph.end type="italics"></emph.end>eſt ut <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>in (<emph type="italics"></emph>DEqXPS/PE<emph.end type="italics"></emph.end>) & vis particulæ ad <lb></lb>diſtantiam <emph type="italics"></emph>PF<emph.end type="italics"></emph.end>exercita conjunctim, hoc eſt (ex Hypotheſi) ut <lb></lb><emph type="italics"></emph>DNXDd,<emph.end type="italics"></emph.end>ſeu area evaneſcens <emph type="italics"></emph>DNnd.<emph.end type="italics"></emph.end>Sunt igitur laminarum <lb></lb>omnium vires in corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>exercitæ, ut areæ omnes <emph type="italics"></emph>DNnd,<emph.end type="italics"></emph.end>hoc <lb></lb>eſt, Sphæræ vis tota ut area tota <emph type="italics"></emph>ABNA. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note161"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi vis centripeta, ad particulas ſingulas tendens, <lb></lb>eadem ſemper maneat in omnibus diſtantiis, & fiat <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ut <lb></lb>(<emph type="italics"></emph>DEqXPS/PE<emph.end type="italics"></emph.end>): erit vis tota qua corpuſculum a Sphæra attrahitur, <lb></lb>ut area <emph type="italics"></emph>ABNA.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si particularum vis centripeta ſit reciproce ut diſtantia <lb></lb>corpuſculi a ſe attracti, & fiat <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>DEqXPS/PEq<emph.end type="italics"></emph.end>): erit vis qua <lb></lb>corpuſculum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>a Sphæra tota attrahitur ut area <emph type="italics"></emph>ABNA.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si particularum vis centripeta ſit reciproce ut cubus di<lb></lb>ſtantiæ corpuſculi a ſe attracti, & fiat <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>DEqXPS/PEqq<emph.end type="italics"></emph.end>): erit <lb></lb>vis qua corpuſculum a tota Sphæra attrahitur ut area <emph type="italics"></emph>ABNA.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Et univerſaliter ſi vis centripeta ad ſingulas Sphæræ <lb></lb>particulas tendens ponatur eſſe reciproce ut quantitas V, fiat au<lb></lb>tem <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>DEqXPS/PEXV<emph.end type="italics"></emph.end>); erit vis qua corpuſculum a Sphæra tota <lb></lb>attrahitur ut area <emph type="italics"></emph>ABNA.<emph.end type="italics"></emph.end><pb xlink:href="039/01/214.jpg" pagenum="186"></pb><arrow.to.target n="note162"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note162"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXI. PROBLEMA XLI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Stantibus jam poſitis, menſuranda est Area<emph.end type="italics"></emph.end>ABNA.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>A puncto <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ducatur recta <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>Sphæram tangens in <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>& ad <lb></lb>axem <emph type="italics"></emph>PAB<emph.end type="italics"></emph.end>demiſſa normali <emph type="italics"></emph>HI,<emph.end type="italics"></emph.end>biſecetur <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>in <emph type="italics"></emph>L;<emph.end type="italics"></emph.end>& erit <lb></lb>(per Prop. </s> <s>12, Lib. </s> <s>2. Elem.) <emph type="italics"></emph>PEq<emph.end type="italics"></emph.end>æquale <emph type="italics"></emph>PSq + SEq<emph.end type="italics"></emph.end>+ <lb></lb>2<emph type="italics"></emph>PSD.<emph.end type="italics"></emph.end>Eſt autem <emph type="italics"></emph>SEq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>SHq<emph.end type="italics"></emph.end>(ob ſimilitudinem triangu<lb></lb>lorum <emph type="italics"></emph>SPH, SHI<emph.end type="italics"></emph.end>) æquale rectangulo <emph type="italics"></emph>PSI.<emph.end type="italics"></emph.end>Ergo <emph type="italics"></emph>PEq<emph.end type="italics"></emph.end>æquale <lb></lb>eſt contento ſub <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PS+SI<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>SD,<emph.end type="italics"></emph.end>hoc eſt, ſub <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>& <lb></lb>2<emph type="italics"></emph>LS<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>SD,<emph.end type="italics"></emph.end>id eſt, ſub <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>& 2<emph type="italics"></emph>LD.<emph.end type="italics"></emph.end>Porro <emph type="italics"></emph>DE quad<emph.end type="italics"></emph.end>æquale <lb></lb>eſt <emph type="italics"></emph>SEq-SDq,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>SEq -LSq<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>SLD-LDq,<emph.end type="italics"></emph.end>id eſt, <lb></lb>2<emph type="italics"></emph>SLD-LDq-ALB.<emph.end type="italics"></emph.end>Nam <emph type="italics"></emph>LSq-SEq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>LSq-SAq<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.214.1.jpg" xlink:href="039/01/214/1.jpg"></figure><lb></lb>(per Prop. </s> <s>6, Lib. </s> <s>2. Elem.) æquatur rectangulo <emph type="italics"></emph>ALB.<emph.end type="italics"></emph.end>Scriba<lb></lb>tur itaque 2<emph type="italics"></emph>SLD -LDq -ALB<emph.end type="italics"></emph.end>pro <emph type="italics"></emph>DEq<emph.end type="italics"></emph.end>; & quantitas <lb></lb>(<emph type="italics"></emph>DEqXPS/PEXV<emph.end type="italics"></emph.end>), quæ ſecundum Corollarium quartum Propoſitionis <lb></lb>præcedentis eſt ut longitudo ordinatim applicatæ <emph type="italics"></emph>DN,<emph.end type="italics"></emph.end>reſolvet <lb></lb>ſeſe in tres partes (2<emph type="italics"></emph>SLDXPS/PE<emph.end type="italics"></emph.end>XV)-(<emph type="italics"></emph>LDqXPS/PE<emph.end type="italics"></emph.end>XV)-(<emph type="italics"></emph>ALBXPS/PE<emph.end type="italics"></emph.end>XV): <lb></lb>ubi ſi pro V ſcribatur ratio inverſa vis centripetæ, & pro <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>me<lb></lb>dium proportionale inter <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>& 2<emph type="italics"></emph>LD<emph.end type="italics"></emph.end>; tres illæ partes evadent <lb></lb>ordinatim applicatæ linearum totidem curvarum, quarum areæ per <lb></lb>Methodos vulgatas innoteſcunt. <emph type="italics"></emph><expan abbr="q.">que</expan> E. F.<emph.end type="italics"></emph.end><pb xlink:href="039/01/215.jpg" pagenum="187"></pb><arrow.to.target n="note163"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note163"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>1. Si vis centripeta ad ſingulas Sphæræ particulas ten<lb></lb>dens ſit reciproce ut diſtantia; pro V ſcribe diſtantiam <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>; dein <lb></lb>2<emph type="italics"></emph>PSXLD<emph.end type="italics"></emph.end>pro <emph type="italics"></emph>PEq,<emph.end type="italics"></emph.end>& fiet <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SL-1/2LD-(ALB/2LD).<emph.end type="italics"></emph.end><lb></lb>Pone <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>æqualem duplo ejus 2<emph type="italics"></emph>SL-LD-(ALB/LD)<emph.end type="italics"></emph.end>: & ordinatæ <lb></lb>pars data 2<emph type="italics"></emph>SL<emph.end type="italics"></emph.end>ducta in longitudinem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>deſcribet aream rectan<lb></lb>gulam 2<emph type="italics"></emph>SLXAB<emph.end type="italics"></emph.end>; & pars indefinita <emph type="italics"></emph>LD<emph.end type="italics"></emph.end>ducta normaliter in <lb></lb>eandem longitudinem per motum continuum, ea lege ut inter mo<lb></lb>vendum creſcendo vel decreſcendo æquetur ſemper longitudini <lb></lb><emph type="italics"></emph>LD,<emph.end type="italics"></emph.end>deſcribet aream (<emph type="italics"></emph>LBq-LAq<emph.end type="italics"></emph.end>/2), id eſt, aream <emph type="italics"></emph>SLXAB<emph.end type="italics"></emph.end>; quæ <lb></lb>ſubducta de area priore 2<emph type="italics"></emph>SLXAB<emph.end type="italics"></emph.end>relinquit aream <emph type="italics"></emph>SLXAB.<emph.end type="italics"></emph.end><lb></lb>Pars autem tertia (<emph type="italics"></emph>ALB/LD<emph.end type="italics"></emph.end>) ducta itidem per motum localem norma<lb></lb>liter in eandem longitudinem, deſcribet <lb></lb><figure id="id.039.01.215.1.jpg" xlink:href="039/01/215/1.jpg"></figure><lb></lb>aream Hyperbolicam; quæ ſubducta de <lb></lb>area <emph type="italics"></emph>SLXAB<emph.end type="italics"></emph.end>relinquet aream quæſitam <lb></lb><emph type="italics"></emph>ABNA.<emph.end type="italics"></emph.end>Unde talis emergit Proble<lb></lb>matis conſtructio. </s> <s>Ad puncta <emph type="italics"></emph>L, A, B<emph.end type="italics"></emph.end><lb></lb>erige perpendicula <emph type="italics"></emph>Ll, Aa, Bb,<emph.end type="italics"></emph.end>quorum <lb></lb><emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>LB,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Bb<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>LA<emph.end type="italics"></emph.end>æquetur. </s> <s><lb></lb>Aſymptotis <emph type="italics"></emph>Ll, LB,<emph.end type="italics"></emph.end>per puncta <emph type="italics"></emph>a, b<emph.end type="italics"></emph.end>de<lb></lb>ſcribatur Hyperbola <emph type="italics"></emph>ab.<emph.end type="italics"></emph.end>Et acta chor<lb></lb>da <emph type="italics"></emph>ba<emph.end type="italics"></emph.end>claudet aream <emph type="italics"></emph>aba<emph.end type="italics"></emph.end>areæ quæſitæ <lb></lb><emph type="italics"></emph>ABNA<emph.end type="italics"></emph.end>æqualem. </s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>2. Si vis centripeta ad ſingulas Sphæræ particulas ten<lb></lb>dens ſit reciproce ut cubus diſtantiæ, vel (quod perinde eſt) ut cubus <lb></lb>ille applicatus ad planum quodvis datum; ſcribe (<emph type="italics"></emph>PEcub/2ASq<emph.end type="italics"></emph.end>) pro V, <lb></lb>dein 2<emph type="italics"></emph>PSXLD<emph.end type="italics"></emph.end>pro <emph type="italics"></emph>PEq<emph.end type="italics"></emph.end>; & fiet <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>(SLXASq/PSXLD)-(ASq/2PS) <lb></lb>-(ALBXASq/2PSXLDq),<emph.end type="italics"></emph.end>id eſt (ob continue proportionales <emph type="italics"></emph>PS, AS, SI<emph.end type="italics"></emph.end>) <lb></lb>ut <emph type="italics"></emph>(LSI/LD)-1/2SI-(ALBXSI/2LDq).<emph.end type="italics"></emph.end>Si ducantur hujus partes tres <lb></lb>in longitudinem <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>prima (<emph type="italics"></emph>LSI/LD<emph.end type="italics"></emph.end>) generabit aream Hyper-</s></p><pb xlink:href="039/01/216.jpg" pagenum="188"></pb> <p type="main"> <s><arrow.to.target n="note164"></arrow.to.target>bolicam; ſecunda 1/2<emph type="italics"></emph>SI<emph.end type="italics"></emph.end>aream 1/2<emph type="italics"></emph>ABXSI<emph.end type="italics"></emph.end>; tertia (<emph type="italics"></emph>ALBXSI/2LDq<emph.end type="italics"></emph.end>) are<lb></lb>am <emph type="italics"></emph>(ALBXSI/2LA)-(ALBXSI/2LB),<emph.end type="italics"></emph.end>id eſt 1/2<emph type="italics"></emph>ABXSI.<emph.end type="italics"></emph.end>De prima ſub<lb></lb>ducatur ſumma ſecundæ & tertiæ, & <lb></lb><figure id="id.039.01.216.1.jpg" xlink:href="039/01/216/1.jpg"></figure><lb></lb>manebit area quæſita <emph type="italics"></emph>ABNA.<emph.end type="italics"></emph.end>Un<lb></lb>de talis emergit Problematis conſtru<lb></lb>ctio. </s> <s>Ad puncta <emph type="italics"></emph>L, A, S, B<emph.end type="italics"></emph.end>erige <lb></lb>perpendicula <emph type="italics"></emph>Ll, Aa, Ss, Bb,<emph.end type="italics"></emph.end>quo<lb></lb>rum <emph type="italics"></emph>Ss<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>SI<emph.end type="italics"></emph.end>æquetur, perque pun<lb></lb>ctum <emph type="italics"></emph>s<emph.end type="italics"></emph.end>Aſymptotis <emph type="italics"></emph>Ll, LB<emph.end type="italics"></emph.end>deſcri<lb></lb>batur Hyperbola <emph type="italics"></emph>asb<emph.end type="italics"></emph.end>occurrens per<lb></lb>pendiculis <emph type="italics"></emph>Aa, Bb<emph.end type="italics"></emph.end>in <emph type="italics"></emph>a<emph.end type="italics"></emph.end>& <emph type="italics"></emph>b<emph.end type="italics"></emph.end>; & rect<lb></lb>angulum 2<emph type="italics"></emph>ASI<emph.end type="italics"></emph.end>ſubductum de area <lb></lb>Hyperbolica <emph type="italics"></emph>AasbB<emph.end type="italics"></emph.end>reliquet aream <lb></lb>quæſitam <emph type="italics"></emph>ABNA.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note164"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>3. Si Vis centripeta, ad ſingulas Sphæræ particulas <lb></lb>tendens, decreſcit in quadruplicata ratione diſtantiæ a particulis; <lb></lb>ſcribe (<emph type="italics"></emph>PEqq/2AScub<emph.end type="italics"></emph.end>) pro V, dein √2<emph type="italics"></emph>PSXLD<emph.end type="italics"></emph.end>pro <emph type="italics"></emph>PE,<emph.end type="italics"></emph.end>& fiet <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>(SIqXSL/√2SI)X(1/√LDc),-(SIq/2√2SI)X(1/√LD),-(SIqXALB/2√2SI)X(1/√LDqc).<emph.end type="italics"></emph.end><lb></lb>Cujus tres partes ductæ in longitudinem <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>producunt areas tot<lb></lb>idem, <emph type="italics"></emph>viz. (2SIqXSL/√2SI<emph.end type="italics"></emph.end>) in <emph type="italics"></emph>(1/√LA)-(1/√LB); (SIq/√2SI)<emph.end type="italics"></emph.end>in <emph type="italics"></emph>√LB-√LA<emph.end type="italics"></emph.end>; <lb></lb>& (<emph type="italics"></emph>SIqXALB/3√2SI<emph.end type="italics"></emph.end>) in <emph type="italics"></emph>(1/√LAcub)-(1/√LBcub).<emph.end type="italics"></emph.end>Et hæ poſt debitam redu<lb></lb>ctionem fiunt <emph type="italics"></emph>(2SIqXSL/LI), SIq,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SIq+(2SIcub/3LI).<emph.end type="italics"></emph.end>Hæ vero, ſub<lb></lb>ctis poſterioribus de priore, evadunt (<emph type="italics"></emph>4SIcub/3LI<emph.end type="italics"></emph.end>). Igitur vis tota, qua <lb></lb>corpuſculum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Sphæræ centrum trahitur, eſt ut <emph type="italics"></emph>(SIcub/PI),<emph.end type="italics"></emph.end>id eſt, <lb></lb>reciproce ut <emph type="italics"></emph>PS cubXPI. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Eadem Methodo determinari poteſt Attractio corpuſculi ſiti in<lb></lb>tra Sphæram, ſed expeditius per Theorema ſequens. <pb xlink:href="039/01/217.jpg" pagenum="189"></pb><arrow.to.target n="note165"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note165"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXII. THEOREMA XLI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>In Sphæra centro<emph.end type="italics"></emph.end>S <emph type="italics"></emph>intervallo<emph.end type="italics"></emph.end>SA <emph type="italics"></emph>deſcripta, ſi capiantur<emph.end type="italics"></emph.end>SI, SA, <lb></lb>SP <emph type="italics"></emph>continue proportionales: dico quod corpuſculi intra Sphæ<lb></lb>ram in loco quovis<emph.end type="italics"></emph.end>I <emph type="italics"></emph>attractio est ad attractionem ipſius extra <lb></lb>Sphæram in loco<emph.end type="italics"></emph.end>P, <emph type="italics"></emph>in ratione compoſita ex ſubduplicata ratione <lb></lb>diſtantiarum a centro<emph.end type="italics"></emph.end>IS, PS <emph type="italics"></emph>& ſubduplicata ratione virium <lb></lb>centripetarum, in locis illis<emph.end type="italics"></emph.end>P <emph type="italics"></emph>&<emph.end type="italics"></emph.end>I, <emph type="italics"></emph>ad centrum tendentium.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Ut ſi vires centripetæ particularum Sphæræ ſint reciproce ut di<lb></lb>ſtantiæ corpuſculi a ſe attracti; vis, qua corpuſculum ſitum in <emph type="italics"></emph>I<emph.end type="italics"></emph.end><lb></lb>trahitur a Sphæra tota, erit ad vim qua trahitur in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>in ratione <lb></lb><figure id="id.039.01.217.1.jpg" xlink:href="039/01/217/1.jpg"></figure><lb></lb>compoſita ex ſubduplicata ratione diſtantiæ <emph type="italics"></emph>SI<emph.end type="italics"></emph.end>ad diſtantiam <emph type="italics"></emph>SP<emph.end type="italics"></emph.end><lb></lb>& ratione ſubduplicata vis centripetæ in loco <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>a particula aliqua <lb></lb>in centro oriundæ, ad vim centripetam in loco <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ab eadem in cen<lb></lb>tro particula oriundam, id eſt, ratione ſubduplicata diſtantiarum <lb></lb><emph type="italics"></emph>SI, SP<emph.end type="italics"></emph.end>ad invicem reciproce. </s> <s>Hæ duæ rationes ſubduplicatæ <lb></lb>componunt rationem æqualitatis, & propterea attractiones in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>a Sphæra tota factæ æquantur. </s> <s>Simili computo, ſi vires particu<lb></lb>larum Sphæræ ſunt reciproce in duplicata ratione diſtantiarum, col<lb></lb>ligetur quod attractio in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>ſit ad attractionem in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ut diſtantia <emph type="italics"></emph>SP<emph.end type="italics"></emph.end><lb></lb>ad Sphæræ ſemidiametrum <emph type="italics"></emph>SA:<emph.end type="italics"></emph.end>Si vires illæ ſunt reciproce in tr<lb></lb>plicata ratione diſtantiarum, attractiones in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erunt ad invi-<pb xlink:href="039/01/218.jpg" pagenum="190"></pb><arrow.to.target n="note166"></arrow.to.target>cem ut <emph type="italics"></emph>SP quad<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SA quad:<emph.end type="italics"></emph.end>Si in quadruplicata, ut <emph type="italics"></emph>SP cub<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>SA cub.<emph.end type="italics"></emph.end>Unde cum attractio in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>in hoc ultimo caſu, inventa <lb></lb>fuit reciproce ut <emph type="italics"></emph>PS cubXPI,<emph.end type="italics"></emph.end>attractio in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>erit reciproce ut <lb></lb><emph type="italics"></emph>SA cubXPI,<emph.end type="italics"></emph.end>id eſt (ob datum <emph type="italics"></emph>SA cub<emph.end type="italics"></emph.end>) reciproce ut <emph type="italics"></emph>PI.<emph.end type="italics"></emph.end>Et <lb></lb>ſimilis eſt progreſſus in infinitum. </s> <s>Theorema vero ſic demon<lb></lb>ſtratur. </s></p> <p type="margin"> <s><margin.target id="note166"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Stantibus jam ante conſtructis, & exiſtente corpore in loco <lb></lb>quovis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ordinatim applicata <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>inventa fuit ut (<emph type="italics"></emph>DEqXPS/PEXV<emph.end type="italics"></emph.end>). <lb></lb>Ergo ſi agatur <emph type="italics"></emph>IE,<emph.end type="italics"></emph.end>ordinata illa ad alium quemvis locum <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>mu<lb></lb>tatis mutandis, evadet ut (<emph type="italics"></emph>DEqXIS/IEXV<emph.end type="italics"></emph.end>). Pone vires centripetas, e <lb></lb>Sphæræ puncto quovis <emph type="italics"></emph>E<emph.end type="italics"></emph.end>manantes, eſſe ad invicem in diſtantiis <lb></lb><emph type="italics"></emph>IE, PE,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PE<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>ad <emph type="italics"></emph>IE<emph type="sup"></emph>n<emph.end type="sup"></emph.end>,<emph.end type="italics"></emph.end>(ubi numerus <emph type="italics"></emph>n<emph.end type="italics"></emph.end>deſignet indicem <lb></lb>poteſtatum <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IE<emph.end type="italics"></emph.end>) & ordinatæ illæ fient ut (<emph type="italics"></emph>DEqXPS/PEXPE<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>) & <lb></lb>(<emph type="italics"></emph>DEqXIS/IEXIE<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>), quarum ratio ad invicem eſt ut <emph type="italics"></emph>PSXIEXIE<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>ISXPEXPE<emph type="sup"></emph>n<emph.end type="sup"></emph.end>.<emph.end type="italics"></emph.end>Quoniam ob ſimilia triangula <emph type="italics"></emph>SPE, SEI,<emph.end type="italics"></emph.end>fit <lb></lb><emph type="italics"></emph>IE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>IS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SE<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>SA<emph.end type="italics"></emph.end>; pro ratione <emph type="italics"></emph>IE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ſcribe <lb></lb>rationem <emph type="italics"></emph>IS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SA<emph.end type="italics"></emph.end>; & ordinatarum ratio evadet <emph type="italics"></emph>PSXIE<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>SAXPE<emph type="sup"></emph>n<emph.end type="sup"></emph.end>.<emph.end type="italics"></emph.end>Sed <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SA<emph.end type="italics"></emph.end>ſubduplicata eſt ratio diſtantiarum <lb></lb><emph type="italics"></emph>PS, SI<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>IE<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PE<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>ſubduplicata eſt ratio virium in diſtan<lb></lb>tiis <emph type="italics"></emph>PS, IS.<emph.end type="italics"></emph.end>Ergo ordinatæ, & propterea areæ quas ordinatæ <lb></lb>deſcribunt, hiſque proportionales attractiones, ſunt in ratione com<lb></lb>poſita ex ſubduplicatis illis rationibus. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXIII. PROBLEMA XLII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire vim qua corpuſculum in centro Sphæræ locatum ad ejus <lb></lb>Segmentum quodcunque attrahitur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>P<emph.end type="italics"></emph.end>corpus in centro Sphæræ, & <emph type="italics"></emph>RBSD<emph.end type="italics"></emph.end>Segmentum ejus <lb></lb>plano <emph type="italics"></emph>RDS<emph.end type="italics"></emph.end>& ſuperficie Sphærica <emph type="italics"></emph>RBS<emph.end type="italics"></emph.end>contentum. </s> <s>Superfi<lb></lb>cie Sphærica <emph type="italics"></emph>EFG<emph.end type="italics"></emph.end>centro <emph type="italics"></emph>P<emph.end type="italics"></emph.end>deſcripta ſecetur <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>in <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>ac di<lb></lb>ſtinguatur Segmentum in partes <emph type="italics"></emph>BREFGS, FEDG.<emph.end type="italics"></emph.end>Sit <lb></lb>autem ſuperficies illa non pure Mathematica, ſed Phyſica, pro<lb></lb>funditatem habens quam minimam. </s> <s>Nominetur iſta profundi-<pb xlink:href="039/01/219.jpg" pagenum="191"></pb><arrow.to.target n="note167"></arrow.to.target>tas O, & erit hæc ſuperficies (per de<lb></lb><figure id="id.039.01.219.1.jpg" xlink:href="039/01/219/1.jpg"></figure><lb></lb>monſtrata <emph type="italics"></emph>Archimedis<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>PFXDFXO.<emph.end type="italics"></emph.end><lb></lb>Ponamus præterea vires attractivas par<lb></lb>ticularum Sphæræ eſſe reciproce ut <lb></lb>diſtantiarum dignitas illa cujus Index <lb></lb>eſt <emph type="italics"></emph>n<emph.end type="italics"></emph.end>; & vis qua ſuperficies <emph type="italics"></emph>FE<emph.end type="italics"></emph.end>trahit <lb></lb>corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erit ut (<emph type="italics"></emph>DFXO/PF<emph type="sup"></emph>n-1<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>). Huic pro<lb></lb>portionale ſit perpendiculum <emph type="italics"></emph>FN<emph.end type="italics"></emph.end>duc<lb></lb>tum in O; & area curvilinea <emph type="italics"></emph>BDLIB,<emph.end type="italics"></emph.end><lb></lb>quam ordinatim applicata <emph type="italics"></emph>FN<emph.end type="italics"></emph.end>in lon<lb></lb>gitudinem <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>per motum continuum <lb></lb>ducta deſcribit, erit ut vis tota qua <lb></lb>Segmentum totum <emph type="italics"></emph>RBSD<emph.end type="italics"></emph.end>trahit corpus <emph type="italics"></emph>P. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note167"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXIV. PROBLEMA XLIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire vim qua corpuſculum, extra centrum Sphæræ in axe Seg<lb></lb>menti cujuſvis locatum, attrahitur ab eodem Segmento.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>A Segmento <emph type="italics"></emph>EBK<emph.end type="italics"></emph.end>trahatur corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>(Vide Fig. </s> <s>Prop. </s> <s>LXXIX, <lb></lb>LXXX, LXXXI) in ejus axe <emph type="italics"></emph>ADB<emph.end type="italics"></emph.end>locatum. </s> <s>Centro <emph type="italics"></emph>P<emph.end type="italics"></emph.end>interval<lb></lb>lo <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>deſcribatur ſuperficies Sphærica <emph type="italics"></emph>EFK,<emph.end type="italics"></emph.end>qua diſtinguatur <lb></lb>Segmentum in partes duas <emph type="italics"></emph>EBKF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EFKD.<emph.end type="italics"></emph.end>Quæratur vis par<lb></lb>tis prioris per Prop. </s> <s>LXXXI, & vis partis poſterioris per Prop. </s> <s><lb></lb>LXXXIII; & ſumma virium erit vis Segmenti totius <emph type="italics"></emph>EBKD. <lb></lb><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Explicatis attractionibus corporum Sphærieorum, jam pergere <lb></lb>liceret ad Leges attractionum aliorum quorundam ex particulis at<lb></lb>tractivis ſimiliter conſtantium corporum; ſed iſta particulatim <lb></lb>tractare minus ad inſtitutum ſpectat. </s> <s>Suffecerit Propoſitiones <lb></lb>quaſdam generaliores de viribus hujuſmodi corporum, deque mo<lb></lb>tibus inde oriundis, ob earum in rebus Philoſophicis aliqualem <lb></lb>uſum, ſubjungere. <pb xlink:href="039/01/220.jpg" pagenum="192"></pb><arrow.to.target n="note168"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note168"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Corporum non Sphærieorum viribus attactivis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXV. THEOREMA XLII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corporis attracti, ubi attrahenti contiguum est, attractio longe <lb></lb>fortior ſit, quam cum vel minimo intervallo ſeparantur ab in<lb></lb>vicem: vires particularum trahentis, in receſſu corporis attrac<lb></lb>ti, decreſcunt in ratione pluſquam duplicata diſtantiarum a <lb></lb>particulis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi vires decreſcunt in ratione duplicata diſtantiarum a par<lb></lb>ticulis; attractio verſus corpus Sphæricum, propterea quod (per <lb></lb>Prop. </s> <s>LXXIV) ſit reciproce ut quadratum diſtantiæ attracti corpo<lb></lb>ris a centro Sphæræ, haud ſenſibiliter augebitur ex contactu; atque <lb></lb>adhuc minus augebitur ex contactu, ſi attractio in receſſu corporis <lb></lb>attracti decreſcat in ratione minore. </s> <s>Patet igitur Propoſitio de <lb></lb>Sphæris attractivis. </s> <s>Et par eſt ratio Orbium Sphærieorum conca<lb></lb>vorum corpora externa trahentium. </s> <s>Et multo magis res conſtat in <lb></lb>Orbibus corpora interius conſtituta trahentibus, cum attractiones <lb></lb>paſſim per Orbium cavitates ab attractionibus contrariis (per Prop. </s> <s><lb></lb>LXX) tollantur, ideoque vel in ipſo contactu nullæ ſunt. </s> <s>Quod <lb></lb>ſi Sphæris hiſce Orbibuſque Sphæricis partes quælibet a loco con<lb></lb>tactus remotæ auferantur, & partes novæ ubivis addantur: mu<lb></lb>tari poſſunt figuræ horum corporum attractivorum pro lubitu, nec <lb></lb>tamen partes additæ vel ſubductæ, cum ſint a loco contactus re<lb></lb>motæ, augebunt notabiliter attractionis exceſſum qui ex contactu <lb></lb>oritur. </s> <s>Conſtat igitur Propoſitio de corporibus Figurarum om<lb></lb>nium. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/221.jpg" pagenum="193"></pb><arrow.to.target n="note169"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note169"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXVI. THEOREMA XLIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si particularum, ex quibus corpus attractivum componitur, vires <lb></lb>in receſſu corporis attracti decreſcunt in triplicata vel pluſquam <lb></lb>triplicata ratione diſtantiarum a particulis: attractio longe for<lb></lb>tior erit in contactu, quam cum attrahens & attractum inter<lb></lb>vallo vel minimo ſeparantur ab invicem.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam attractionem in acceſſu attracti corpuſculi ad hujuſmodi <lb></lb>Sphæram trahentem augeri in infinitum, conſtat per ſolutionem Pro<lb></lb>blematis XLI, in Exemplo ſecundo ac tertio exhibitam. </s> <s>Idem, per <lb></lb>Exempla illa & Theorema XLI inter ſe collata, facile colligitur <lb></lb>de attractionibus corporum verſus Orbes concavo-convexos, ſive <lb></lb>corpora attracta collocentur extra Orbes, ſive intra in eorum cavi<lb></lb>tatibus. </s> <s>Sed & addendo vel auferendo his Sphæris & Orbibus ubi<lb></lb>vis extra locum contactus materiam quamlibet attractivam, eo ut <lb></lb>corpora attractiva induant figuram quamvis aſſignatam, conſtabit <lb></lb>Propoſitio de corporibus univerſis. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXVII. THEOREMA XLIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpora duo ſibi invicem ſimilia, & ex materia æqualiter attra<lb></lb>ctiva conſtantia, ſeorſim attrahant corpuſcula ſibi ipſis proporti<lb></lb>onalia & ad ſe ſimiliter poſita: attractiones acceleratrices cor<lb></lb>puſculorum in corpora tota erunt ut attractiones acceleratrices <lb></lb>corpuſculorum in eorum particulas totis proportionales & in to<lb></lb>tis ſimiliter poſitas.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi corpora diſtinguantur in particulas, quæ ſint totis pro<lb></lb>portionales & in totis ſimiliter ſitæ; erit, ut attractio in particulam <lb></lb>quamlibet unius corporis ad attractionem in particulam correſpon<lb></lb>dentem in corpore altero, ita attractiones in particulas ſingulas <lb></lb>primi corporis ad attractiones in alterius particulas ſingulas correſ<lb></lb>pondentes; & componendo, ita attractio in totum primum corpus <lb></lb>ad attractionem in totum ſecundum. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Ergo ſi vires attractivæ particularum, augendo diſtan<lb></lb>tias corpuſculorum attractorum, decreſcant in ratione dignitatis <pb xlink:href="039/01/222.jpg" pagenum="194"></pb><arrow.to.target n="note170"></arrow.to.target>cujuſvis diſtantiarum: attractiones acceleratrices in corpora tota <lb></lb>erunt ut corpora directe & diſtantiarum dignitates illæ inverſe. </s> <s>Ut <lb></lb>ſi vires particularum decreſcant in ratione duplicata diſtantiarum <lb></lb>a corpuſculis attractis, corpora autem ſint ut <emph type="italics"></emph>A cub.<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B cub.<emph.end type="italics"></emph.end>ad<lb></lb>eoque tum corporum latera cubica, tum corpuſculorum attracto<lb></lb>rum diſtantiæ a corporibus, ut <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B:<emph.end type="italics"></emph.end>attractiones acceleratri<lb></lb>ces in corpora erunt ut (<emph type="italics"></emph>Acub./Aquad.<emph.end type="italics"></emph.end>) & (<emph type="italics"></emph>Bcub./Bquad.<emph.end type="italics"></emph.end>) id eſt, ut corporum la<lb></lb>tera illa cubica <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B.<emph.end type="italics"></emph.end>Si vires particularum decreſcant in ra<lb></lb>tione triplicata diſtantiarum a corpuſculis attractis; attractiones <lb></lb>acceleratrices in corpora tota erunt ut (<emph type="italics"></emph>Acub./Acub.<emph.end type="italics"></emph.end>) & (<emph type="italics"></emph>Bcub./Bcub.<emph.end type="italics"></emph.end>), id eſt, æqua<lb></lb>les. </s> <s>Si vires decreſcant in ratione quadruplicata; attractiones in <lb></lb>corpora erunt ut (<emph type="italics"></emph>Acub./Aqq.<emph.end type="italics"></emph.end>) & (<emph type="italics"></emph>Bcub./Bqq.<emph.end type="italics"></emph.end>) id eſt, reciproce ut latera cubi<lb></lb>ca <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B.<emph.end type="italics"></emph.end>Et ſic in cæteris. </s></p> <p type="margin"> <s><margin.target id="note170"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Unde viciſſim, ex viribus quibus corpora ſimilia tra<lb></lb>hunt corpuſcula ad ſe ſimiliter poſita, colligi poteſt ratio decre<lb></lb>menti virium particularum attractivarum in receſſu corpuſculi at<lb></lb>tracti; ſi modo decrementum illud ſit directe vel inverſe in ratione <lb></lb>aliqua diſtantiarum. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXVIII. THEOREMA XLV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si particularum æqualium Corporis cujuſcunque vires attractivæ <lb></lb>ſint ut diſtantiæ loeorum a particulis: vis corporis totius ten<lb></lb>det ad ipſius centrum gravitatis; & eadem erit cum vi Globi <lb></lb>ex materia conſimili & æquali conſtantis & centrum habentis <lb></lb>in ejus centro gravitatis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Corporis <emph type="italics"></emph>RSTV<emph.end type="italics"></emph.end>particulæ <emph type="italics"></emph>A, <lb></lb>B<emph.end type="italics"></emph.end>trahant corpuſculum aliquod <lb></lb><figure id="id.039.01.222.1.jpg" xlink:href="039/01/222/1.jpg"></figure><lb></lb><emph type="italics"></emph>Z<emph.end type="italics"></emph.end>viribus quæ, ſi particulæ æ<lb></lb>quantur inter ſe, ſint ut diſtan<lb></lb>tiæ <emph type="italics"></emph>AZ, BZ<emph.end type="italics"></emph.end>; ſin particulæ ſta<lb></lb>tuantur inæquales, ſint ut hæ par<lb></lb>ticulæ in diſtantias ſuas <emph type="italics"></emph>AZ, BZ<emph.end type="italics"></emph.end><lb></lb>reſpective ductæ. </s> <s>Et exponan<lb></lb>tur hæ vires per contenta illa <lb></lb><emph type="italics"></emph>AXAZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BXBZ.<emph.end type="italics"></emph.end>Jungatur <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end><lb></lb>& ſecetur ea in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>AG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BG<emph.end type="italics"></emph.end>ut particula <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad particulam <emph type="italics"></emph>A<emph.end type="italics"></emph.end>; <pb xlink:href="039/01/223.jpg" pagenum="195"></pb>& erit <emph type="italics"></emph>G<emph.end type="italics"></emph.end>commune centrum gravitatis particularum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B.<emph.end type="italics"></emph.end>Vis <lb></lb><arrow.to.target n="note171"></arrow.to.target><emph type="italics"></emph>AXAZ<emph.end type="italics"></emph.end>(per Legum Corol.2.) reſolvitur in vires <emph type="italics"></emph>AXGZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AXAG<emph.end type="italics"></emph.end><lb></lb>& vis <emph type="italics"></emph>BXBZ<emph.end type="italics"></emph.end>in vires <emph type="italics"></emph>BXGZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BXBG.<emph.end type="italics"></emph.end>Vires autem <emph type="italics"></emph>AXAG<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>BXBG,<emph.end type="italics"></emph.end>ob proportionales <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AG,<emph.end type="italics"></emph.end>æquantur; <lb></lb>adeoque cum dirigantur in partes contrarias, ſe mutuo deſtruunt. </s> <s><lb></lb>Reſtant vires <emph type="italics"></emph>AXGZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BXGZ.<emph.end type="italics"></emph.end>Tendunt hæ ab Z verſus cen<lb></lb>trum <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>& vim —<emph type="italics"></emph>A+BXGZ<emph.end type="italics"></emph.end>componunt; hoc eſt, vim eandem ac <lb></lb>ſi particulæ attractivæ <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B<emph.end type="italics"></emph.end>conſiſterent in eorum communi gra<lb></lb>vitatis centro <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>Globum ibi componentes. </s></p> <p type="margin"> <s><margin.target id="note171"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s>Eodem argumento, ſi adjungatur particula tertia <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>& compo<lb></lb>natur hujus vis cum vi —<emph type="italics"></emph>A+BXGZ<emph.end type="italics"></emph.end>tendente ad centrum <emph type="italics"></emph>G<emph.end type="italics"></emph.end>; vis <lb></lb>inde oriunda tendet ad commune centrum gravitatis Globi illius <emph type="italics"></emph>G<emph.end type="italics"></emph.end><lb></lb>& particulæ <emph type="italics"></emph>C<emph.end type="italics"></emph.end>; hoc eſt, ad commune centrum gravitatis trium par<lb></lb>ticularum <emph type="italics"></emph>A, B, C<emph.end type="italics"></emph.end>; & eadem erit ac ſi Globus & particula <emph type="italics"></emph>C<emph.end type="italics"></emph.end>conſi<lb></lb>ſterent in centro illo communi, Globum majorem ibi componentes. </s> <s><lb></lb>Et ſic pergitur in infinitum. </s> <s>Eadem eſt igitur vis tota particula<lb></lb>rum omnium corporis cujuſcunque <emph type="italics"></emph>RSTV<emph.end type="italics"></emph.end>ac ſi corpus illud, ſer<lb></lb>vato gravitatis centro, figuram Globi indueret. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc motus corporis attracti <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>idem erit ac ſi corpus <lb></lb>attrahens <emph type="italics"></emph>RSTV<emph.end type="italics"></emph.end>eſſet Sphæricum: & propterea ſi corpus illud <lb></lb>attrahens vel quieſcat, vel progrediatur uniformiter in directum; <lb></lb>corpus attractum movebitur in Ellipſi centrum habente in attra<lb></lb>hentis centro gravitatis. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LXXXIX. THEOREMA XLVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Corpora ſint plura ex particulis æqualibus conſtantia, quarum vi<lb></lb>res ſunt ut diſtantiæ loeorum a ſingulis: vis ex omnium viri<lb></lb>bus compoſita, qua corpuſculum quodcunque trahitur, tendet ad <lb></lb>trahentium commune centrum gravitatis, & eadem erit ac ſi <lb></lb>trahentia illa, ſervato gravitatis centro communi, coirent & in <lb></lb>Globum formarentur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Demonſtratur eodem modo, atque Propoſitio ſuperior. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Ergo motus corporis attracti idem erit ac ſi corpora tra<lb></lb>hentia, ſervato communi gravitatis centro, coirent & in Globum <lb></lb>formarentur. </s> <s>Ideoque ſi corporum trahentium commune gravita<lb></lb>tis centrum vel quieſcit, vel progreditur uniformiter in linea recta: <lb></lb>corpus attractum movebitur in Ellipſi, centrum habente in com<lb></lb>muni illo trahentium centro gravitatis. <pb xlink:href="039/01/224.jpg" pagenum="196"></pb><arrow.to.target n="note172"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note172"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XC. PROBLEMA XLIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si ad ſingula Circuli cujuſcunque puncta tendant vires æquales cen<lb></lb>tripetæ, decreſcentes in quacunQ.E.D.ſtantiarum ratione: inve<lb></lb>nire vim qua corpuſculum attrahitur ubivis poſitum in recta <lb></lb>quæ plano Circuli ad centrum ejus perpendiculariter inſiſtit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Centro <emph type="italics"></emph>A<emph.end type="italics"></emph.end>intervallo quovis <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>in plano cui recta <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>per<lb></lb>pendicularis eſt, deſcribi intelligatur Circulus; & invenienda ſit vis <lb></lb>qua corpuſculum quodvis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in eundem attrahitur. </s> <s>A Circuli puncto <lb></lb>quovis <emph type="italics"></emph>E<emph.end type="italics"></emph.end>ad corpuſculum attractum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>agatur recta <emph type="italics"></emph>PE:<emph.end type="italics"></emph.end>In re<lb></lb>cta <emph type="italics"></emph>PA<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>PF<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>æ<lb></lb><figure id="id.039.01.224.1.jpg" xlink:href="039/01/224/1.jpg"></figure><lb></lb>qualis, & erigatur normalis <emph type="italics"></emph>FK,<emph.end type="italics"></emph.end><lb></lb>quæ ſit ut vis qua punctum <emph type="italics"></emph>E<emph.end type="italics"></emph.end>tra<lb></lb>hit corpuſculum <emph type="italics"></emph>P.<emph.end type="italics"></emph.end>Sitque <emph type="italics"></emph>IKL<emph.end type="italics"></emph.end><lb></lb>curva linea quam punctum <emph type="italics"></emph>K<emph.end type="italics"></emph.end>per<lb></lb>petuo tangit. </s> <s>Occurrat eadem Cir<lb></lb>culi plano in <emph type="italics"></emph>L.<emph.end type="italics"></emph.end>In <emph type="italics"></emph>PA<emph.end type="italics"></emph.end>capiatur <lb></lb><emph type="italics"></emph>PH<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>PD,<emph.end type="italics"></emph.end>& erigatur per<lb></lb>pendiculum <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>curvæ prædictæ <lb></lb>occurrens in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>; & erit corpuſ<lb></lb>culi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>attractio in Circulum ut area <lb></lb><emph type="italics"></emph>AHIL<emph.end type="italics"></emph.end>ducta in altitudinem <emph type="italics"></emph>AP. <lb></lb><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Etenim in <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>capiatur linea quam minima <emph type="italics"></emph>Ee.<emph.end type="italics"></emph.end>Jungatur <emph type="italics"></emph>Pe,<emph.end type="italics"></emph.end><lb></lb>& in <emph type="italics"></emph>PE, PA<emph.end type="italics"></emph.end>capiantur <emph type="italics"></emph>PC, Pf<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>Pe<emph.end type="italics"></emph.end>æquales. </s> <s>Et quoniam vis, <lb></lb>qua annuli punctum quodvis <emph type="italics"></emph>E<emph.end type="italics"></emph.end>trahit ad ſe corpus <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ponitur eſſe <lb></lb>ut <emph type="italics"></emph>FK,<emph.end type="italics"></emph.end>& inde vis qua punctum illud trahit corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>eſt ut <lb></lb>(<emph type="italics"></emph>APXFK/PE<emph.end type="italics"></emph.end>), & vis qua annulus totus trahit corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut <lb></lb>annulus & (<emph type="italics"></emph>APXFK/PE<emph.end type="italics"></emph.end>) conjunctim; annulus autem iſte eſt ut rectan<lb></lb>gulum ſub radio <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>& latitudine <emph type="italics"></emph>Ee,<emph.end type="italics"></emph.end>& hoc rectangulum (ob pro<lb></lb>portionales <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AE, Ee<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>) æquatur rectangulo <emph type="italics"></emph>PEXCE<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>PEXFf<emph.end type="italics"></emph.end>; erit vis qua annulus iſte trahit corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>verſus <lb></lb><emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PEXFf<emph.end type="italics"></emph.end>& (<emph type="italics"></emph>APXFK/PE<emph.end type="italics"></emph.end>) conjunctim, id eſt, ut contentum <lb></lb><emph type="italics"></emph>FfXFKXAP,<emph.end type="italics"></emph.end>ſive ut area <emph type="italics"></emph>FKkf<emph.end type="italics"></emph.end>ducta in <emph type="italics"></emph>AP.<emph.end type="italics"></emph.end>Et propterea <lb></lb>ſumma virium, quibus annuli omnes in Circulo, qui centro <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& in-<pb xlink:href="039/01/225.jpg" pagenum="197"></pb>tervallo <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>deſcribitur, trahunt corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>eſt ut area <lb></lb><arrow.to.target n="note173"></arrow.to.target>tota <emph type="italics"></emph>AHIKL<emph.end type="italics"></emph.end>ducta in <emph type="italics"></emph>AP. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note173"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi vires punctorum decreſcunt in duplicata di<lb></lb>ſtantiarum ratione, hoc eſt, ſi ſit <emph type="italics"></emph>FK<emph.end type="italics"></emph.end>ut (1/<emph type="italics"></emph>PFquad.<emph.end type="italics"></emph.end>), atque adeo a<lb></lb>rea <emph type="italics"></emph>AHIKL<emph.end type="italics"></emph.end>ut (1/<emph type="italics"></emph>PA<emph.end type="italics"></emph.end>-1/<emph type="italics"></emph>PH<emph.end type="italics"></emph.end>); erit attractio corpuſculi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Circu<lb></lb>lum ut (1-<emph type="italics"></emph>PA/PH<emph.end type="italics"></emph.end>), id eſt, ut (<emph type="italics"></emph>AH/PH<emph.end type="italics"></emph.end>). </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et univerſaliter, ſi vires punctorum ad diſtantias D ſint <lb></lb>reciproce ut diſtantiarum dignitas quælibet D<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, hoc eſt, ſi ſit <emph type="italics"></emph>FK<emph.end type="italics"></emph.end><lb></lb>ut (1/D<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>), adeoque area <emph type="italics"></emph>AHIKL<emph.end type="italics"></emph.end>ut (1/<emph type="italics"></emph>PA<emph type="sup"></emph>n-1<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>-1/<emph type="italics"></emph>PH<emph type="sup"></emph>n-1<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>); erit attra<lb></lb>ctio corpuſculi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Circulum ut (1/<emph type="italics"></emph>PA<emph type="sup"></emph>n-2<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>-<emph type="italics"></emph>PA/PH<emph type="sup"></emph>n-1<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>). </s></p> <p type="main"> <s><emph type="italics"></emph>Corol<emph.end type="italics"></emph.end>3. Et ſi diameter Circuli augeatur in infinitum, & nume<lb></lb>rus <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ſit unitate major; attractio corpuſculi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in planum totum <lb></lb>infinitum erit reciproce ut <emph type="italics"></emph>PA<emph type="sup"></emph>n-2<emph.end type="sup"></emph.end>,<emph.end type="italics"></emph.end>propterea quod terminus al<lb></lb>ter (<emph type="italics"></emph>PA/PH<emph type="sup"></emph>n-1<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>) evaneſcet. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XCI. PROBLEMA XLV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Invenire attractionem corpuſculi ſiti in axe Solidi rotundi, ad cujus <lb></lb>puncta ſingula tendunt vires æquales centripetæ in quacunque <lb></lb>diſtantiarum ratione decreſcentes.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>In Solidum <emph type="italics"></emph>ADEFG<emph.end type="italics"></emph.end>tra<lb></lb><figure id="id.039.01.225.1.jpg" xlink:href="039/01/225/1.jpg"></figure><lb></lb>hatur corpuſculum <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſitum in <lb></lb>ejus axe <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Circulo quoli<lb></lb>bet <emph type="italics"></emph>RFS<emph.end type="italics"></emph.end>ad hunc axem per<lb></lb>pendiculari ſecetur hoc Solidum, <lb></lb>& in ejus diametro <emph type="italics"></emph>FS,<emph.end type="italics"></emph.end>in pla<lb></lb>no aliquo <emph type="italics"></emph>PALKB<emph.end type="italics"></emph.end>per axem <lb></lb>tranſeunte, capiatur (per Prop. </s> <s><lb></lb>XC) longitudo <emph type="italics"></emph>FK<emph.end type="italics"></emph.end>vi qua cor<lb></lb>puſculum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in circulum illum <lb></lb>attrahitur proportionalis. </s> <s>Tangat autem punctum <emph type="italics"></emph>K<emph.end type="italics"></emph.end>curvam line<lb></lb>am <emph type="italics"></emph>LKI,<emph.end type="italics"></emph.end>planis extimorum circulorum <emph type="italics"></emph>AL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BI<emph.end type="italics"></emph.end>occurrentem in <lb></lb><emph type="italics"></emph>L<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I<emph.end type="italics"></emph.end>; & erit attractio corpuſculi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Solidum ut area <emph type="italics"></emph>LABI. <lb></lb><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end><pb xlink:href="039/01/226.jpg" pagenum="198"></pb><arrow.to.target n="note174"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note174"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Unde ſi Solidum <lb></lb><figure id="id.039.01.226.1.jpg" xlink:href="039/01/226/1.jpg"></figure><lb></lb>Cylindrus ſit, parallelogrammo <lb></lb><emph type="italics"></emph>ADEB<emph.end type="italics"></emph.end>circa axem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>revo<lb></lb>luto deſcriptus, & vires centri<lb></lb>petæ in ſingula ejus puncta ten<lb></lb>dentes ſint reciproce ut quadra<lb></lb>ta diſtantiarum a punctis: erit <lb></lb>attractio corpuſculi <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in hunc <lb></lb>Cylindrum ut <emph type="italics"></emph>AB-PE+PD.<emph.end type="italics"></emph.end><lb></lb>Nam ordinatim applicata <emph type="italics"></emph>FK<emph.end type="italics"></emph.end><lb></lb>(per Corol. </s> <s>1. Prop. </s> <s>XC) erit ut 1-(<emph type="italics"></emph>PF/PR<emph.end type="italics"></emph.end>). Hujus pars 1 ducta in lon<lb></lb>gitudinem <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>deſcribit aream 1X<emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; & pars altera (<emph type="italics"></emph>PF/PR<emph.end type="italics"></emph.end>) ducta <lb></lb>in longitudinem <emph type="italics"></emph>PB,<emph.end type="italics"></emph.end>deſcribit aream 1 in —(<emph type="italics"></emph>PE-AD<emph.end type="italics"></emph.end>) (id quod <lb></lb>ex curvæ <emph type="italics"></emph>LIK<emph.end type="italics"></emph.end>quadratura facile oſtendi poteſt:) & ſimiliter pars <lb></lb>eadem ducta in longitudinem <emph type="italics"></emph>PA<emph.end type="italics"></emph.end>deſcribit aream 1 in —(<emph type="italics"></emph>PD-AD<emph.end type="italics"></emph.end>), <lb></lb>ductaQ.E.I. ipſarum <emph type="italics"></emph>PB, PA<emph.end type="italics"></emph.end>differentiam <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>deſcribit arearum <lb></lb>differentiam 1 in —(<emph type="italics"></emph>PE-PD<emph.end type="italics"></emph.end>). De contento primo 1X<emph type="italics"></emph>AB<emph.end type="italics"></emph.end>aufe<lb></lb>ratur contentum poſtremum 1 in —(<emph type="italics"></emph>PE-PD<emph.end type="italics"></emph.end>), & reſtabit area <emph type="italics"></emph>LABI<emph.end type="italics"></emph.end><lb></lb>æqualis 1 in —(<emph type="italics"></emph>AB-PE+PD<emph.end type="italics"></emph.end>). Ergo vis, huic areæ proportiona<lb></lb>lis, eſt ut <emph type="italics"></emph>AB-PE+PD.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Hinc etiam <lb></lb><figure id="id.039.01.226.2.jpg" xlink:href="039/01/226/2.jpg"></figure><lb></lb>vis innoteſcit qua Sphæ<lb></lb>rois <emph type="italics"></emph>AGBCD<emph.end type="italics"></emph.end>attrahit <lb></lb>corpus quodvis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>exte<lb></lb>rius in axe ſuo <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ſi<lb></lb>tum. </s> <s>Sit <emph type="italics"></emph>NKRM<emph.end type="italics"></emph.end>Se<lb></lb>ctio Conica cujus ordi<lb></lb>natim applicata <emph type="italics"></emph>ER,<emph.end type="italics"></emph.end>ipſi <lb></lb><emph type="italics"></emph>PE<emph.end type="italics"></emph.end>perpendicularis, æ<lb></lb>quetur ſemper longitu<lb></lb>dini <emph type="italics"></emph>PD,<emph.end type="italics"></emph.end>quæ ducitur <lb></lb>ad punctum illud <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>in <lb></lb>quo applicata iſta Sphæroidem ſecat. </s> <s>A Sphæroidis verticibus <emph type="italics"></emph>A, B<emph.end type="italics"></emph.end><lb></lb>ad ejus axem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>erigantur perpendicula <emph type="italics"></emph>AK, BM<emph.end type="italics"></emph.end>ipſis <emph type="italics"></emph>AP, BP<emph.end type="italics"></emph.end><lb></lb>æqualia reſpective, & propterea Sectioni Conicæ occurrentia in <emph type="italics"></emph>K<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>M<emph.end type="italics"></emph.end>; & jungatur <emph type="italics"></emph>KM<emph.end type="italics"></emph.end>auferens ab eadem ſegmentum <emph type="italics"></emph>KMRK.<emph.end type="italics"></emph.end><lb></lb>Sit autem Sphæroidis centrum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>& ſemidiameter maxima <emph type="italics"></emph>SC:<emph.end type="italics"></emph.end>& vis <pb xlink:href="039/01/227.jpg" pagenum="199"></pb>qua Sphærois trahit corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erit ad vim qua Sphæra, diametro <emph type="italics"></emph>AB<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><arrow.to.target n="note175"></arrow.to.target>deſcripta, trahit idem corpus, ut (<emph type="italics"></emph>ASXCSq-PSXKMRK/PSq+CSq-ASq<emph.end type="italics"></emph.end>) <lb></lb>ad (<emph type="italics"></emph>AS cub/3PS quad<emph.end type="italics"></emph.end>). Et eodem computandi fundamento invenire licet <lb></lb>vires ſegmentorum Sphæroidis. </s></p> <p type="margin"> <s><margin.target id="note175"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Quod ſi corpuſculum intra Sphæroidem, in data qua<lb></lb>vis ejuſdem diametro, collocetur; attractio erit ut ipſius diſtantia a <lb></lb>centro. </s> <s>Id quod facilius colligetur hoc argumento. </s> <s>Sit <emph type="italics"></emph>AGOF<emph.end type="italics"></emph.end><lb></lb>Sphærois attrahens, <emph type="italics"></emph>S<emph.end type="italics"></emph.end>centrum ejus & <emph type="italics"></emph>P<emph.end type="italics"></emph.end>corpus attractum. </s> <s>Per <lb></lb>corpus illud <emph type="italics"></emph>P<emph.end type="italics"></emph.end>agantur tum ſemidiameter <emph type="italics"></emph>SPA,<emph.end type="italics"></emph.end>tum rectæ duæ <lb></lb>quævis <emph type="italics"></emph>DE, FG<emph.end type="italics"></emph.end>Sphæroidi hinc inde occurrentes in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>E, F<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>G:<emph.end type="italics"></emph.end>Sintque <emph type="italics"></emph>PCM, HLN<emph.end type="italics"></emph.end>ſuperficies Sphæroidum duarum in<lb></lb>teriorum, exteriori ſimilium & concentricarum, quarum prior tranſ<lb></lb>eat per corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& ſecet rectas <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>in <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>poſterior <lb></lb>ſecet eaſdem rectas in <emph type="italics"></emph>H, I<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K, L.<emph.end type="italics"></emph.end>Habeant autem Sphæroides <lb></lb>omnes axem communem, & erunt rect<lb></lb><figure id="id.039.01.227.1.jpg" xlink:href="039/01/227/1.jpg"></figure><lb></lb>arum partes hinc inde interceptæ <emph type="italics"></emph>DP<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>BE, FP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CG, DH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IE, FK<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>LG<emph.end type="italics"></emph.end>ſibi mutuo æquales; propterea <lb></lb>quod rectæ <emph type="italics"></emph>DE, PB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>biſecan<lb></lb>tur in eodem puncto, ut & rectæ <emph type="italics"></emph>FG, <lb></lb>PC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>KL.<emph.end type="italics"></emph.end>Concipe jam <emph type="italics"></emph>DPF, <lb></lb>EPG<emph.end type="italics"></emph.end>deſignare Conos oppoſitos, an<lb></lb>gulis verticalibus <emph type="italics"></emph>DPF, EPG<emph.end type="italics"></emph.end>infi<lb></lb>nite parvis deſcriptos, & lineas etiam <lb></lb><emph type="italics"></emph>DH, EI<emph.end type="italics"></emph.end>infinite parvas eſſe; & Conorum particulæ Sphæroidum <lb></lb>ſuperficiebus abſciſſæ <emph type="italics"></emph>DHKF, GLIE,<emph.end type="italics"></emph.end>ob æqualitatem linearum <lb></lb><emph type="italics"></emph>DH, EI,<emph.end type="italics"></emph.end>erunt ad invicem ut quadrata diſtantiarum ſuarum a <lb></lb>corpuſculo <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& propterea corpuſculum illud æqualiter trahent. </s> <s><lb></lb>Et pari ratione, ſi ſuperficiebus Sphæroidum innumerarum ſimilium <lb></lb>concentricarum & axem communem habentium dividantur ſpatia <lb></lb><emph type="italics"></emph>DPF, EGCB<emph.end type="italics"></emph.end>in particulas, hæ omnes utrinque æqualiter tra<lb></lb>hent corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in partes contrarias. </s> <s>Æquales igitur ſunt vires <lb></lb>Coni <emph type="italics"></emph>DPF<emph.end type="italics"></emph.end>& ſegmenti Conici <emph type="italics"></emph>EGCB,<emph.end type="italics"></emph.end>& per contrarietatem ſe <lb></lb>mutuo deſtruunt. </s> <s>Et par eſt ratio virium materiæ omnis extra Sphæ<lb></lb>roidem intimam <emph type="italics"></emph>PCBM.<emph.end type="italics"></emph.end>Trahitur igitur corpus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>a ſola Sphæ<lb></lb>roide intima <emph type="italics"></emph>PCBM,<emph.end type="italics"></emph.end>& propterea (per Corol. </s> <s>3. Prop. </s> <s>LXXII) at<lb></lb>tractio ejus eſt ad vim, qua corpus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>trahitur a Sphæroide tota <lb></lb><emph type="italics"></emph>AGOD,<emph.end type="italics"></emph.end>ut diſtantia <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad diſtantiam <emph type="italics"></emph>AS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/228.jpg" pagenum="200"></pb><arrow.to.target n="note176"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note176"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XCII. PROBLEMA XLVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Dato Corpore attractivo, invenire rationem decrementi virium cen<lb></lb>tripetarum in ejus puncta ſingula tendentium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>E Corpore dato formanda eſt Sphæra vel Cylindrus aliave figu<lb></lb>ra regularis, cujus lex attractionis, cuivis decrementi rationi con<lb></lb>gruens (per Prop. </s> <s>LXXX, LXXXI, & XCI) inveniri poteſt. </s> <s>Dein fa<lb></lb>ctis experimentis invenienda eſt vis attractionis in diverſis diſtan<lb></lb>tiis, & lex attractionis in totum inde patefacta dabit rationem de<lb></lb>crementi virium partium ſingularum, quam invenire oportuit. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XCIII. THEOREMA XLVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Solidum ex una parte planum, ex reliquis autem partibus infiNI<lb></lb>tum, conſtet ex particulis æqualibus æqualiter attractivis, qua<lb></lb>rum vires in receſſu a Solido decreſcunt in ratione poteſtatis cu<lb></lb>juſvis diſtantiarum pluſquam quadraticæ, & vi Solidi totius cor<lb></lb>puſculum ad utramvis plani partem conſtitutum trahatur: dico <lb></lb>quod Solidi vis illa attractiva, in receſſu ab ejus ſuperficie pla<lb></lb>na, decreſcet in ratione poteſtatis, cujus latus est diſtantia cor<lb></lb>puſculi a plano, & Index ternario minor quam Index poteſta<lb></lb>tis diſtantiarum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Sit <emph type="italics"></emph>LGl<emph.end type="italics"></emph.end>planum <lb></lb><figure id="id.039.01.228.1.jpg" xlink:href="039/01/228/1.jpg"></figure><lb></lb>quo Solidum terminatur. </s> <s><lb></lb>Jaceat Solidum autem ex <lb></lb>parte plani hujus verſus <lb></lb><emph type="italics"></emph>I,<emph.end type="italics"></emph.end>inque plana innumera <lb></lb><emph type="italics"></emph>mHM, nIN,<emph.end type="italics"></emph.end>&c. </s> <s>ipſi <emph type="italics"></emph>GL<emph.end type="italics"></emph.end><lb></lb>parallela reſolvatur. </s> <s>Et <lb></lb>primo collocetur corpus at<lb></lb>tractum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>extra Solidum. </s> <s><lb></lb>Agatur autem <emph type="italics"></emph>CGHI<emph.end type="italics"></emph.end>pla<lb></lb>nis illis innumeris perpendicularis, & decreſcant vires attractivæ <lb></lb>punctorum Solidi in ratione poteſtatis diſtantiarum, cujus index ſit <lb></lb>numerus <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ternario non minor. </s> <s>Ergo (per Corol. </s> <s>3. Prop. </s> <s>XC) <pb xlink:href="039/01/229.jpg" pagenum="201"></pb>vis qua planum quodvis <emph type="italics"></emph>mHM<emph.end type="italics"></emph.end>trahit punctum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>eſt reciproce ut <lb></lb><arrow.to.target n="note177"></arrow.to.target><emph type="italics"></emph>CH<emph type="sup"></emph>n-2<emph.end type="sup"></emph.end>.<emph.end type="italics"></emph.end>In plano <emph type="italics"></emph>mHM<emph.end type="italics"></emph.end>capiatur longitudo <emph type="italics"></emph>HM<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>CH<emph type="sup"></emph>n-2<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>re<lb></lb>ciproce proportionalis, & erit vis illa ut <emph type="italics"></emph>HM.<emph.end type="italics"></emph.end>Similiter in planis ſin<lb></lb>gulis <emph type="italics"></emph>lGL, nIN, oKO,<emph.end type="italics"></emph.end>&c. </s> <s>capiantur longitudines <emph type="italics"></emph>GL, IN, KO,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>ipſis <emph type="italics"></emph>CG<emph type="sup"></emph>n-2<emph.end type="sup"></emph.end>, CI<emph type="sup"></emph>n-2<emph.end type="sup"></emph.end>, CK<emph type="sup"></emph>n-2<emph.end type="sup"></emph.end>,<emph.end type="italics"></emph.end>&c. </s> <s>reciproce proportionales; & vi<lb></lb>res planorum eorundem erunt ut longitudines captæ, adeoque <lb></lb>ſumma virium ut ſumma longitudinum, hoc eſt, vis Solidi totius ut <lb></lb>area <emph type="italics"></emph>GLOK<emph.end type="italics"></emph.end>in infinitum verſus <emph type="italics"></emph>OK<emph.end type="italics"></emph.end>producta. </s> <s>Sed area illa (per <lb></lb>notas quadraturarum methodos) eſt reciproce ut <emph type="italics"></emph>CG<emph type="sup"></emph>n-3<emph.end type="sup"></emph.end>,<emph.end type="italics"></emph.end>& prop<lb></lb>terea vis Solidi totius eſt reciproce ut <emph type="italics"></emph>CG<emph type="sup"></emph>n-3<emph.end type="sup"></emph.end>. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note177"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Collocetur jam corpuſculum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ex parte plani <emph type="italics"></emph>lGL<emph.end type="italics"></emph.end>in<lb></lb>tra Solidum, & capiatur diſtantia <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>æqualis diſtantiæ <emph type="italics"></emph>CG.<emph.end type="italics"></emph.end>Et So<lb></lb>lidi pars <emph type="italics"></emph>LGloKO,<emph.end type="italics"></emph.end>planis parallelis <emph type="italics"></emph>lGL, oKO<emph.end type="italics"></emph.end>terminata, cor<lb></lb>puſculum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>in medio ſitum nullam in partem trahet, contrariis op<lb></lb>poſitorum punctorum actionibus ſe mutuo per æqualitatem tollenti<lb></lb>bus. </s> <s>Proinde corpuſculum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ſola vi Solidi ultra planum <emph type="italics"></emph>OK<emph.end type="italics"></emph.end>ſiti tra<lb></lb>hitur. </s> <s>Hæc autem vis (per Caſum primum) eſt reciproce ut <emph type="italics"></emph>CK<emph type="sup"></emph>n-3<emph.end type="sup"></emph.end>,<emph.end type="italics"></emph.end><lb></lb>hoc eſt (ob æquales <emph type="italics"></emph>CG, CK<emph.end type="italics"></emph.end>) reciproce ut <emph type="italics"></emph>CG<emph type="sup"></emph>n-3<emph.end type="sup"></emph.end>. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi Solidum <emph type="italics"></emph>LGIN<emph.end type="italics"></emph.end>planis duobus infinitis pa<lb></lb>rallelis <emph type="italics"></emph>LG, IN<emph.end type="italics"></emph.end>utrinque terminetur; innoteſcit ejus vis attra<lb></lb>ctiva, ſubducendo de vi attractiva Solidi totius infiniti <emph type="italics"></emph>LGKO<emph.end type="italics"></emph.end><lb></lb>vim attractivam partis ulterioris <emph type="italics"></emph>NICO,<emph.end type="italics"></emph.end>in infinitum verſus <emph type="italics"></emph>KO<emph.end type="italics"></emph.end><lb></lb>productæ. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si Solidi hujus infiniti pars ulterior, quando attractio e<lb></lb>jus collata cum attractione partis citerioris nullius pene eſt momen<lb></lb>ti, rejiciatur: attractio partis illius citerioris augendo diſtantiam de<lb></lb>creſcet quam proxime in ratione poteſtatis <emph type="italics"></emph>CG<emph type="sup"></emph>n-3<emph.end type="sup"></emph.end>.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et hinc ſi corpus quodvis finitum & ex una parte pla<lb></lb>num trahat corpuſculum e regione medii illius plani, & diſtantia <lb></lb>inter corpuſculum & planum collata cum dimenſionibus corpo<lb></lb>ris attrahentis perexigua ſit, conſtet autem corpus attrahens ex <lb></lb>particulis homogeneis, quarum vires attractivæ decreſcunt in <lb></lb>ratione poteſtatis cujuſvis pluſquam quadruplicatæ diſtantiarum; <lb></lb>vis attractiva corporis totius decreſcet quamproxime in ratione <lb></lb>poteſtatis, cujus latus ſit diſtantia illa perexigua, & Index terna<lb></lb>rio minor quam Index poteſtatis prioris. </s> <s>De corpore ex particulis <lb></lb>conſtante, quarum vires attractivæ decreſcunt in ratione poteſtatis <lb></lb>triplicatæ diſtantiarum, aſſertio non valet; propterea quod, in hoc <lb></lb>caſu, attractio partis illius ulterioris corporis infiniti in Corollario <lb></lb>ſecundo, ſemper eſt infinite major quam attractio partis citerioris. <pb xlink:href="039/01/230.jpg" pagenum="202"></pb><arrow.to.target n="note178"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note178"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si corpus aliquod perpendiculariter verſus planum datum tra<lb></lb>hatur, & ex data lege attractionis quæratur motus corporis: Sol<lb></lb>vetur Problema quærendo (per Prop. </s> <s>XXXIX) motum corporis recta <lb></lb>deſcendentis ad hoc planum, & (per Legum Corol. </s> <s>2.) componen<lb></lb>do motum iſtum cum uniformi motu, ſecundum lineas eidem plano <lb></lb>parallelas facto. </s> <s>Et contra, ſi quæratur Lex attractionis in planum <lb></lb>ſecundum lineas perpendiculares factæ, ea conditione ut corpus at<lb></lb>tractum in data quacunque curva linea moveatur, ſolvetur Proble<lb></lb>ma operando ad exemplum Problematis tertii. </s></p> <p type="main"> <s>Operationes autem contrahi ſolent reſolvendo ordinatim appli<lb></lb>catas in Series convergentes. </s> <s>Ut ſi ad baſem A in angulo quovis <lb></lb>dato ordinatim applicetur longitudo B, quæ ſit ut baſis dignitas <lb></lb>quælibet A<emph type="sup"></emph><emph type="italics"></emph>m/n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>; & quæratur vis qua corpus, ſecundum poſitionem <lb></lb>ordinatim applicatæ, vel in baſem attractum vel a baſi fugatum, <lb></lb>moveri poſſit in curva linea quam ordinatim applicata termi<lb></lb>no ſuo ſuperiore ſemper attingit: Suppono baſem augeri parte <lb></lb>quam minima O, & ordinatim applicatam —(A+O)<emph type="italics"></emph>m/n<emph.end type="italics"></emph.end>reſolvo in <lb></lb>Seriem infinitam A<emph type="sup"></emph><emph type="italics"></emph>m/n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>+<emph type="italics"></emph>m/n<emph.end type="italics"></emph.end>OA<emph type="sup"></emph>(<emph type="italics"></emph>m-n/n<emph.end type="italics"></emph.end>)<emph.end type="sup"></emph.end>+(<emph type="italics"></emph>mm-mn/2nn<emph.end type="italics"></emph.end>) OOA<emph type="sup"></emph>(<emph type="italics"></emph>m-2n/n<emph.end type="italics"></emph.end>)<emph.end type="sup"></emph.end> &c. </s> <s>at<lb></lb>que hujus termino in quo O duarum eſt dimenſionum, id eſt, ter<lb></lb>mino (<emph type="italics"></emph>mm-mn/2nn<emph.end type="italics"></emph.end>) OOA<emph type="sup"></emph>(<emph type="italics"></emph>m-2n/n<emph.end type="italics"></emph.end>)<emph.end type="sup"></emph.end> vim proportionalem eſſe ſuppono. </s> <s>Eſt <lb></lb>igitur vis quæſita ut (<emph type="italics"></emph>mm-mn/nn<emph.end type="italics"></emph.end>)A<emph type="sup"></emph>(<emph type="italics"></emph>m-2n/n<emph.end type="italics"></emph.end>)<emph.end type="sup"></emph.end>, vel quod perinde eſt, ut <lb></lb>(<emph type="italics"></emph>mm-mn/nn<emph.end type="italics"></emph.end>)B<emph type="sup"></emph>(<emph type="italics"></emph>m-2n/m<emph.end type="italics"></emph.end>)<emph.end type="sup"></emph.end>. </s> <s>Ut ſi ordinatim applicata Parabolam attingat, <lb></lb>exiſtente <emph type="italics"></emph>m<emph.end type="italics"></emph.end>=2, & <emph type="italics"></emph>n<emph.end type="italics"></emph.end>=1: fiet vis ut data 2B°, adeoQ.E.D.bi<lb></lb>tur. </s> <s>Data igitur vi corpus movebitur in Parabola, quemad<lb></lb>modum <emph type="italics"></emph>Galilæus<emph.end type="italics"></emph.end>demonſtravit. </s> <s>Quod ſi ordinatim applicata <lb></lb>Hyperbolam attingat, exiſtente <emph type="italics"></emph>m<emph.end type="italics"></emph.end>=o-1, & <emph type="italics"></emph>n<emph.end type="italics"></emph.end>=1; fiet vis ut <lb></lb>2A<emph type="sup"></emph>-3<emph.end type="sup"></emph.end> ſeu 2B<emph type="sup"></emph>3<emph.end type="sup"></emph.end>: adeoque vi, quæ ſit ut cubus ordinatim applicatæ, <lb></lb>corpus movebitur in Hyperbola. </s> <s>Sed miſſis hujuſmodi Propoſiti<lb></lb>onibus, pergo ad alias quaſdam de Motu, quas nondum attigi. <pb xlink:href="039/01/231.jpg" pagenum="203"></pb><arrow.to.target n="note179"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note179"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Motu corporum minimorum, quæ Viribus centripetis ad ſingulas <lb></lb>magni alicujus corporis partes tendentibus agitantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XCIV. THEOREMA XLVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Media duo ſimilaria, ſpatio planis parallelis utrinque terminato, <lb></lb>diſtinguantur ab invicem, & corpus in tranſitu per hoc ſpatium <lb></lb>attrahatur vel impellatur perpendiculariter verſus Medium alter<lb></lb>utrum, neque ulla alia vi agitetur vel impediatur: Sit autem <lb></lb>attractio, in æqualibus ab utroque plano diſtantiis ad eandem <lb></lb>ipſius partem captis, ubique eadem: dico quod ſinus incidentiæ <lb></lb>in planum alterutrum erit ad ſinum emergentiæ ex plano altero <lb></lb>in ratione data.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Sunto <emph type="italics"></emph>Aa, Bb<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.231.1.jpg" xlink:href="039/01/231/1.jpg"></figure><lb></lb>plana duo parallela. </s> <s>Inci<lb></lb>dat corpus in planum pri<lb></lb>us <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ſecundum lineam <lb></lb><emph type="italics"></emph>GH,<emph.end type="italics"></emph.end>ac toto ſuo per ſpati<lb></lb>um intermedium tranſitu <lb></lb>attrahatur vel impellatur <lb></lb>verſus Medium inciden<lb></lb>tiæ, eaque actione deſcri<lb></lb>bat lineam curvam <emph type="italics"></emph>HI,<emph.end type="italics"></emph.end>& <lb></lb>emergat ſecundum line<lb></lb>am <emph type="italics"></emph>IK.<emph.end type="italics"></emph.end>Ad planum emer<lb></lb>gentiæ <emph type="italics"></emph>Bb<emph.end type="italics"></emph.end>erigatur per<lb></lb>pendiculum <emph type="italics"></emph>IM,<emph.end type="italics"></emph.end>occur<lb></lb>rens tum lineæ inciden<lb></lb>tiæ <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>productæ in <emph type="italics"></emph>M,<emph.end type="italics"></emph.end><lb></lb>tum plano incidentiæ <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>in <emph type="italics"></emph>R<emph.end type="italics"></emph.end>; & linea emergentiæ <emph type="italics"></emph>KI<emph.end type="italics"></emph.end>producta <lb></lb>occurrat <emph type="italics"></emph>HM<emph.end type="italics"></emph.end>in <emph type="italics"></emph>L.<emph.end type="italics"></emph.end>Centro <emph type="italics"></emph>L<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>LI<emph.end type="italics"></emph.end>deſcribatur Circulus, <pb xlink:href="039/01/232.jpg" pagenum="204"></pb><arrow.to.target n="note180"></arrow.to.target>ſecans tam <emph type="italics"></emph>HM<emph.end type="italics"></emph.end>in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end>quam <emph type="italics"></emph>MI<emph.end type="italics"></emph.end>productam in <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>& primo <lb></lb>ſi attractio vel impulſus ponatur uniformis, erit (ex demonſtratis <lb></lb><emph type="italics"></emph>Galilæi<emph.end type="italics"></emph.end>) curva <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>Parabola, cujus hæc eſt proprietas, ut rectan<lb></lb>gulum ſub dato latere recto & linea <emph type="italics"></emph>IM<emph.end type="italics"></emph.end>æquale ſit <emph type="italics"></emph>HM<emph.end type="italics"></emph.end>quadrato; <lb></lb>ſed & linea <emph type="italics"></emph>HM<emph.end type="italics"></emph.end>biſecabitur in <emph type="italics"></emph>L.<emph.end type="italics"></emph.end>Unde ſi ad <emph type="italics"></emph>MI<emph.end type="italics"></emph.end>demittatur <lb></lb>perpendiculum <emph type="italics"></emph>LO,<emph.end type="italics"></emph.end>æ<lb></lb><figure id="id.039.01.232.1.jpg" xlink:href="039/01/232/1.jpg"></figure><lb></lb>quales erunt <emph type="italics"></emph>MO, OR<emph.end type="italics"></emph.end>; <lb></lb>& additis æqualibus <emph type="italics"></emph>ON, <lb></lb>OI,<emph.end type="italics"></emph.end>fient totæ æquales <lb></lb><emph type="italics"></emph>MN, IR.<emph.end type="italics"></emph.end>Proinde cum <lb></lb><emph type="italics"></emph>IR<emph.end type="italics"></emph.end>detur, datur etiam <lb></lb><emph type="italics"></emph>MN<emph.end type="italics"></emph.end>; eſtque rectangu<lb></lb>lum <emph type="italics"></emph>NMI<emph.end type="italics"></emph.end>ad rectangu<lb></lb>lum ſub latere recto & <lb></lb><emph type="italics"></emph>IM,<emph.end type="italics"></emph.end>hoc eſt, ad <emph type="italics"></emph>HMq,<emph.end type="italics"></emph.end><lb></lb>in data ratione. </s> <s>Sed rect<lb></lb>angulum <emph type="italics"></emph>NMI<emph.end type="italics"></emph.end>æquale <lb></lb>eſt rectangulo <emph type="italics"></emph>PMQ,<emph.end type="italics"></emph.end>id <lb></lb>eſt, differentiæ quadrato<lb></lb>rum <emph type="italics"></emph>MLq,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PLq<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>LIq<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>HMq<emph.end type="italics"></emph.end>datam <lb></lb>rationem habet ad ſui ipſius quartam partem <emph type="italics"></emph>MLq:<emph.end type="italics"></emph.end>ergo datur <lb></lb>ratio <emph type="italics"></emph>MLq-LIq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MLq,<emph.end type="italics"></emph.end>& diviſim, ratio <emph type="italics"></emph>LIq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MLq,<emph.end type="italics"></emph.end>& <lb></lb>ratio dimidiata <emph type="italics"></emph>LI<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ML.<emph.end type="italics"></emph.end>Sed in omni triangulo <emph type="italics"></emph>LMI,<emph.end type="italics"></emph.end>ſinus <lb></lb>angulorum ſunt proportionales lateribus oppoſitis. </s> <s>Ergo datur <lb></lb>ratio ſinus anguli incidentiæ <emph type="italics"></emph>LMR<emph.end type="italics"></emph.end>ad ſinum anguli emergen<lb></lb>tiæ <emph type="italics"></emph>LIR. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note180"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p><figure id="id.039.01.232.2.jpg" xlink:href="039/01/232/2.jpg"></figure> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Tranſeat jam corpus ſucceſſive per ſpatia plura paralle<lb></lb>lis planis terminata, <emph type="italics"></emph>AabB, BbcC,<emph.end type="italics"></emph.end>&c. </s> <s>& agitetur vi quæ ſit in <pb xlink:href="039/01/233.jpg" pagenum="205"></pb>ſingulis ſeparatim uniformis, at in diverſis diverſa; & per jam de<lb></lb><arrow.to.target n="note181"></arrow.to.target>monſtrata, ſinus incidentiæ in planum primum <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>erit ad ſinum <lb></lb>emergentiæ ex plano ſecundo <emph type="italics"></emph>Bb,<emph.end type="italics"></emph.end>in data ratione; & hic ſinus, <lb></lb>qui eſt ſinus incidentiæ in planum ſecundum <emph type="italics"></emph>Bb,<emph.end type="italics"></emph.end>erit ad ſinum <lb></lb>emergentiæ ex plano tertio <emph type="italics"></emph>Cc,<emph.end type="italics"></emph.end>in data ratione; & hic ſinus ad <lb></lb>ſinum emergentiæ ex plano quarto <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>in data ratione; & ſic in <lb></lb>infinitum: & ex æquo, ſinus incidentiæ in planum primum ad ſi<lb></lb>num emergentiæ ex plano ultimo in data ratione. </s> <s>Minuantur jam <lb></lb>planorum intervalla & augeatur numerus in infinitum, eo ut attra<lb></lb>ctionis vel impulſus actio, ſecundum legem quamcunque aſſignatam, <lb></lb>continua reddatur; & ratio ſinus incidentiæ in planum primum ad <lb></lb>ſinum emergentiæ ex plano ultimo, ſemper data exiſtens, etiam<lb></lb>num dabitur. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note181"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XCV. THEOREMA XLIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis; dico quod velocitas corporis ante incidentiam eſt <lb></lb>ad ejus velocitatem poſt emergentiam, ut ſinus emergentiæ ad <lb></lb>ſinum incidentiæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Capiantur <emph type="italics"></emph>AH, Id<emph.end type="italics"></emph.end>æquales, & erigantur perpendicula <emph type="italics"></emph>AG, dK<emph.end type="italics"></emph.end><lb></lb>occurrentia lineis incidentiæ & emergentiæ <emph type="italics"></emph>GH, IK,<emph.end type="italics"></emph.end>in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K.<emph.end type="italics"></emph.end><lb></lb>In <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>TH<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>IK,<emph.end type="italics"></emph.end>& ad planum <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>demittatur <lb></lb>normaliter <emph type="italics"></emph>Tv.<emph.end type="italics"></emph.end>Et (per Legum Corol. </s> <s>2) diſtinguatur motus cor<lb></lb>poris in duos, unum planis <emph type="italics"></emph>Aa, Bb, Cc,<emph.end type="italics"></emph.end>&c. </s> <s>perpendicularem, al<lb></lb>terum iiſdem parallelum. </s> <s>Vis attractionis vel impulſus, agendo ſe<lb></lb>cundum lineas perpendiculares, nil mutat motum ſecundum paralle<lb></lb>las, & propterea corpus hoc motu conficiet æqualibus temporibus <lb></lb>æqualia illa ſecundum parallelas intervalla, quæ ſunt inter lineam <lb></lb><emph type="italics"></emph>AG<emph.end type="italics"></emph.end>& punctum <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>interque punctum <emph type="italics"></emph>I<emph.end type="italics"></emph.end>& lineam <emph type="italics"></emph>dK<emph.end type="italics"></emph.end>; hoc eſt, <lb></lb>æqualibus temporibus deſcribet lineas <emph type="italics"></emph>GH, IK.<emph.end type="italics"></emph.end>Proinde velo<lb></lb>citas ante incidentiam eſt ad velocitatem poſt emergentiam, ut <lb></lb><emph type="italics"></emph>GH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>TH,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Id<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>vH,<emph.end type="italics"></emph.end>hoc eſt <lb></lb>(reſpectu radii <emph type="italics"></emph>TH<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>) ut ſinus emergentiæ ad ſinum inci<lb></lb>dentiæ. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/234.jpg" pagenum="206"></pb><arrow.to.target n="note182"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note182"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XCVI. THEOREMA L.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis & quod motus ante incidentiam velocior ſit quam <lb></lb>poſtea: dico quod corpus, inclinando lineam incidentiæ, refle<lb></lb>ctetur tandem, & angulus reflexionis fiet æqualis angulo inci<lb></lb>dentiæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam concipe corpus inter parallela plana <emph type="italics"></emph>Aa, Bb, Cc,<emph.end type="italics"></emph.end>&c. </s> <s>de<lb></lb>ſcribere arcus Parabolicos, ut ſupra; ſintque arcus illi <emph type="italics"></emph>HP, PQ, <lb></lb>QR,<emph.end type="italics"></emph.end>&c. </s> <s>Et ſit ea lineæ incidentiæ <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>obliquitas ad planum pri<lb></lb>mum <emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end>ut ſinus incidentiæ ſit ad radium circuli, cujus eſt ſinus, <lb></lb>in ea ratione quam habet idem ſinus incidentiæ ad ſinum emer<lb></lb>gentiæ ex plano <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>in ſpatium <emph type="italics"></emph>DdeE:<emph.end type="italics"></emph.end>& ob ſinum emergen<lb></lb>tiæ jam factum æqualem radio, angulus emergentiæ erit rectus, ad<lb></lb>eoque linea emergentiæ coincidet cum plano <emph type="italics"></emph>Dd.<emph.end type="italics"></emph.end>Perveniat cor<lb></lb>pus ad hoc planum in puncto <emph type="italics"></emph>R<emph.end type="italics"></emph.end>; & quoniam linea emergentiæ <lb></lb>coincidit cum eodem <lb></lb><figure id="id.039.01.234.1.jpg" xlink:href="039/01/234/1.jpg"></figure><lb></lb>plano, perſpicuum eſt <lb></lb>quod corpus non po<lb></lb>teſt ultra pergere ver<lb></lb>ſus planum <emph type="italics"></emph>Ee.<emph.end type="italics"></emph.end>Sed <lb></lb>nec poteſt idem perge<lb></lb>re in linea emergentiæ <lb></lb><emph type="italics"></emph>Rd,<emph.end type="italics"></emph.end>propterea quod <lb></lb>perpetuo attrahitur vel impellitur verſus Medium incidentiæ. </s> <s>Re<lb></lb>vertetur itaQ.E.I.ter plana <emph type="italics"></emph>Cc, Dd,<emph.end type="italics"></emph.end>deſcribendo arcum Parabolæ <lb></lb><emph type="italics"></emph>QRq,<emph.end type="italics"></emph.end>cujus vertex principalis (juxta demonſtrata <emph type="italics"></emph>Galilæi<emph.end type="italics"></emph.end>) eſt in <lb></lb><emph type="italics"></emph>R<emph.end type="italics"></emph.end>; ſecabit planum <emph type="italics"></emph>Cc<emph.end type="italics"></emph.end>in eodem angulo in <emph type="italics"></emph>q,<emph.end type="italics"></emph.end>ac prius in <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>; dein <lb></lb>pergendo in arcubus parabolicis <emph type="italics"></emph>qp, ph,<emph.end type="italics"></emph.end>&c. </s> <s>arcubus prioribus <lb></lb><emph type="italics"></emph>QP, PH<emph.end type="italics"></emph.end>ſimilibus & æqualibus, ſecabit reliqua plana in iiſdem <lb></lb>angulis in <emph type="italics"></emph>p, h,<emph.end type="italics"></emph.end>&c. </s> <s>ac prius in <emph type="italics"></emph>P, H,<emph.end type="italics"></emph.end>&c. </s> <s>emergetque tandem ea<lb></lb>dem obliquitate in <emph type="italics"></emph>h,<emph.end type="italics"></emph.end>qua incidit in <emph type="italics"></emph>H.<emph.end type="italics"></emph.end>Concipe jam planorum <lb></lb><emph type="italics"></emph>Aa, Bb, Cc, Dd, Ee,<emph.end type="italics"></emph.end>&c. </s> <s>intervalla in infinitum minui & nume<lb></lb>rum augeri, eo ut actio attractionis vel impulſus ſecundum legem <lb></lb>quamcunque aſſignatam continua reddatur; & angulus emergen<lb></lb>tiæ ſemper angulo incidentiæ æqualis exiſtens, eidem etiamnum <lb></lb>manebit æqualis. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/235.jpg" pagenum="207"></pb><arrow.to.target n="note183"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note183"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Harum attractionum haud multum diſſimiles ſunt Lucis reflexi<lb></lb>ones & refractiones, factæ ſecundum datam Secantium rationem, ut <lb></lb>invenit <emph type="italics"></emph>Snellius,<emph.end type="italics"></emph.end>& per conſequens ſecundum datam Sinuum ratio<lb></lb>nem, ut expoſuit <emph type="italics"></emph>Carteſius.<emph.end type="italics"></emph.end>Namque Lucem ſucceſſive propagari <lb></lb>& ſpatio quaſi ſeptem vel octo minutorum primorum a Sole ad <lb></lb>Terram venire, jam conſtat per Phænomena Satellitum <emph type="italics"></emph>Jovis,<emph.end type="italics"></emph.end>Ob<lb></lb>ſervationibus diverſorum Aſtronomorum confirmata. </s> <s>Radii autem <lb></lb>in aere exiſtentes (uti dudum <emph type="italics"></emph>Grimaldus,<emph.end type="italics"></emph.end>luce per foramen in te<lb></lb>nebroſum cubiculum admiſſa, invenit, & ipſe quoque expertus <lb></lb>ſum) in tranſitu ſuo prope corporum vel opaeorum vel perſpicuo<lb></lb>rum angulos (quales ſunt nummorum ex auro, argento & ære cu<lb></lb>ſorum termini rectanguli circulares, & cultrorum, lapidum aut fra<lb></lb>ctorum vitrorum acies) incurvantur circum corpora, quaſi attracti <lb></lb>in eadem; & ex his radiis, qui in tranſitu illo propius accedunt <lb></lb>ad corpora incurvantur magis, qua<lb></lb><figure id="id.039.01.235.1.jpg" xlink:href="039/01/235/1.jpg"></figure><lb></lb>ſi magis attracti, ut ipſe etiam dili<lb></lb>genter obſervavi. </s> <s>In figura deſig<lb></lb>nat <emph type="italics"></emph>s<emph.end type="italics"></emph.end>aciem cultri vel cunei cujuſvis <lb></lb><emph type="italics"></emph>AsB<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>gowog, fnunf, emtme, <lb></lb>dlsld,<emph.end type="italics"></emph.end>ſunt radii, arcubus <emph type="italics"></emph>owo, <lb></lb>nun, mtm, lsl<emph.end type="italics"></emph.end>verſus cultrum <lb></lb>incurvati; idque magis vel mi<lb></lb>nus pro diſtantia eorum a cultro. </s> <s><lb></lb>Cum autem talis incurvatio radio<lb></lb>rum fiat in aere extra cultrum, de<lb></lb>bebunt etiam radii, qui incidunt in cultrum, prius incurvari in aere <lb></lb>quam cultrum attingunt. </s> <s>Et par eſt ratio incidentium in vitrum. </s> <s><lb></lb>Fit igitur refractio, non in puncto incidentiæ, ſed paulatim per <lb></lb>continuam incurvationem radiorum, factam partim in aere ante<lb></lb>quam attingunt vitrum, partim (ni fallor) in vitro, poſtquam illud <lb></lb>ingreſſi ſunt: uti in radiis <emph type="italics"></emph>ckzkc, biyib, ahxha<emph.end type="italics"></emph.end>incidentibus ad <lb></lb><emph type="italics"></emph>r, q, p,<emph.end type="italics"></emph.end>& inter <emph type="italics"></emph>k<emph.end type="italics"></emph.end>& <emph type="italics"></emph>z, i<emph.end type="italics"></emph.end>& <emph type="italics"></emph>y, h<emph.end type="italics"></emph.end>& <emph type="italics"></emph>x<emph.end type="italics"></emph.end>incurvatis, delineatum eſt. </s> <s><lb></lb>Igitur ob analogiam quæ eſt inter propagationem radiorum lucis <lb></lb>& progreſſum corporum, viſum eſt Propoſitiones ſequentes in uſus <lb></lb>Opticos ſubjungere; interea de natura radiorum (utrum ſint cor<lb></lb>pora necne) nihil omnino diſputans, ſed Trajectorias corporum <lb></lb>Trajectoriis radiorum perſimiles ſolummodo determinans. <pb xlink:href="039/01/236.jpg" pagenum="208"></pb><arrow.to.target n="note184"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note184"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XCVII. PROBLEMA XLVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſito quod ſinus incidentiæ in ſuperficiem aliquam ſit ad ſinum e<lb></lb>mergentiæ in data ratione, quodQ.E.I.curvatio viæ corporum <lb></lb>juxta ſuperficiem illam fiat in ſpatio breviſſimo, quod ut pun<lb></lb>ctum conſiderari poſſit; determinare ſuperficiem quæ corpuſcula <lb></lb>omnia de loco dato ſucceſſive manantia convergere faciat ad <lb></lb>alium locum datum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>A<emph.end type="italics"></emph.end>locus a quo corpuſcula divergunt; <emph type="italics"></emph>B<emph.end type="italics"></emph.end>locus in quem con<lb></lb>vergere debent; <emph type="italics"></emph>CDE<emph.end type="italics"></emph.end>curva linea quæ circa axem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>revoluta <lb></lb>deſcribat ſuperficiem quæſitam; <emph type="italics"></emph>D, E<emph.end type="italics"></emph.end>curvæ illius puncta duo quæ<lb></lb>vis; & <emph type="italics"></emph>EF, EG<emph.end type="italics"></emph.end>perpendicula in corporis vias <emph type="italics"></emph>AD, DB<emph.end type="italics"></emph.end>demiſſa. </s> <s><lb></lb>Accedat punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ad punctum <emph type="italics"></emph>E<emph.end type="italics"></emph.end>; & lineæ <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>qua <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>au<lb></lb>getur, ad lineam <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>qua <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>diminuitur, ratio ultima erit ea<lb></lb>dem quæ ſinus incidentiæ ad ſinum emergentiæ. </s> <s>Datur ergo ratio <lb></lb><figure id="id.039.01.236.1.jpg" xlink:href="039/01/236/1.jpg"></figure><lb></lb>incrementi lineæ <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>ad decrementum lineæ <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>; & propterea <lb></lb>ſi in axe <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ſumatur ubivis punctum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>per quod curva <emph type="italics"></emph>CDE<emph.end type="italics"></emph.end><lb></lb>tranſire debet, & capiatur ipſius <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>incrementum <emph type="italics"></emph>CM,<emph.end type="italics"></emph.end>ad ipſius <lb></lb><emph type="italics"></emph>BC<emph.end type="italics"></emph.end>decrementum <emph type="italics"></emph>CN<emph.end type="italics"></emph.end>in data illa ratione; centriſque <emph type="italics"></emph>A, B,<emph.end type="italics"></emph.end>& in<lb></lb>tervallis <emph type="italics"></emph>AM, BN<emph.end type="italics"></emph.end>deſcribantur circuli duo ſe mutuo ſecantes in <lb></lb><emph type="italics"></emph>D:<emph.end type="italics"></emph.end>punctum illud <emph type="italics"></emph>D<emph.end type="italics"></emph.end>tanget curvam quæſitam <emph type="italics"></emph>CDE,<emph.end type="italics"></emph.end>eandemque <lb></lb>ubivis tangendo determinabit. <emph type="italics"></emph><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Faciendo autem ut punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>B<emph.end type="italics"></emph.end>nunc abeat in in<lb></lb>finitum, nunc migret ad alteras partes puncti <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>habebuntur Fi<lb></lb>guræ illæ omnes quas <emph type="italics"></emph>Carteſius<emph.end type="italics"></emph.end>in Optica & Geometria ad Refra<lb></lb>ctiones expoſuit. </s> <s>Quarum inventionem cum <emph type="italics"></emph>Carteſius<emph.end type="italics"></emph.end>maximi <lb></lb>fecerit & ſtudioſe celaverit, viſum fuit hac propoſitione expo<lb></lb>nere. </s></p><pb xlink:href="039/01/237.jpg" pagenum="209"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si corpus in ſuperficiem quamvis <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end>ſecundum lineam <lb></lb><arrow.to.target n="note185"></arrow.to.target>rectam <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>lege quavis ductam incidens, emergat ſecundum aliam <lb></lb>quamvis rectam <emph type="italics"></emph>DK,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.237.1.jpg" xlink:href="039/01/237/1.jpg"></figure><lb></lb>& a puncto <emph type="italics"></emph>C<emph.end type="italics"></emph.end>duci in<lb></lb>telligantur Lineæ curvæ <lb></lb><emph type="italics"></emph>CP, CQ<emph.end type="italics"></emph.end>ipſis <emph type="italics"></emph>AD, DK<emph.end type="italics"></emph.end><lb></lb>ſemper perpendiculares: <lb></lb>erunt incrementa linea<lb></lb>rum <emph type="italics"></emph>PD, QD,<emph.end type="italics"></emph.end><expan abbr="atq;">atque</expan> ad<lb></lb>eo lineæ ipſæ <emph type="italics"></emph>PD, QD,<emph.end type="italics"></emph.end><lb></lb>incrementis iſtis genitæ, <lb></lb>ut ſinus incidentiæ & e<lb></lb>mergentiæ ad invicem: <lb></lb>& contra. </s></p> <p type="margin"> <s><margin.target id="note185"></margin.target>LIBER <lb></lb>PRIMUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XCVIII. PROBLEMA XLVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, & circa axem<emph.end type="italics"></emph.end>AB <emph type="italics"></emph>deſcripta ſuperficie quacunque <lb></lb>attractiva<emph.end type="italics"></emph.end>CD, <emph type="italics"></emph>regulari vel irregulari, per quam corpora de <lb></lb>loco dato<emph.end type="italics"></emph.end>A <emph type="italics"></emph>exeuntia tranſire debent: invenire ſuperficiem ſe<lb></lb>cundam attractivam<emph.end type="italics"></emph.end>EF, <emph type="italics"></emph>quæ corpora illa ad locum datum<emph.end type="italics"></emph.end>B <lb></lb><emph type="italics"></emph>convergere faciat.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Juncta <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ſecet ſuperficiem primam in <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& ſecundam in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end><lb></lb>puncto <emph type="italics"></emph>D<emph.end type="italics"></emph.end>utcunque aſſumpto. </s> <s>Et poſito ſinu incidentiæ in ſuper<lb></lb>ficiem primam ad ſinum emergentiæ ex eadem, & ſinu emergentiæ <lb></lb>e ſuperficie ſecunda ad ſinum incidentiæ in eandem, ut quantitas <lb></lb>aliqua data M ad aliam datam N; produc tum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>BG<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ut M-N ad N, tum <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>AG,<emph.end type="italics"></emph.end>tum <lb></lb>etiam <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>K<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DH<emph.end type="italics"></emph.end>ut N ad M. </s> <s>Junge <emph type="italics"></emph>KB,<emph.end type="italics"></emph.end>& <lb></lb>centro <emph type="italics"></emph>D<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>DH<emph.end type="italics"></emph.end>deſcribe circulum occurrentem <emph type="italics"></emph>KB<emph.end type="italics"></emph.end>pro<lb></lb>ductæ in <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>ipſique <emph type="italics"></emph>DL<emph.end type="italics"></emph.end>parallelam age <emph type="italics"></emph>BF:<emph.end type="italics"></emph.end>& punctum <emph type="italics"></emph>F<emph.end type="italics"></emph.end>tan<lb></lb>get Lineam <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>quæ circa axem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>revoluta deſcribet ſuperfi<lb></lb>ciem quæſitam. <emph type="italics"></emph><expan abbr="q.">que</expan> E. F.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam concipe Lineas <emph type="italics"></emph>CP, CQ<emph.end type="italics"></emph.end>ipſis <emph type="italics"></emph>AD, DF<emph.end type="italics"></emph.end>reſpective, & Li<lb></lb>neas <emph type="italics"></emph>ER, ES<emph.end type="italics"></emph.end>ipſis <emph type="italics"></emph>FB, FD<emph.end type="italics"></emph.end>ubique perpendiculares eſſe, adeoque <lb></lb><emph type="italics"></emph>QS<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ſemper æqualem; & erit (per Corol. </s> <s>2. Prop. </s> <s>XCVII) <lb></lb><emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>QD<emph.end type="italics"></emph.end>ut M ad N, adeoque ut <emph type="italics"></emph>DL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>FB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FK<emph.end type="italics"></emph.end>; <pb xlink:href="039/01/238.jpg" pagenum="210"></pb><arrow.to.target n="note186"></arrow.to.target>& diviſim ut <emph type="italics"></emph>DL-FP<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>PH-PD-FB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FD<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>FQ-QD<emph.end type="italics"></emph.end>; <lb></lb>& compoſite ut <emph type="italics"></emph>PH-FB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FQ,<emph.end type="italics"></emph.end>id eſt (ob æquales <emph type="italics"></emph>PH<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>CG, QS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CE) <lb></lb><figure id="id.039.01.238.1.jpg" xlink:href="039/01/238/1.jpg"></figure><lb></lb>CE+BG-FR<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>CE-FS.<emph.end type="italics"></emph.end>Verum (ob <lb></lb>proportionales <emph type="italics"></emph>BG<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>CE<emph.end type="italics"></emph.end>& M-N ad N) <lb></lb>eſt etiam <emph type="italics"></emph>CE+BG<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>CE<emph.end type="italics"></emph.end>ut M ad N: adeoque <lb></lb>diviſim <emph type="italics"></emph>FR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FS<emph.end type="italics"></emph.end>ut <lb></lb>M ad N, & propterea per <lb></lb>Corol. </s> <s>2. Prop. </s> <s>XCVII, <lb></lb>ſuperficies <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>cogit cor<lb></lb>pus, in ipſam ſecundum lineam <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>incidens, pergere in linea <emph type="italics"></emph>FR<emph.end type="italics"></emph.end><lb></lb>ad locum <emph type="italics"></emph>B. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note186"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Eadem methodo pergere liceret ad ſuperficies tres vel plures. </s> <s><lb></lb>Ad uſus autem Opticos maxime accommodatæ ſunt figuræ Sphæ<lb></lb>ricæ. </s> <s>Si Perſpicillorum vitra Objectiva ex vitris duobus Sphæri<lb></lb>ce figuratis & Aquam inter ſe claudentibus conflentur; fieri poteſt <lb></lb>ut a refractionibus Aquæ errores refractionum, quæ fiunt in vitro<lb></lb>rum ſuperficiebus extremis, ſatis accurate corrigantur. </s> <s>Talia au<lb></lb>tem vitra Objectiva vitris Ellipticis & Hyperbolicis præferenda <lb></lb>ſunt, non ſolum quod facilius & accuratius formari poſſint, ſed <lb></lb>etiam quod Penicillos radiorum extra axem vitri ſitos accurativs <lb></lb>refringant. </s> <s>Verum tamen diverſa diverſorum radiorum Refrangi<lb></lb>bilitas impedimento eſt, quo minus Optica per Figuras vel Sphæ<lb></lb>ricas vel alias quaſcunque perfici poſſit. </s> <s>Niſi corrigi poſſint er<lb></lb>rores illinc oriundi, labor omnis in cæteris corrigendis imperite <lb></lb>collocabitur. <pb xlink:href="039/01/239.jpg" pagenum="211"></pb><arrow.to.target n="note187"></arrow.to.target></s></p></subchap2></subchap1><subchap1><subchap2> <p type="margin"> <s><margin.target id="note187"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>DE <lb></lb>MOTU CORPORUM <lb></lb>LIBER SECUNDUS.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="center"></emph>SECTIO I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Motu Corporum quibus reſiſtitur in ratione <lb></lb>Velocitatis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO I. THEOREMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Corporis, cui reſiſtitur in ratione velocitatis, motus ex reſiſtentia <lb></lb>amiſſus eſt ut ſpatium movendo confectum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>NAm cum motus ſingulis temporis particulis æqualibus amiſſus <lb></lb>ſit ut velocitas, hoc eſt, ut itineris confecti particula: erit, <lb></lb>componendo, motus toto tempore amiſſus ut iter totum. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Igitur ſi corpus, gravitate omni deſtitutum, in ſpatiis libe<lb></lb>ris ſola vi inſita moveatur; ac detur tum motus totus ſub initio, tum <lb></lb>etiam motus reliquus poſt ſpatium aliquod confectum: dabitur ſpa<lb></lb>tium totum quod corpus infinito tempore deſcribere poteſt. </s> <s>Erit <lb></lb>enim ſpatium illud ad ſpatium jam deſcriptum, ut motus totus ſub <lb></lb>initio ad motus illius partem amiſſam. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Quantitates differentiis ſuis proportionales, ſunt continue propor<lb></lb>tionales.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></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></lb>fiet A ad B ut B ad C & C ad D, &c. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/240.jpg" pagenum="212"></pb><arrow.to.target n="note188"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note188"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO II. THEOREMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Corpori reſiſtitur in ratione velocitatis, & idem ſola vi inſita <lb></lb>per Medium ſimilare moveatur, ſumantur autem tempora æqua<lb></lb>lia: velocitates in principiis ſingulorum temporum ſunt in pro<lb></lb>greſſione Geometrica, & ſpatia ſingulis temporibus deſcripta <lb></lb>ſunt ut velocitates.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Dividatur tempus in particulas æquales; & ſi ipſis parti<lb></lb>cularum initiis agat vis reſiſtentiæ impulſo unico, quæ ſit ut velo<lb></lb>citas: erit decrementum velocitatis ſingulis temporis particulis ut <lb></lb>eadem velocitas. </s> <s>Sunt ergo velocitates differentiis ſuis proportio<lb></lb>nales, & propterea (per Lem. </s> <s>I. Lib. </s> <s>II.) continue proportionales. </s> <s><lb></lb>Proinde ſi ex æquali particularum numero componantur tempora <lb></lb>quælibet æqualia, erunt velocitates ipſis temporum initiis, ut ter<lb></lb>mini in progreſſione continua, qui per ſaltum capiuntur, omiſſo <lb></lb>paſſim æquali terminorum intermediorum numero. </s> <s>Componuntur <lb></lb>autem horum terminorum rationes ex æqualibus rationibus termi<lb></lb>norum intermediorum æqualiter repetitis, & propterea ſunt æqua<lb></lb>les. </s> <s>Igitur velocitates, his terminis proportionales, ſunt in pro<lb></lb>greſſione Geometrica. </s> <s>Minuantur jam æquales illæ temporum par<lb></lb>ticulæ, & augeatur earum numerus in infinitum, eo ut reſiſtentiæ <lb></lb>impulſus reddatur continuus; & velocitates in principiis æqualium <lb></lb>temporum, ſemper continue proportionales, erunt in hoc etiam <lb></lb>caſu continue proportionales. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Et diviſim velocitatum differentiæ, hoc eſt, earum partes <lb></lb>ſingulis temporibus amiſſæ, ſunt ut totæ: Spatia autem ſingulis <lb></lb>temporibus deſcripta ſunt ut velocitatum partes amiſſæ, (per Prop. </s> <s><lb></lb>I. </s> <s>Lib II.) & propterea etiam ut totæ. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc ſi Aſymptotis rectangulis <emph type="italics"></emph>ADC, CH<emph.end type="italics"></emph.end>deſcribatur <lb></lb>Hyperbola <emph type="italics"></emph>BG,<emph.end type="italics"></emph.end>ſintque <emph type="italics"></emph>AB, DG<emph.end type="italics"></emph.end>ad Aſymptoton <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>perpen<lb></lb>diculares, & exponatur tum corporis velocitas tum reſiſtentia Me<lb></lb>dii, ipſo motus initio, per lineam quam<lb></lb><figure id="id.039.01.240.1.jpg" xlink:href="039/01/240/1.jpg"></figure><lb></lb>vis datam <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>elapſo autem tempore ali<lb></lb>quo per lineam indefinitam <emph type="italics"></emph>DC:<emph.end type="italics"></emph.end>exponi <lb></lb>poteſt tempus per aream <emph type="italics"></emph>ABGD,<emph.end type="italics"></emph.end>& ſpa<lb></lb>tium eo tempore deſcriptum per lineam <lb></lb><emph type="italics"></emph>AD.<emph.end type="italics"></emph.end>Nam ſi area illa per motum puncti <lb></lb><emph type="italics"></emph>D<emph.end type="italics"></emph.end>augeatur uniformiter ad modum tempo-<pb xlink:href="039/01/241.jpg" pagenum="213"></pb>ris, decreſcet recta <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>in ratione Geometrica ad modum veloci<lb></lb><arrow.to.target n="note189"></arrow.to.target>tatis, & partes rectæ <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>æqualibus temporibus deſcriptæ decre<lb></lb>ſcent in eadem ratione. </s></p> <p type="margin"> <s><margin.target id="note189"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO III. PROBLEMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corporis, cui dum in Medio ſimilari recta aſcendit vel deſcendit, <lb></lb>reſiſtitur in ratione velocitatis, quodque ab uniformi gravitate <lb></lb>urgetur, definire motum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Corpore aſcendente, ex<lb></lb><figure id="id.039.01.241.1.jpg" xlink:href="039/01/241/1.jpg"></figure><lb></lb>ponatur gravitas per datum <lb></lb>quodvis rectangulum <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>& <lb></lb>reſiſtentia Medii initio aſ<lb></lb>cenſus per rectangulum <emph type="italics"></emph>BD<emph.end type="italics"></emph.end><lb></lb>ſumptum ad contrarias par<lb></lb>tes. </s> <s>Aſymptotis rectangulis <lb></lb><emph type="italics"></emph>AC, CH,<emph.end type="italics"></emph.end>per punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>de<lb></lb>ſcribatur Hyperbola ſecans per<lb></lb>pendicula <emph type="italics"></emph>DE, de<emph.end type="italics"></emph.end>in <emph type="italics"></emph>G, g;<emph.end type="italics"></emph.end>& <lb></lb>corpus aſcendendo, tempore <emph type="italics"></emph>DGgd,<emph.end type="italics"></emph.end>deſcribet ſpatium <emph type="italics"></emph>EGge,<emph.end type="italics"></emph.end>tem<lb></lb>pore <emph type="italics"></emph>DGBA<emph.end type="italics"></emph.end>ſpatium aſcenſus totius <emph type="italics"></emph>EGB<emph.end type="italics"></emph.end>; tempore <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>2<emph type="italics"></emph>G<emph.end type="italics"></emph.end>2<emph type="italics"></emph>D<emph.end type="italics"></emph.end><lb></lb>ſpatium deſcenſus <emph type="italics"></emph>BF<emph.end type="italics"></emph.end>2<emph type="italics"></emph>G,<emph.end type="italics"></emph.end>atque tempore 2<emph type="italics"></emph>D<emph.end type="italics"></emph.end>2<emph type="italics"></emph>G<emph.end type="italics"></emph.end>2<emph type="italics"></emph>g<emph.end type="italics"></emph.end>2<emph type="italics"></emph>d<emph.end type="italics"></emph.end>ſpatium <lb></lb>deſcenſus 2<emph type="italics"></emph>GF<emph.end type="italics"></emph.end>2<emph type="italics"></emph>e<emph.end type="italics"></emph.end>2<emph type="italics"></emph>g<emph.end type="italics"></emph.end>: & velocitates corporis (reſiſtentiæ Medii <lb></lb>proportionales) in horum temporum periodis erunt <emph type="italics"></emph>ABED, <lb></lb>ABed,<emph.end type="italics"></emph.end>nulla, <emph type="italics"></emph>ABF<emph.end type="italics"></emph.end>2<emph type="italics"></emph>D, AB<emph.end type="italics"></emph.end>2<emph type="italics"></emph>e<emph.end type="italics"></emph.end>2<emph type="italics"></emph>d<emph.end type="italics"></emph.end>reſpective; atque maxima <lb></lb>velocitas, quam corpus deſcendendo poteſt acquirere, erit <emph type="italics"></emph>BC.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Reſolvatur enim rectan<lb></lb><figure id="id.039.01.241.2.jpg" xlink:href="039/01/241/2.jpg"></figure><lb></lb>gulum <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>in rectangula <lb></lb>innumera <emph type="italics"></emph>Ak, Kl, Lm, Mn,<emph.end type="italics"></emph.end><lb></lb>&c. </s> <s>quæ ſint ut incrementa <lb></lb>velocitatum æqualibus tot<lb></lb>idem temporibus facta; & e<lb></lb>runt nihil, <emph type="italics"></emph>Ak, Al, Am, An,<emph.end type="italics"></emph.end><lb></lb>&c. </s> <s>ut velocitates totæ, at<lb></lb>que adeo (per Hypotheſin) <lb></lb>ut reſiſtentiæ Medii princi<lb></lb>pio ſingulorum temporum <lb></lb>æqualium. </s> <s>Fiat <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>ABHC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ABkK,<emph.end type="italics"></emph.end>ut vis gra<lb></lb>vitatis ad reſiſtentiam in principio temporis ſecundi, deque vi gravi-<pb xlink:href="039/01/242.jpg" pagenum="214"></pb><arrow.to.target n="note190"></arrow.to.target>tatis ſubducantur reſiſtentiæ, & manebunt <emph type="italics"></emph>ABHC, KkHC, LlHC, <lb></lb>NnHC,<emph.end type="italics"></emph.end>&c. </s> <s>ut vires abſolutæ quibus corpus in principio ſingu<lb></lb>lorum temporum urgetur, atque adeo (per motus Legem 11) ut <lb></lb>incrementa velocitatum, id eſt, ut rectangula <emph type="italics"></emph>Ak, Kl, Lm, Mn,<emph.end type="italics"></emph.end>&c; <lb></lb>& propterea (per Lem. </s> <s>I. Lib. </s> <s>II) in progreſſione Geometrica. </s> <s>Qua<lb></lb>re ſi rectæ <emph type="italics"></emph>Kk, Ll, Mm, Nn,<emph.end type="italics"></emph.end>&c. </s> <s>productæ occurrant Hyperbolæ <lb></lb>in <emph type="italics"></emph>q, r, s, t,<emph.end type="italics"></emph.end>&c. </s> <s>erunt areæ <emph type="italics"></emph>ABqK, KqrL, LrsM, MstN,<emph.end type="italics"></emph.end>&c. <lb></lb></s> <s>æquales, adeoque tum temporibus tum viribus gravitatis ſemper <lb></lb>æqualibus analogæ. </s> <s>Eſt autem area <emph type="italics"></emph>ABqK<emph.end type="italics"></emph.end>(per Corol. </s> <s>3. Lem. </s> <s>VII, <lb></lb>& Lem. </s> <s>VIII, Lib. </s> <s>I) ad aream <emph type="italics"></emph>Bkq<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Kq<emph.end type="italics"></emph.end>ad 1/2 <emph type="italics"></emph>kq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad 1/2 <emph type="italics"></emph>AK,<emph.end type="italics"></emph.end><lb></lb>hoc eſt, ut vis gravitatis ad reſiſtentiam in medio temporis primi. </s> <s><lb></lb>Et ſimili argumento areæ <lb></lb><figure id="id.039.01.242.1.jpg" xlink:href="039/01/242/1.jpg"></figure><lb></lb><emph type="italics"></emph>qKLr, rLMs, sMNt,<emph.end type="italics"></emph.end>&c. <lb></lb></s> <s>ſunt ad areas <emph type="italics"></emph>qklr, rlms, <lb></lb>smnt,<emph.end type="italics"></emph.end>&c. </s> <s>ut vires gravi<lb></lb>tatis ad reſiſtentias in me<lb></lb>dio temporis ſecundi, ter<lb></lb>tii, quarti, &c. </s> <s>Proinde cum <lb></lb>areæ æquales <emph type="italics"></emph>BAKq, qKLr, <lb></lb>rLMs, sMNt,<emph.end type="italics"></emph.end>&c. </s> <s>ſint vi<lb></lb>ribus gravitatis analogæ, e<lb></lb>runt areæ <emph type="italics"></emph>Bkq, qklr, rlms, <lb></lb>smnt,<emph.end type="italics"></emph.end>&c. </s> <s>reſiſtentiis in mediis ſingulorum temporum, hoc eſt (per <lb></lb>Hypotheſin) velocitatibus, atque adeo deſcriptis ſpatiis analogæ. </s> <s><lb></lb>Sumantur analogarum ſummæ, & erunt areæ <emph type="italics"></emph>Bkq, Blr, Bms, Bnt,<emph.end type="italics"></emph.end><lb></lb>&c. </s> <s>ſpatiis totis deſcriptis analogæ; necnon areæ <emph type="italics"></emph>ABqK, ABrL, <lb></lb>ABsM, ABtN,<emph.end type="italics"></emph.end>&c. </s> <s>temporibus. </s> <s>Corpus igitur inter deſcenden<lb></lb>dum, tempore quovis <emph type="italics"></emph>ABrL,<emph.end type="italics"></emph.end>deſcribit ſpatium <emph type="italics"></emph>Blr,<emph.end type="italics"></emph.end>& tempore <lb></lb><emph type="italics"></emph>LrtN<emph.end type="italics"></emph.end>ſpatium <emph type="italics"></emph>rlnt. </s> <s>Q.E.D.<emph.end type="italics"></emph.end>Et ſimilis eſt demonſtratio motus <lb></lb>expoſiti in aſcenſu. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note190"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur velocitas maxima, quam corpus cadendo poteſt <lb></lb>acquirere, eſt ad velocitatem dato quovis tempore acquiſitam, ut<lb></lb>vis data gravitatis qua perpetuo urgetur, ad vim reſiſtentiæ qua in<lb></lb>fine temporis illius impeditur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Tempore autem aucto in progreſſione Arithmetica, ſumma<lb></lb>velocitatis illius maximæ ac velocitatis in aſcenſu (atque etiam earun <lb></lb>dem differentia in deſcenſu) decreſcit in progreſſione Geometrica. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Sed & differentiæ ſpatiorum, quæ in æqualibus tempo <lb></lb>rum differentiis deſcribuntur, decreſcunt in eadem progreſſion <lb></lb>Geometrica. </s></p><pb xlink:href="039/01/243.jpg" pagenum="215"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Spatium vero a corpore deſcriptum differentia eſt duo<lb></lb><arrow.to.target n="note191"></arrow.to.target>rum ſpatiorum, quorum alterum eſt ut tempus ſumptum ab initio <lb></lb>deſcenſus, & alterum ut velocitas, quæ etiam ipſo deſcenſus initio <lb></lb>æquantur inter ſe. </s></p> <p type="margin"> <s><margin.target id="note191"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO IV. PROBLEMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſito quod vis gravitatis in Medio aliquo ſimilari uniformis ſit, <lb></lb>ac tendat perpendiculariter ad planum Horizontis; definire mo<lb></lb>tum Projectilis in eodem, reſiſtentiam velocitati proportionalem <lb></lb>patientis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Eloco quovis <emph type="italics"></emph>D<emph.end type="italics"></emph.end>egrediatur Pro<lb></lb><figure id="id.039.01.243.1.jpg" xlink:href="039/01/243/1.jpg"></figure><lb></lb>jectile ſecundum lineam quam<lb></lb>vis rectam <emph type="italics"></emph>DP,<emph.end type="italics"></emph.end>& per longitu<lb></lb>dinem <emph type="italics"></emph>DP<emph.end type="italics"></emph.end>exponatur ejuſdem <lb></lb>velocitas ſub initio motus. </s> <s>A <lb></lb>puncto <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ad lineam Horizonta<lb></lb>lem <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>demittatur perpendi<lb></lb>culum <emph type="italics"></emph>PC,<emph.end type="italics"></emph.end>& ſecetur <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>in <emph type="italics"></emph>A<emph.end type="italics"></emph.end><lb></lb>ut ſit <emph type="italics"></emph>DA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ut reſiſtentia <lb></lb>Medii, ex motu in altitudinem <lb></lb>ſub initio orta, ad vim gravi<lb></lb>tatis; vel (quod perinde eſt) ut <lb></lb>ſit rectangulum ſub <emph type="italics"></emph>DA<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DP<emph.end type="italics"></emph.end><lb></lb>ad rectangulum ſub <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CP<emph.end type="italics"></emph.end><lb></lb>ut reſiſtentia tota ſub initio mo<lb></lb>tus ad vim gravitatis. </s> <s>Aſymptotis <lb></lb><emph type="italics"></emph>DC, CP,<emph.end type="italics"></emph.end>deſcribatur Hyperbo<lb></lb>la quævis <emph type="italics"></emph>GTBS<emph.end type="italics"></emph.end>ſecans perpen<lb></lb>dicula <emph type="italics"></emph>DG, AB<emph.end type="italics"></emph.end>in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B<emph.end type="italics"></emph.end>; & <lb></lb>compleatur parallelogrammum <lb></lb><emph type="italics"></emph>DGKC,<emph.end type="italics"></emph.end>cujus latus <emph type="italics"></emph>GK<emph.end type="italics"></emph.end>ſecet <lb></lb><emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in <emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end>Capiatur linea N in <lb></lb>ratione ad <emph type="italics"></emph>QB<emph.end type="italics"></emph.end>qua <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>ſit ad <lb></lb><emph type="italics"></emph>CP<emph.end type="italics"></emph.end>; & ad rectæ <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>pun<lb></lb>ctum quodvis <emph type="italics"></emph>R<emph.end type="italics"></emph.end>erecto perpen<lb></lb>diculo <emph type="italics"></emph>RT,<emph.end type="italics"></emph.end>quod Hyperbolæ <lb></lb>in <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>& rectis <emph type="italics"></emph>EH, GK, DP<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>I, t<emph.end type="italics"></emph.end>& <emph type="italics"></emph>V<emph.end type="italics"></emph.end>occurrat; in eo cape <emph type="italics"></emph>Vr<emph.end type="italics"></emph.end>æqualem (<emph type="italics"></emph>tGT<emph.end type="italics"></emph.end>/N), vel quod per-<pb xlink:href="039/01/244.jpg" pagenum="216"></pb><arrow.to.target n="note192"></arrow.to.target>inde eſt, cape <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>æqualem (<emph type="italics"></emph>GTIE<emph.end type="italics"></emph.end>/N); & Projectile tempore <emph type="italics"></emph>DRTG<emph.end type="italics"></emph.end><lb></lb>perveniet ad punctum <emph type="italics"></emph>r,<emph.end type="italics"></emph.end>deſcribens curvam lineam <emph type="italics"></emph>DraF,<emph.end type="italics"></emph.end>quam <lb></lb>punctum <emph type="italics"></emph>r<emph.end type="italics"></emph.end>ſemper tangit, perveniens autem ad maximam altitudi<lb></lb>nem <emph type="italics"></emph>a<emph.end type="italics"></emph.end>in perpendiculo <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>& poſtea ſemper appropinquans ad A<lb></lb>ſymptoton <emph type="italics"></emph>PLC.<emph.end type="italics"></emph.end>Eſtque velocitas ejus in puncto quovis <emph type="italics"></emph>r<emph.end type="italics"></emph.end>ut Cur<lb></lb>væ Tangens <emph type="italics"></emph>rL. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note192"></margin.target>DE MOTU <lb></lb>CORPORUN</s></p> <p type="main"> <s>Eſt enim N ad <emph type="italics"></emph>QB<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>DR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>RV,<emph.end type="italics"></emph.end>adeoque <emph type="italics"></emph>RV<emph.end type="italics"></emph.end><lb></lb>æqualis (<emph type="italics"></emph>DRXQB<emph.end type="italics"></emph.end>/N), & <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>(id eſt <emph type="italics"></emph>RV-Vr<emph.end type="italics"></emph.end>ſeu (<emph type="italics"></emph>DRXQB-tGT<emph.end type="italics"></emph.end>/N)) <lb></lb>æqualis (<emph type="italics"></emph>DRXAB-RDGT<emph.end type="italics"></emph.end>/N). Exponatur jam tempus per are<lb></lb>am <emph type="italics"></emph>RDGT,<emph.end type="italics"></emph.end>& (per Legum <lb></lb><figure id="id.039.01.244.1.jpg" xlink:href="039/01/244/1.jpg"></figure><lb></lb>Corol. </s> <s>2.) diſtinguatur motus <lb></lb>corporis in duos, unum aſcen<lb></lb>ſus, alterum ad latus. </s> <s>Et cum <lb></lb>reſiſtentia ſit ut motus, diſtin<lb></lb>guetur etiam hæc in partes duas <lb></lb>partibus motus proportionales <lb></lb>& contrarias: ideoque longitu<lb></lb>do, a motu ad latus deſcripta, e<lb></lb>rit (per Prop. </s> <s>11. hujus) ut linea <lb></lb><emph type="italics"></emph>DR,<emph.end type="italics"></emph.end>altitudo vero (per Prop. </s> <s><lb></lb>111. hujus) ut area <emph type="italics"></emph>DRXAB <lb></lb>-RDGT,<emph.end type="italics"></emph.end>hoc eſt, ut linea <emph type="italics"></emph>Rr.<emph.end type="italics"></emph.end><lb></lb>Ipſo autem motus initio area <lb></lb><emph type="italics"></emph>RDGT<emph.end type="italics"></emph.end>æqualis eſt rectangulo <lb></lb><emph type="italics"></emph>DRXAQ,<emph.end type="italics"></emph.end>ideoque linea illa <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end><lb></lb>(ſeu (<emph type="italics"></emph>DRXAB-DRXAQ<emph.end type="italics"></emph.end>/N)) <lb></lb>tunc eſt ad <emph type="italics"></emph>DR<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AB-AQ<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>QB<emph.end type="italics"></emph.end>ad N, id eſt, ut <emph type="italics"></emph>CP<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>; atque adeo ut motus <lb></lb>in altitudinem ad motum in <lb></lb>longitudinem ſub initio. </s> <s>Cum <lb></lb>igitur <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>ſemper ſit ut altitu<lb></lb>do, ac <emph type="italics"></emph>DR<emph.end type="italics"></emph.end>ſemper ut longi<lb></lb>tudo, atque <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DR<emph.end type="italics"></emph.end>ſub <lb></lb>initio ut altitudo ad longitudinem: neceſſe eſt ut <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>ſemper ſit ad <lb></lb><emph type="italics"></emph>DR<emph.end type="italics"></emph.end>ut altitudo ad longitudinem, & propterea ut corpus movea<lb></lb>tur in linea <emph type="italics"></emph>DraF,<emph.end type="italics"></emph.end>quam punctum <emph type="italics"></emph>r<emph.end type="italics"></emph.end>perpetuo tangit. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/245.jpg" pagenum="217"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Eſt igitur <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>æqualis (<emph type="italics"></emph>DRXAB<emph.end type="italics"></emph.end>/N)-(<emph type="italics"></emph>RDGT<emph.end type="italics"></emph.end>/N), ideoque <lb></lb><arrow.to.target n="note193"></arrow.to.target>ſi producatur <emph type="italics"></emph>RT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>X<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>RX<emph.end type="italics"></emph.end>æqualis (<emph type="italics"></emph>DRXAB<emph.end type="italics"></emph.end>/N), (id eſt, ſi <lb></lb>compleatur parallelogrammum <emph type="italics"></emph>ACPY,<emph.end type="italics"></emph.end>jungatur <emph type="italics"></emph>DY<emph.end type="italics"></emph.end>ſecans <emph type="italics"></emph>CP<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>Z,<emph.end type="italics"></emph.end>& producatur <emph type="italics"></emph>RT<emph.end type="italics"></emph.end>donec occurrat <emph type="italics"></emph>DY<emph.end type="italics"></emph.end>in <emph type="italics"></emph>X<emph.end type="italics"></emph.end>;) erit <emph type="italics"></emph>Xr<emph.end type="italics"></emph.end>æqua<lb></lb>lis (<emph type="italics"></emph>RDGT<emph.end type="italics"></emph.end>/N), & propterea tempori proportionalis. </s></p> <p type="margin"> <s><margin.target id="note193"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Unde ſi capiantur innumeræ <emph type="italics"></emph>CR<emph.end type="italics"></emph.end>vel, quod perinde eſt, <lb></lb>innumeræ Z<emph type="italics"></emph>X,<emph.end type="italics"></emph.end>in progreſſione Geometrica; erunt totidem <emph type="italics"></emph>Xr<emph.end type="italics"></emph.end>in <lb></lb>progreſſione Arithmetica. </s> <s>Et hinc Curva <emph type="italics"></emph>DraF<emph.end type="italics"></emph.end>per tabulam Lo<lb></lb>garithmorum facile delineatur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si vertice <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>diametro <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>deorſum producta, & La<lb></lb>tere recto quod ſit ad 2<emph type="italics"></emph>DP<emph.end type="italics"></emph.end>ut reſiſtentia tota, ipſo motus initio, <lb></lb>ad vim gravitatis, Parabola conſtruatur: velocitas quacum corpus <lb></lb>exire debet de loco <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ſecundum rectam <emph type="italics"></emph>DP,<emph.end type="italics"></emph.end>ut in Medio uNI<lb></lb>formi reſiſtente deſcribat Curvam <emph type="italics"></emph>DraF,<emph.end type="italics"></emph.end>ea ipſa erit quacum ex<lb></lb>ire debet de eodem loco <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ſecundum eandem rectam <emph type="italics"></emph>DP,<emph.end type="italics"></emph.end>ut <lb></lb>in ſpatio non reſiſtente deſcribat Parabolam. </s> <s>Nam Latus re<lb></lb>ctum Parabolæ hujus, ipſo motus initio, eſt (<emph type="italics"></emph>DVquad./Vr<emph.end type="italics"></emph.end>) & <emph type="italics"></emph>Vr<emph.end type="italics"></emph.end><lb></lb>eſt (<emph type="italics"></emph>tGT<emph.end type="italics"></emph.end>/N) ſeu (<emph type="italics"></emph>DRXTt<emph.end type="italics"></emph.end>/2N). Recta autem quæ, ſi duceretur, Hy<lb></lb>perbolam <emph type="italics"></emph>GTB<emph.end type="italics"></emph.end>tangeret in <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>parallela eſt ipſi <emph type="italics"></emph>DK,<emph.end type="italics"></emph.end>ideoque <lb></lb><emph type="italics"></emph>Tt<emph.end type="italics"></emph.end>eſt (<emph type="italics"></emph>CKXDR/DC<emph.end type="italics"></emph.end>) & N erat (<emph type="italics"></emph>QBXDC/CP<emph.end type="italics"></emph.end>). Et propterea <emph type="italics"></emph>Vr<emph.end type="italics"></emph.end>eſt <lb></lb>(<emph type="italics"></emph>DRqXCKXCP/2DCqXQB<emph.end type="italics"></emph.end>), id eſt, (ob proportionales <emph type="italics"></emph>DR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DC, DV<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>DP<emph.end type="italics"></emph.end>) (<emph type="italics"></emph>DVqXCKXCP/2DPqXQB<emph.end type="italics"></emph.end>), & Latus rectum (<emph type="italics"></emph>DVquad./Vr<emph.end type="italics"></emph.end>) prodit <lb></lb>(2<emph type="italics"></emph>DPqXQB/CKXCP<emph.end type="italics"></emph.end>), id eſt (ob proportionales <emph type="italics"></emph>QB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CK, DA<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>) <lb></lb>(2<emph type="italics"></emph>DPqXDA/ACXCP<emph.end type="italics"></emph.end>), adeoque ad 2 <emph type="italics"></emph>DP,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DPXDA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CPXAC<emph.end type="italics"></emph.end>; hoc <lb></lb>eſt, ut reſiſtentia ad gravitatem. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Unde ſi corpus de loco quovis <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>data cum velocitate, <lb></lb>ſecundum rectam quamvis poſitione datam <emph type="italics"></emph>DP<emph.end type="italics"></emph.end>projiciatur; & re<lb></lb>ſiſtentia Medii ipſo motus initio detur: inveniri poteſt Curva <lb></lb><emph type="italics"></emph>DraF,<emph.end type="italics"></emph.end>quam corpus idem deſcribet. </s> <s>Nam ex data velocitate <pb xlink:href="039/01/246.jpg" pagenum="218"></pb><arrow.to.target n="note194"></arrow.to.target>datur latus rectum Parabolæ, ut <lb></lb>notum eſt. </s> <s>Et ſumendo 2<emph type="italics"></emph>DP<emph.end type="italics"></emph.end><lb></lb>ad latus illud rectum, ut eſt vis <lb></lb>gravitatis ad vim reſiſtentiæ, <lb></lb>datur <emph type="italics"></emph>DP.<emph.end type="italics"></emph.end>Dein ſecando <emph type="italics"></emph>DC<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>CPXAC<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>DPXDA<emph.end type="italics"></emph.end>in eadem illa rati<lb></lb>one gravitatis ad reſiſtentiam, <lb></lb>dabitur punctum <emph type="italics"></emph>A.<emph.end type="italics"></emph.end>Et inde <lb></lb>datur Curva <emph type="italics"></emph>DraF.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note194"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et contra, ſi datur <lb></lb><figure id="id.039.01.246.1.jpg" xlink:href="039/01/246/1.jpg"></figure><lb></lb>Curva <emph type="italics"></emph>DraF,<emph.end type="italics"></emph.end>dabitur & ve<lb></lb>locitas corporis & reſiſtentia <lb></lb>Medii in locis ſingulis <emph type="italics"></emph>r.<emph.end type="italics"></emph.end>Nam <lb></lb>ex data ratione <emph type="italics"></emph>CPXAC<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>DPXDA,<emph.end type="italics"></emph.end>datur tum reſiſten<lb></lb>tia Medii ſub initio motus, tum <lb></lb>latus rectum Parabolæ: & inde <lb></lb>datur etiam velocitas ſub initio <lb></lb>motus. </s> <s>Deinde ex longitudine <lb></lb>tangentis <emph type="italics"></emph>rL,<emph.end type="italics"></emph.end>datur & huic <lb></lb>proportionalis velocitas, & ve<lb></lb>locitati proportionalis reſiſten<lb></lb>tia in loco quovis <emph type="italics"></emph>r.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Cum autem longitu<lb></lb>do 2<emph type="italics"></emph>DP<emph.end type="italics"></emph.end>ſit ad latus rectum <lb></lb>Parabolæ ut gravitas ad reſiſtentiam in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>; & ex aucta velocitate <lb></lb>augeatur reſiſtentia in eadem ratione, at latus rectum Parabolæ au<lb></lb>geatur in ratione illa duplicata: patet longitudinem 2<emph type="italics"></emph>DP<emph.end type="italics"></emph.end>augeri <lb></lb>in ratione illa ſimplici, adeoque velocitati ſemper proportionalem <lb></lb>eſſe, neque ex angulo <emph type="italics"></emph>CDP<emph.end type="italics"></emph.end>mutato augeri vel minui, niſi mu<lb></lb>tetur quoque velocitas. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Unde liquet methodus determinandi Curvam <emph type="italics"></emph>DraF<emph.end type="italics"></emph.end><lb></lb>ex Phænomenis quamproxime, & inde colligendi reſiſtentiam & <lb></lb>velocitatem quacum corpus projicitur. </s> <s>Projiciantur corpora duo <lb></lb>ſimilia & æqualia eadem cum velocitate, de loco <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ſecundum <lb></lb>angulos diverſos <emph type="italics"></emph>GDP, cDp<emph.end type="italics"></emph.end>(minuſcularum literarum locis ſub<lb></lb>intellectis) & cognoſcantur loca <emph type="italics"></emph>F, f,<emph.end type="italics"></emph.end>abi incidunt in horizontale <lb></lb>planum <emph type="italics"></emph>DC.<emph.end type="italics"></emph.end>Tum, aſſumpta quacunque longitudine pro <emph type="italics"></emph>DP<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>Dp,<emph.end type="italics"></emph.end>fingatur quod reſiſtentia in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ſit ad gravitatem in ra-<pb xlink:href="039/01/247.jpg" pagenum="219"></pb>tione qualibet, & exponatur ratio illa per longitudinem quamvis <lb></lb><arrow.to.target n="note195"></arrow.to.target><emph type="italics"></emph>SM.<emph.end type="italics"></emph.end>Deinde per computationem, ex longitudine illa aſſumpta <lb></lb><emph type="italics"></emph>DP,<emph.end type="italics"></emph.end>inveniantur longitudines <emph type="italics"></emph>DF, Df,<emph.end type="italics"></emph.end>ac de ratione (<emph type="italics"></emph>Ef/DF<emph.end type="italics"></emph.end>) per <lb></lb>calculum inventa, auferatur ratio eadem <lb></lb><figure id="id.039.01.247.1.jpg" xlink:href="039/01/247/1.jpg"></figure><lb></lb>per experimentum inventa, & exponatur <lb></lb>differentia per perpendiculum <emph type="italics"></emph>MN.<emph.end type="italics"></emph.end>Idem <lb></lb>fac iterum ac tertio, aſſumendo ſemper <lb></lb>novam reſiſtentiæ ad gravitatem rationem <lb></lb><emph type="italics"></emph>SM,<emph.end type="italics"></emph.end>& colligendo novam differentiam <lb></lb><emph type="italics"></emph>MN.<emph.end type="italics"></emph.end>Ducantur autem differentiæ affirmativæ ad unam partem <lb></lb>rectæ <emph type="italics"></emph>SM,<emph.end type="italics"></emph.end>& negativæ ad alteram; & per puncta <emph type="italics"></emph>N, N, N<emph.end type="italics"></emph.end>agatur <lb></lb>ourva regularis <emph type="italics"></emph>NNN<emph.end type="italics"></emph.end>ſecans rectam <emph type="italics"></emph>SMMM<emph.end type="italics"></emph.end>in <emph type="italics"></emph>X,<emph.end type="italics"></emph.end>& erit <emph type="italics"></emph>SX<emph.end type="italics"></emph.end><lb></lb>vera ratio reſiſtentiæ ad gravitatem, quam invenire oportuit. </s> <s>Ex <lb></lb>hac ratione colligenda eſt longitudo <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>per calculum; & longi<lb></lb>tudo quæ ſit ad aſſumptam longitudinem <emph type="italics"></emph>DP,<emph.end type="italics"></emph.end>at longitudo <emph type="italics"></emph>DF<emph.end type="italics"></emph.end><lb></lb>per experimentum cognita ad longitudinem <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>modo inventam, <lb></lb>erit vera longitudo <emph type="italics"></emph>DP.<emph.end type="italics"></emph.end>Qua inventa, habetur tum Curva linea <lb></lb><emph type="italics"></emph>DraF<emph.end type="italics"></emph.end>quam corpus deſcribit, tum corporis velocitas & reſiſten<lb></lb>tia in locis ſingulis. </s></p> <p type="margin"> <s><margin.target id="note195"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cæterum, reſiſtentiam corporum eſſe in ratione velocitatis, Hy<lb></lb>potheſis eſt magis Mathematica quam Naturalis. </s> <s>Obtinet hæc ra<lb></lb>tio quamproxime ubi corpora in Mediis rigore aliquo præditis tar<lb></lb>diſſime moventur. </s> <s>In Mediis antem quæ rigore omni vacant re<lb></lb>ſiſtentiæ corporum ſunt in duplicata ratione velocitatum. </s> <s>Etenim <lb></lb>actione corporis velocioris communicatur eidem Medii quantitati, <lb></lb>tempore minore, motus major in ratione majoris velocitatis; ad<lb></lb>eoque tempore æquali (ob majorem Medii quantitatem perturba<lb></lb>tam) communicatur motus in duplicata ratione major; eſt que re<lb></lb>ſiſtentia (per motus Legem II & III) ut motus communicatus. </s> <s><lb></lb>Videamus igitur quades oriantur motus ex hac lege Reſiſtentiæ. <pb xlink:href="039/01/248.jpg" pagenum="220"></pb><arrow.to.target n="note196"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note196"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De motu Corporum quibus reſiſtitur in duplicata ra<lb></lb>tione Velocitatum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO V. THEOREMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Corpori reſiſiitur in velocitatis ratione duplicata, & idem ſola <lb></lb>vi inſita per Medium ſimilare movetur; tempora vero ſuman<lb></lb>tur in progreſſione Geometrica a minoribus terminis ad majores <lb></lb>pergente: dico quod velocitates initio ſingulorum temporum <lb></lb>ſunt in eadem progreſſione Geometrica inverſe, & quod ſpatia <lb></lb>ſunt æqualia quæ ſingulis temporibus deſcribuntur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam quoniam quadrato velocita<lb></lb><figure id="id.039.01.248.1.jpg" xlink:href="039/01/248/1.jpg"></figure><lb></lb>tis proportionalis eſt reſiſtentia Me<lb></lb>dii, & reſiſtentiæ proportionale eſt <lb></lb>decrementum velocitatis; ſi tempus <lb></lb>in particulas innumeras æquales divi<lb></lb>datur, quadrata velocitatum ſingulis <lb></lb>temporum initiis erunt velocitatum <lb></lb>earundem differentiis proportionalia. </s> <s><lb></lb>Sunto temporis particulæ illæ <emph type="italics"></emph>AK, <lb></lb>KL, LM,<emph.end type="italics"></emph.end>&c. </s> <s>in recta <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ſumptæ, <lb></lb>& erigantur perpendicula <emph type="italics"></emph>AB, Kk, <lb></lb>Ll, Mm,<emph.end type="italics"></emph.end>&c. </s> <s>Hyperbolæ <emph type="italics"></emph>BklmG,<emph.end type="italics"></emph.end><lb></lb>centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>Aſymptotis rectangulis <emph type="italics"></emph>CD, CH<emph.end type="italics"></emph.end>deſcriptæ, occurrentia <lb></lb>in <emph type="italics"></emph>B, k, t, m,<emph.end type="italics"></emph.end>&c. </s> <s>& erit <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CA,<emph.end type="italics"></emph.end>& diviſim <lb></lb><emph type="italics"></emph>AB-Kk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CA,<emph.end type="italics"></emph.end>& viciſſim <emph type="italics"></emph>AB-Kk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AK<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CA,<emph.end type="italics"></emph.end>adeoque ut <emph type="italics"></emph>ABXKk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ABXCA.<emph.end type="italics"></emph.end>Unde, cum <lb></lb><emph type="italics"></emph>AK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ABXCA<emph.end type="italics"></emph.end>dentur, erit <emph type="italics"></emph>AB-Kk<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>ABXKk<emph.end type="italics"></emph.end>; & ultimo, <lb></lb>ubi coeunt <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Kk,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph><expan abbr="ABq.">ABque</expan><emph.end type="italics"></emph.end>Et ſimili argumento erunt <emph type="italics"></emph>Kk-Ll, <lb></lb>Ll-Mm,<emph.end type="italics"></emph.end>&c. </s> <s>ut <emph type="italics"></emph>Kkq, Llq,<emph.end type="italics"></emph.end>&c. </s> <s>Linearum igitur <emph type="italics"></emph>AB, Kk, Ll, Mm<emph.end type="italics"></emph.end><pb xlink:href="039/01/249.jpg" pagenum="221"></pb>quadrata ſunt ut earundem differentiæ; & idcirco cum quadrata ve<lb></lb><arrow.to.target n="note197"></arrow.to.target>locitatum fuerint etiam ut ipſarum differentiæ, ſimilis erit amba<lb></lb>rum progreſſio. </s> <s>Quo demonſtrato, conſequens eſt etiam ut areæ <lb></lb>his lineis deſcriptæ ſint in progreſſione conſimili cum ſpatiis quæ <lb></lb>velocitatibus deſcribuntur. </s> <s>Ergo ſi velocitas initio primi tempo<lb></lb>ris <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>exponatur per lineam <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>& velocitas initio ſecundi <emph type="italics"></emph>KL<emph.end type="italics"></emph.end><lb></lb>per lineam <emph type="italics"></emph>Kk,<emph.end type="italics"></emph.end>& longitudo primo tempore deſcripta per aream <lb></lb><emph type="italics"></emph>AKkB<emph.end type="italics"></emph.end>; velocitates omnes ſubſequentes exponentur per lineas <lb></lb>ſubſequentes <emph type="italics"></emph>Ll, Mm,<emph.end type="italics"></emph.end>&c. </s> <s>& longitudines deſcriptæ per areas <lb></lb><emph type="italics"></emph>Kl, Lm,<emph.end type="italics"></emph.end>&c. </s> <s>Et compoſite, ſi tempus totum exponatur per ſum<lb></lb>mam partium ſuarum <emph type="italics"></emph>AM,<emph.end type="italics"></emph.end>longitudo tota deſcripta exponetur per <lb></lb>ſummam partium ſuarum <emph type="italics"></emph>AMmB.<emph.end type="italics"></emph.end>Concipe jam tempus <emph type="italics"></emph>AM<emph.end type="italics"></emph.end>ita <lb></lb>dividi in partes <emph type="italics"></emph>AK, KL, LM,<emph.end type="italics"></emph.end>&c. </s> <s>ut ſint <emph type="italics"></emph>CA, CK, CL, CM,<emph.end type="italics"></emph.end><lb></lb>&c. </s> <s>in progreſſione Geometrica; & erunt partes illæ in eadem pro<lb></lb>greſſione, & velocitates <emph type="italics"></emph>AB, Kk, Ll, Mm,<emph.end type="italics"></emph.end>&c. </s> <s>in progreſſione ea<lb></lb>dem inverſa, atque ſpatia deſcripta <emph type="italics"></emph>Ak, Kl, Lm,<emph.end type="italics"></emph.end>&c. </s> <s>æqualia. <lb></lb><emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note197"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Pater ergo quod, ſi tempus exponatur per Aſymptoti <lb></lb>partem quamvis <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>& velocitas in principio temporis per ordi<lb></lb>natim applicatam <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; velocitas in fine temporis exponetur per <lb></lb>ordinatam <emph type="italics"></emph>DG,<emph.end type="italics"></emph.end>& ſpatium totum deſcriptum per aream Hyper<lb></lb>bolicam adjacentem <emph type="italics"></emph>ABGD<emph.end type="italics"></emph.end>; necnon ſpatium quod corpus ali<lb></lb>quod eodem tempore <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>velocitate prima <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>in Medio non <lb></lb>reſiſtente deſcribere poſſet, per rectangulum <emph type="italics"></emph>ABXAD.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Unde datur ſpatium in Medio reſiſtente deſcriptum, ca<lb></lb>piendo illud ad ſpatium quod velocitate uniformi <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in medio non <lb></lb>reſiſtente ſimul deſcribi poſſet, ut eſt area Hyperbolica <emph type="italics"></emph>ABGD<emph.end type="italics"></emph.end><lb></lb>ad rectangulum <emph type="italics"></emph>ABXAD.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Datur etiam reſiſtentia Medii, ſtatuendo eam ipſo mo<lb></lb>tus initio æqualem eſſe vi uniformi centripetæ, quæ in cadente cor<lb></lb>pore, tempore <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>in Medio non reſiſtente, generare poſſet velo<lb></lb>citatem <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Nam ſi ducatur <emph type="italics"></emph>BT<emph.end type="italics"></emph.end>quæ tangat Hyperbolam in <emph type="italics"></emph>B,<emph.end type="italics"></emph.end><lb></lb>& occurrat Aſymptoto in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>; recta <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>æqualis erit ipſi <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>& <lb></lb>tempus exponet quo reſiſtentia prima uniformiter continuata tolle<lb></lb>re poſſet velocitatem totam <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol<emph.end type="italics"></emph.end>4. Et inde datur etiam proportio hujus reſiſtentiæ ad vim <lb></lb>gravitatis, aliamve quamvis datam vim centripetam. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et viceverſa, ſi datur proportio reſiſtentiæ ad datam <lb></lb>quamvis vim centripetam; datur tempus <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>quo vis centripeta <lb></lb>reſiſtentiæ æqualis generare poſſit velocitatem quamvis <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; & in-<pb xlink:href="039/01/250.jpg" pagenum="222"></pb><arrow.to.target n="note198"></arrow.to.target>de datur punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>per quod Hyperbola, Aſymptoris <emph type="italics"></emph>CH, CD,<emph.end type="italics"></emph.end><lb></lb>deſcribi debet; ut & ſpatium <emph type="italics"></emph>ABGD,<emph.end type="italics"></emph.end>quod corpus incipiendo <lb></lb>motum ſuum cum velocitate illa <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>tempore quovis <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>in Me<lb></lb>dio ſimilari reſiſtente deſcribere poteſt. </s></p> <p type="margin"> <s><margin.target id="note198"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO VI. THEOREMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corpora Spherica homogemea & æqualia, reſiſtentiis in duplicata <lb></lb>ratione velocitatum impedita, & ſolis viribus inſitis incitata, <lb></lb>temporibus quæ ſunt reciproce ut velocitates ſub initio, deſcri<lb></lb>bunt ſemper æqualia ſpatia, & amittunt partes velocitatum pro<lb></lb>portionales totis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Aſymptotis rectangulis <emph type="italics"></emph>CD, <lb></lb><figure id="id.039.01.250.1.jpg" xlink:href="039/01/250/1.jpg"></figure><lb></lb>CH<emph.end type="italics"></emph.end>deſcripta Hyperbola qua<lb></lb>vis <emph type="italics"></emph>BbEe<emph.end type="italics"></emph.end>ſecante perpendicula <lb></lb><emph type="italics"></emph>AB, ab, DE, de,<emph.end type="italics"></emph.end>in <emph type="italics"></emph>B, b, E, e,<emph.end type="italics"></emph.end><lb></lb>exponantur velocitates initi<lb></lb>ales per perpendicula <emph type="italics"></emph>AB, <lb></lb>DE,<emph.end type="italics"></emph.end>& tempora per lineas <lb></lb><emph type="italics"></emph>Aa, Dd.<emph.end type="italics"></emph.end>Eſt ergo ut <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>ita (per Hypotheſin) <emph type="italics"></emph>DE<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>& ita (ex natura Hy<lb></lb>perbolæ) <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>; & com<lb></lb>ponendo, ita <emph type="italics"></emph>Ca<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Cd.<emph.end type="italics"></emph.end>Ergo <lb></lb>areæ <emph type="italics"></emph>ABba, DEed,<emph.end type="italics"></emph.end>hoc eſt, ſpatia deſcripta æquamtur inter ſe, <lb></lb>& velocitates primæ <emph type="italics"></emph>AB, DE<emph.end type="italics"></emph.end>ſunt ultimis <emph type="italics"></emph>ab, de,<emph.end type="italics"></emph.end>& propterea <lb></lb>(dividendo) partibus etiam ſuis amiſſis <emph type="italics"></emph>AB-ab, DE-de<emph.end type="italics"></emph.end>pro<lb></lb>portionales. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO VII. THEOREMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corpora Sphærica quibus reſiſtitur in duplicata ratione velocitatum, <lb></lb>temporibus quæ ſunt ut motus primi directe & reſiſtentiæ pri<lb></lb>mæ inverſe, amittent partes motuum proportionales totis, & <lb></lb>ſpatia deſcribent temporibus iſtis in velocitates primas ductis <lb></lb>proportionalia.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Namque motuum partes amiſſæ ſunt ut reſiſtentiæ & tempora <pb xlink:href="039/01/251.jpg" pagenum="223"></pb>conjunctim. </s> <s>Igitur ut partes illæ ſint totis proportionales, debe<lb></lb><arrow.to.target n="note199"></arrow.to.target>bit reſiſtentia & tempus conjunctim eſſe ut motus. </s> <s>Proinde tem<lb></lb>pus erit ut motus directe & reſiſtentia inverſe. </s> <s>Quare temporam <lb></lb>particulis in ea ratione ſumptis, corpora amittent ſemper parti<lb></lb>culas motuum proportionales totis, adeoque retinebunt velocita<lb></lb>tes in ratione prima. </s> <s>Et ob datam velocitatum rationem, deſcri<lb></lb>bent ſemper ſpatia quæ ſunt ut velocitates primæ & tempora con<lb></lb>junctim. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note199"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur ſi æquivelocibus corporibus reſiſtitur in duplicata <lb></lb>ratione diametrorum: Globi homogenei quibuſcunque cum velocita<lb></lb>tibus moti, deſcribendo ſpatia diametris ſuis proportionalia, amit<lb></lb>tent partes motuum proportionales totis. </s> <s>Motus enim Globi cu<lb></lb>juſque erit ut ejus velocitas & Maſſa conjunctim, id eſt, ut veloci<lb></lb>tas & cubus diametri; reſiſtentia (per Hypotheſin) erit ut quadra<lb></lb>tum diametri & quadratum velocitatis conjunctim; & tempus (per <lb></lb>hanc Propoſitionem) eſt in ratione priore directe & ratione poſte<lb></lb>riore inverſe, id eſt, ut diameter directe & velocitas inverſe; ad<lb></lb>eoque ſpatium (tempori & velocitati proportionale) eſt ut dia<lb></lb>meter. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si æquivelocibus corporibus reſiſtitur in ratione ſeſquial<lb></lb>tera diametrorum: Globi homogenei quibuſcunque cum velocitati<lb></lb>bus moti, deſcribendo ſpatia in ſeſquialtera ratione diametrorum, <lb></lb>amittent partes motuum proportionales totis. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et univerſaliter, ſi æquivelocibus corporibus reſiſtitur in <lb></lb>ratione dignitatis cujuſcunQ.E.D.ametrorum: ſpatia quibus Globi <lb></lb>homogenei, quibuſcunque cum velocitatibus moti, amittent partes <lb></lb>motuum proportionales totis, erunt ut cubi diametrorum ad digNI<lb></lb>tatem illam applicati. </s> <s>Sunto diametri D & E; & ſi reſiſtentiæ, <lb></lb>ubi velocitates æquales ponuntur, ſint ut D<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> & E<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>: ſpatia quibus <lb></lb>Globi quibuſcunque cum velocitatibus moti, amitteus partes mo<lb></lb>tuum proportionales totis, erunt ut D<emph type="sup"></emph>3-<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> & E<emph type="sup"></emph>3-<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>. </s> <s>Igitur deſcri<lb></lb>bendo ſpatia ipſis D<emph type="sup"></emph>3-<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> & E<emph type="sup"></emph>3-<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> proportionalia, retinebunt veloci<lb></lb>tates in eadem ratione ad invicem ac ſub initio. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Quod ſi Globi non ſint homogenei, ſpatium a Globo <lb></lb>denſiore deſcriptum augeri debet in ratione denſitatis. </s> <s>Motus <lb></lb>enim, ſub pari velocitare, major eſt in ratione denſitatis, & tempus <lb></lb>(per hanc Propoſitionem) augetur in ratione motus directe, ac <lb></lb>ſpatium deſcriptum in ratione temporis. <pb xlink:href="039/01/252.jpg" pagenum="224"></pb><arrow.to.target n="note200"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note200"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et ſi Globi moveantur in Mediis diverſis; ſpatium in <lb></lb>Medio, quod cæteris paribus magis reſiſtit, diminuendum erit in <lb></lb>ratione majoris reſiſtentiæ. </s> <s>Tempus enim (per hanc Propoſitio<lb></lb>nem) diminuetur in ratione reſiſtentiæ auctæ, & ſpatium in ra<lb></lb>tione temporis. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Momentum Genitæ æquatur Momentis laterum ſingulorum gene<lb></lb>rantium in eorundem laterum indices dignitatum & coefficien<lb></lb>tia continue ductis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Genitam voco quantitatem omnem quæ ex lateribus vel termi<lb></lb>nis quibuſcunque, in Arithmetica per multiplicationem, diviſionem, <lb></lb>& extractionem radicum; in Geometria per inventionem vel con<lb></lb>tentorum & laterum, vel extremarum & mediarum proportionalium, <lb></lb>abſque additione & ſubductione generatur. </s> <s>Ejuſmodi quantita<lb></lb>tes ſunt Facti, Quoti, Radices, Rectangula, Quadrata, Cubi, Latera <lb></lb>quadrata, Latera cubica, & ſimiles. </s> <s>Has quantitates ut indeterminatas <lb></lb>& inſtabiles, & quaſi motu fluxuve perpetuo creſcentes vel decre<lb></lb>ſcentes, hic conſidero; & earum incrementa vel decrementa momen<lb></lb>tanea ſub nomine Momentorum intelligo: ita ut incrementa pro <lb></lb>momentis addititiis ſeu affirmativis, ac decrementa pro ſubductitiis <lb></lb>ſeu negativis habeantur. </s> <s>Cave tamen intellexeris particulas fiNI<lb></lb>tas. </s> <s>Particulæ finitæ non ſunt momenta, ſed quantitates ipſæ ex <lb></lb>momentis genitæ. </s> <s>Intelligenda ſunt principia jamjam naſcentia fi<lb></lb>nitarum magnitudinum. </s> <s>Neque enim ſpectatur in hoc Lemmate <lb></lb>magnitudo momentorum, ſed prima naſcentium proportio. </s> <s>Eo<lb></lb>dem recidit ſi loco momentorum uſurpentur vel velocitates incre<lb></lb>mentorum ac decrementorum, (quas etiam motus, mutationes <lb></lb>& fluxiones quantitatum nominare licet) vel finitæ quævis quanti<lb></lb>tates velocitatibus hiſce proportionales. </s> <s>Lateris autem cujuſque <lb></lb>generantis Coefficiens eſt quantitas, quæ oritur applicando GeNI<lb></lb>tam ad hoc latus. </s></p> <p type="main"> <s>Igitur ſenſus Lemmatis eſt, ut, ſi quantitatum quarumcunque <lb></lb>perpetuo motu creſcentium vel decreſcentium A, B, C, &c. </s> <s>mo<lb></lb>menta, vel mutationum velocitates dicantur <emph type="italics"></emph>a, b, c,<emph.end type="italics"></emph.end>&c. </s> <s>momentum <lb></lb>vel mutatio geniti rectanguli AB fuerit <emph type="italics"></emph>a<emph.end type="italics"></emph.end>B+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A, & geniti con<lb></lb>tenti ABC momentum fuerit <emph type="italics"></emph>a<emph.end type="italics"></emph.end>BC+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>AC+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>AB: & genitarum <pb xlink:href="039/01/253.jpg" pagenum="225"></pb>dignitatum A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>, A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>, A<emph type="sup"></emph>4<emph.end type="sup"></emph.end>, A<emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>, A<emph type="sup"></emph>1/3<emph.end type="sup"></emph.end>, A<emph type="sup"></emph>1/3<emph.end type="sup"></emph.end>, A<emph type="sup"></emph>2/3<emph.end type="sup"></emph.end>, A<emph type="sup"></emph>-1<emph.end type="sup"></emph.end>, A<emph type="sup"></emph>-2<emph.end type="sup"></emph.end>, & A<emph type="sup"></emph>-1/2<emph.end type="sup"></emph.end> momenta </s></p> <p type="main"> <s><arrow.to.target n="note201"></arrow.to.target>2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A, 3<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>, 4<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>, 1/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-1/2<emph.end type="sup"></emph.end>, 3/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>, 1/3<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-2/3<emph.end type="sup"></emph.end>, 2/3<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-1/3<emph.end type="sup"></emph.end>, -<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-2<emph.end type="sup"></emph.end>, <lb></lb>-2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-3<emph.end type="sup"></emph.end>, & -1/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-1/2<emph.end type="sup"></emph.end> reſpective. </s> <s>Et generaliter, ut dignitatis <lb></lb>cujuſcunque A<emph type="sup"></emph><emph type="italics"></emph>n/m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> momentum fuerit <emph type="italics"></emph>n/m a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>(<emph type="italics"></emph>n-m/m<emph.end type="italics"></emph.end>)<emph.end type="sup"></emph.end>. </s> <s>Item ut Genitæ <lb></lb>A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>B momentum fuerit 2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>AB+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>; & Genitæ A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>B<emph type="sup"></emph>4<emph.end type="sup"></emph.end>C<emph type="sup"></emph>2<emph.end type="sup"></emph.end> momen<lb></lb>tum 3<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>B<emph type="sup"></emph>4<emph.end type="sup"></emph.end>C<emph type="sup"></emph>2<emph.end type="sup"></emph.end>+4<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>B<emph type="sup"></emph>3<emph.end type="sup"></emph.end>C<emph type="sup"></emph>2<emph.end type="sup"></emph.end>+2<emph type="italics"></emph>c<emph.end type="italics"></emph.end>A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>B<emph type="sup"></emph>4<emph.end type="sup"></emph.end>C; & Genitæ (A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>/B<emph type="sup"></emph>2<emph.end type="sup"></emph.end>) ſi<lb></lb>ve A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>B<emph type="sup"></emph>-2<emph.end type="sup"></emph.end> momentum 3<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>B<emph type="sup"></emph>-2<emph.end type="sup"></emph.end>-2<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>B<emph type="sup"></emph>-3<emph.end type="sup"></emph.end>: & ſic in cæteris. </s> <s><lb></lb>Demonſtratur vero Lemma in hunc modum. </s></p> <p type="margin"> <s><margin.target id="note201"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Rectangulum quodvis motu perpetuo auctum AB, <lb></lb>ubi de lateribus A & B deerant momentorum dimidia 1/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>& 1/2<emph type="italics"></emph>b,<emph.end type="italics"></emph.end><lb></lb>fuit A-1/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>in B-1/2<emph type="italics"></emph>b,<emph.end type="italics"></emph.end>ſeu AB-1/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>B-1/2<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A+1/4<emph type="italics"></emph>ab<emph.end type="italics"></emph.end>; & quam pri<lb></lb>mum latera A & B alteris momentorum dimidiis aucta ſunt, eva<lb></lb>dit A+1/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>in B+1/2<emph type="italics"></emph>b<emph.end type="italics"></emph.end>ſeu AB+1/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>B+1/2<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A+1/4<emph type="italics"></emph>ab.<emph.end type="italics"></emph.end>De hoc rectan<lb></lb>gulo ſubducatur rectangulum prius, & manebit exceſſus <emph type="italics"></emph>a<emph.end type="italics"></emph.end>B+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A. </s> <s><lb></lb>Igitur laterum incrementis totis <emph type="italics"></emph>a<emph.end type="italics"></emph.end>& <emph type="italics"></emph>b<emph.end type="italics"></emph.end>generatur rectanguli incre<lb></lb>mentum <emph type="italics"></emph>a<emph.end type="italics"></emph.end>B+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Ponatur AB ſemper æquale G, & contenti ABC ſeu <lb></lb>GC momentum (per Cas. </s> <s>1.) erit <emph type="italics"></emph>g<emph.end type="italics"></emph.end>C+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>G, id eſt (ſi pro G & <emph type="italics"></emph>g<emph.end type="italics"></emph.end><lb></lb>ſcribantur AB & <emph type="italics"></emph>a<emph.end type="italics"></emph.end>B+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A) <emph type="italics"></emph>a<emph.end type="italics"></emph.end>BC+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>AC+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>AB. </s> <s>Et par eſt ra<lb></lb>tio contenti ſub lateribus quotcunque. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Ponantur latera A, B, C ſibi mutuo ſemper æqualia; & <lb></lb>ipſius A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>, id eſt rectanguli AB, momentum <emph type="italics"></emph>a<emph.end type="italics"></emph.end>B+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>A erit 2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A, ip<lb></lb>ſius autem A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>, id eſt contenti ABC, momentum <emph type="italics"></emph>a<emph.end type="italics"></emph.end>BC+<emph type="italics"></emph>b<emph.end type="italics"></emph.end>AC <lb></lb>+<emph type="italics"></emph>c<emph.end type="italics"></emph.end>AB erit 3<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>. </s> <s>Et eodem argumento momentum dignitatis <lb></lb>cujuſcunque A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> eſt <emph type="italics"></emph>na<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1.<emph.end type="sup"></emph.end> <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>4. Unde cum 1/A in A ſit 1, momentum ipſius 1/A ductum <lb></lb>in A, una cum 1/A ducto in <emph type="italics"></emph>a<emph.end type="italics"></emph.end>erit momentum ipſius 1, id eſt, NI<lb></lb>hil. </s> <s>Proinde momentum ipſius 1/A ſeu ipſius A<emph type="sup"></emph>-1<emph.end type="sup"></emph.end> eſt (-<emph type="italics"></emph>a<emph.end type="italics"></emph.end>/A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>). Et ge<lb></lb>neraliter cum (1/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>) in A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> ſit 1, momentum ipſius (1/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>) ductum in A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end><pb xlink:href="039/01/254.jpg" pagenum="226"></pb><arrow.to.target n="note202"></arrow.to.target>una cum (1/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>) in <emph type="italics"></emph>na<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end> erit nihil. </s> <s>Et propterea momentum ip<lb></lb>ſius (1/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>) ſeu A<emph type="sup"></emph>-<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> erit-(<emph type="italics"></emph>na<emph.end type="italics"></emph.end>/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>+1). <emph type="italics"></emph>q.ED.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note202"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>5. Et cum A<emph type="sup"></emph>1/2<emph.end type="sup"></emph.end> in A<emph type="sup"></emph>1/2<emph.end type="sup"></emph.end> ſit A, momentum ipſius A<emph type="sup"></emph>1/2<emph.end type="sup"></emph.end> ductum in <lb></lb>2A<emph type="sup"></emph>1/2<emph.end type="sup"></emph.end> erit <emph type="italics"></emph>a,<emph.end type="italics"></emph.end>per Cas. </s> <s>3: ideoque momentum ipſius A<emph type="sup"></emph>1/2<emph.end type="sup"></emph.end> erit (<emph type="italics"></emph>a<emph.end type="italics"></emph.end>/2A 1/2) <lb></lb>ſive 1/2<emph type="italics"></emph>a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-1/2<emph.end type="sup"></emph.end>. </s> <s>Et generaliter ſi ponatur A<emph type="sup"></emph><emph type="italics"></emph>m/n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> æquale B, erit A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> æ<lb></lb>quale B<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, ideoque <emph type="italics"></emph>ma<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end> æquale <emph type="italics"></emph>nb<emph.end type="italics"></emph.end>B<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1,<emph.end type="sup"></emph.end> & <emph type="italics"></emph>ma<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-1<emph.end type="sup"></emph.end> æqua<lb></lb>le <emph type="italics"></emph>nb<emph.end type="italics"></emph.end>B<emph type="sup"></emph>-1<emph.end type="sup"></emph.end> ſeu <emph type="italics"></emph>nb<emph.end type="italics"></emph.end>A<emph type="sup"></emph>-<emph type="italics"></emph>m/n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, adeoque <emph type="italics"></emph>m/n a<emph.end type="italics"></emph.end>A<emph type="sup"></emph>(<emph type="italics"></emph>m-n/n<emph.end type="italics"></emph.end>)<emph.end type="sup"></emph.end> æquale <emph type="italics"></emph>b,<emph.end type="italics"></emph.end>id eſt, æquale <lb></lb>momento ipſius A<emph type="sup"></emph><emph type="italics"></emph>m/n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>6. Igitur Genitæ cujuſeunque A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>B<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> momentum eſt mo<lb></lb>mentum ipſius A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> ductum in B<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, una cum momento ipſius B<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> du<lb></lb>cto in A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, id eſt <emph type="italics"></emph>ma<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>B<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>+<emph type="italics"></emph>nb<emph.end type="italics"></emph.end>B<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>m<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>; idque ſive dignita<lb></lb>tum indices <emph type="italics"></emph>m<emph.end type="italics"></emph.end>& <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ſint integri numeri vel fracti, ſive affirmati<lb></lb>vi vel negativi. </s> <s>Et par eſt ratio contenti ſub pluribus dignitati<lb></lb>bus. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc in continue proportionalibus, ſi terminus unus <lb></lb>datur, momenta terminorum reliquorum erunt ut iidem termini <lb></lb>multiplicati per numerum intervallorum inter ipſos & terminum <lb></lb>datum. </s> <s>Sunto A, B, C, D, E, F continue proportionales; & ſi <lb></lb>detur terminus C, momenta reliquorum terminorum erunt inter <lb></lb>ſe ut-2A, -B, D, 2E, 3F. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et ſi in quatuor proportionalibus duæ mediæ dentur, <lb></lb>momenta extremarum erunt ut eædem extremæ. </s> <s>Idem intelligen<lb></lb>dum eſt de lateribus rectanguli cujuſcunQ.E.D.ti. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et ſi ſumma vel differentia duorum quadratorum detur, <lb></lb>momenta laterum erunt reciproce ut latera. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In literis quæ mihi cum Geometra peritiſſimo <emph type="italics"></emph>G.G. Leibnitio<emph.end type="italics"></emph.end>an<lb></lb>nis abhinc decem intercedebant, cum ſignificarem me compotem <lb></lb>eſſe methodi determinandi Maximas & Minimas, ducendi Tangen<lb></lb>tes, & ſimilia peragendi, quæ in terminis ſurdis æque ac in ratio<lb></lb>nalibus procederet, & literis tranſpoſitis hanc ſententiam involven-<pb xlink:href="039/01/255.jpg" pagenum="227"></pb>tibus [<emph type="italics"></emph>Data Æquatione quotcunque Fluentes quantitates invelven-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note203"></arrow.to.target><emph type="italics"></emph>te, Fluxiones invenire, & vice verſa<emph.end type="italics"></emph.end>] eandem celarem: reſcripſit <lb></lb>Vir Clariſſimus ſe quoQ.E.I. ejuſmodi methodum incidiſſe, & me<lb></lb>thodum ſuam communicavit a mea vix abludentem præterquam in <lb></lb>verborum & notarum formulis, & Idea generationis quantitatum. </s> <s><lb></lb>Utriuſque fundamentum continetur in hoc Lemmate. </s></p> <p type="margin"> <s><margin.target id="note203"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO VIII. THEOREMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si corpus in Medio uniformi, Gravitate uniformiter agente, recta <lb></lb>aſcendat vel deſcendat, & ſpatium totum deſcriptum diſtingua<lb></lb>tur in partes æquales, inque principiis ſingularum partium <lb></lb>(addendo reſiſtentiam Medii ad vim gravitatis, quando cor<lb></lb>pus aſcendit, vel ſubducendo ipſam quando corpus deſcendit) <lb></lb>colligantur vires abſolutæ; dico quod vires illæ abſolutæ ſunt <lb></lb>in progreſſione Geometrica.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Exponatur enim vis gravitatis per datam lineam <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>; reſiſten<lb></lb>tia per lineam indefinitam <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>; vis abſoluta in deſcenſu corporis <lb></lb>per differentiam <emph type="italics"></emph>KC<emph.end type="italics"></emph.end>; velocitas corporis per lineam <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>(quæ ſit <lb></lb>media proportionalis inter <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>ideoQ.E.I. ſubduplicata <lb></lb>ratione reſiſtentiæ;) incrementum reſiſtentiæ data temporis particu<lb></lb>la factum per lineolam <emph type="italics"></emph>KL,<emph.end type="italics"></emph.end>& contemporaneum velocitatis incre<lb></lb>mentum per lineolam <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>; & centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>Aſymptotis rectangulis <lb></lb><emph type="italics"></emph>CA, CH<emph.end type="italics"></emph.end>deſcribatur Hyperbola quævis <emph type="italics"></emph>BNS,<emph.end type="italics"></emph.end>erectis perpendi<lb></lb>culis <emph type="italics"></emph>AB, KN, LO, PR, QS<emph.end type="italics"></emph.end>occurrens in <emph type="italics"></emph>B, N, O, R, S.<emph.end type="italics"></emph.end>Quo<lb></lb>niam <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>eſt ut <emph type="italics"></emph>APq,<emph.end type="italics"></emph.end>erit hujus momentum <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>ut illius mo<lb></lb>mentum 2<emph type="italics"></emph>APQ,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>in <emph type="italics"></emph>KC.<emph.end type="italics"></emph.end>Nam velocitatis incre<lb></lb>mentum <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>(per motus Leg.11.) proportionale eſt vi generanti <emph type="italics"></emph>KC.<emph.end type="italics"></emph.end><lb></lb>Componatur ratio ipſius <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>cum ratione ipſius <emph type="italics"></emph>KN,<emph.end type="italics"></emph.end>& fiet rect<lb></lb>angulum <emph type="italics"></emph>KLXKN<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>APXKCXKN<emph.end type="italics"></emph.end>; hoc eſt, ob datum rect<lb></lb>angulum <emph type="italics"></emph>KCXKN,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AP.<emph.end type="italics"></emph.end>Atqui areæ Hyperbolicæ <emph type="italics"></emph>KNOL<emph.end type="italics"></emph.end><lb></lb>ad rectangulum <emph type="italics"></emph>KLXKN<emph.end type="italics"></emph.end>ratio ultima, ubi coeunt puncta <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L,<emph.end type="italics"></emph.end><lb></lb>eſt æqualitatis. </s> <s>Ergo area illa Hyperbolica evaneſcens eſt ut <emph type="italics"></emph>AP.<emph.end type="italics"></emph.end><lb></lb>Componitur igitur area tota Hyperbolica <emph type="italics"></emph>ABOL<emph.end type="italics"></emph.end>ex particulis <lb></lb><emph type="italics"></emph>KNOL<emph.end type="italics"></emph.end>velocitati <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ſemper proportionalibus, & propterea <lb></lb>ſpatio velocitate iſta deſcripto proportionalis eſt. </s> <s>Dividatur jam <lb></lb>area illa in partes æquales <emph type="italics"></emph>ABMI, IMNK, KNOL,<emph.end type="italics"></emph.end>&c. </s> <s>& vi-<pb xlink:href="039/01/256.jpg" pagenum="228"></pb><arrow.to.target n="note204"></arrow.to.target>res abſolutæ <emph type="italics"></emph>AC, IC, KC, LC,<emph.end type="italics"></emph.end>&c. </s> <s>erunt in progreſſione Geo<lb></lb>metrica. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end>Et ſimili argumento, in aſcenſu corporis, ſu<lb></lb>mendo, ad contrariam partem puncti <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>æquales areas <emph type="italics"></emph>ABmi, <lb></lb>imnk, knol,<emph.end type="italics"></emph.end>&c. </s> <s>conſtabit quod vires abſolutæ <emph type="italics"></emph>AC, iC, kC, lC,<emph.end type="italics"></emph.end>&c. <lb></lb></s> <s>ſunt continue proportionales. </s> <s>Ideoque ſi ſpatia omnia in aſcenſu & <lb></lb>deſcenſu capiantur æqualia; omnes vires abſolutæ <emph type="italics"></emph>lC, kC, iC, AC, <lb></lb>IC, KC, LC,<emph.end type="italics"></emph.end>&c. </s> <s>erunt continue proportionales. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note204"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p><figure id="id.039.01.256.1.jpg" xlink:href="039/01/256/1.jpg"></figure> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi ſpatium deſcriptum exponatur per aream Hy<lb></lb>perbolicam <emph type="italics"></emph>ABNK<emph.end type="italics"></emph.end>; exponi poſſunt vis gravitatis, velocitas cor<lb></lb>poris & reſiſtentia Medii per lineas <emph type="italics"></emph>AC, AP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>reſpective; <lb></lb>& vice verſa. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et velocitatis maximæ, quam corpus in infinitum deſcen<lb></lb>dendo poteſt unquam acquirere, exponens eſt linea <emph type="italics"></emph>AC.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Igitur ſi in data aliqua velocitate cognoſcatur reſiſten<lb></lb>tia Medii, invenietur velocitas maxima, ſumendo ipſam ad veloci-<pb xlink:href="039/01/257.jpg" pagenum="229"></pb>tatem illam datam in ſubduplicata ratione, quam habet vis Gravi<lb></lb><arrow.to.target n="note205"></arrow.to.target>tatis ad Medii reſiſtentiam illam cognitam. </s></p> <p type="margin"> <s><margin.target id="note205"></margin.target>LIBER <lb></lb>SECUMDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO IX. THEOREMA VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſitis jam demonſtratis, dico quod ſi Tangentes angulorum ſecto <lb></lb>ris Circularis & ſectoris Hyperbolici ſumantur velocitatibus <lb></lb>proportionales, exiſtente radio juſtæ magnitudinis: erit tempus <lb></lb>omne aſcenſus futuri ut ſector Circuli, & tempus omne deſcen<lb></lb>ſus præteriti ut ſector Hyperbolæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Rectæ <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>qua vis gravitatis exponitur, perpendicularis & æ<lb></lb>qualis ducatur <emph type="italics"></emph>AD.<emph.end type="italics"></emph.end>Centro <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ſemidiametro <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>deſcribatur tum <lb></lb>Circuli quadrans <emph type="italics"></emph>AtE,<emph.end type="italics"></emph.end>tum Hyperbola rectangula <emph type="italics"></emph>AVZ<emph.end type="italics"></emph.end>axem <lb></lb>habens <emph type="italics"></emph>AX,<emph.end type="italics"></emph.end>verticem principalem <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& Aſymptoton <emph type="italics"></emph>DC.<emph.end type="italics"></emph.end>Jun<lb></lb>gantur <emph type="italics"></emph>Dp, DP,<emph.end type="italics"></emph.end>& erit ſector Circularis <emph type="italics"></emph>AtD<emph.end type="italics"></emph.end>ut tempus aſcenſus <lb></lb>omnis futuri; & ſector Hyperbolicus <emph type="italics"></emph>ATD<emph.end type="italics"></emph.end>ut tempus deſcenſus <lb></lb>omnis præteriti. </s> <s>Si modo ſectorum Tangentes <emph type="italics"></emph>Ap, AP<emph.end type="italics"></emph.end>ſint ut <lb></lb>velocitates. </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Agatur enim <emph type="italics"></emph>Dvq<emph.end type="italics"></emph.end>abſcindens ſectoris <emph type="italics"></emph>ADt<emph.end type="italics"></emph.end>& trian<lb></lb>guli <emph type="italics"></emph>ADp<emph.end type="italics"></emph.end>momenta, ſeu particulas quam minimas ſimul deſcrip<lb></lb>tas <emph type="italics"></emph>tDv<emph.end type="italics"></emph.end>& <emph type="italics"></emph><expan abbr="pDq.">pDque</expan><emph.end type="italics"></emph.end>Cum particulæ illæ, ob angulum commu<lb></lb>nem <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ſunt in duplicata ratione laterum, erit particula <emph type="italics"></emph>tDv<emph.end type="italics"></emph.end><lb></lb>ut (<emph type="italics"></emph>qDp/pDquad<emph.end type="italics"></emph.end>). Sed <emph type="italics"></emph>pDquad.<emph.end type="italics"></emph.end>eſt <emph type="italics"></emph>ADquad+Apquad.<emph.end type="italics"></emph.end>id eſt, <lb></lb><emph type="italics"></emph>ADquad+ADXAk<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>ADXCk<emph.end type="italics"></emph.end>; & <emph type="italics"></emph>qDp<emph.end type="italics"></emph.end>eſt 1/2 <emph type="italics"></emph><expan abbr="ADXpq.">ADXpque</expan><emph.end type="italics"></emph.end><lb></lb>Ergo ſectoris particula <emph type="italics"></emph>tDv<emph.end type="italics"></emph.end>eſt ut (<emph type="italics"></emph>pq/Ck<emph.end type="italics"></emph.end>), id eſt, ut velocitatis de<lb></lb>crementum quam minimum <emph type="italics"></emph>pq<emph.end type="italics"></emph.end>directe & vis illa <emph type="italics"></emph>Ck<emph.end type="italics"></emph.end>quæ velo<lb></lb>citatem diminuit inverſe, atque adeo ut particula temporis decre<lb></lb>mento reſpondens. </s> <s>Et componendo fit ſumma particularum om<lb></lb>nium <emph type="italics"></emph>tDv<emph.end type="italics"></emph.end>in ſectore <emph type="italics"></emph>ADt,<emph.end type="italics"></emph.end>ut ſumma particularum temporis <lb></lb>ſingulis velocitatis decreſcentis <emph type="italics"></emph>Ap<emph.end type="italics"></emph.end>particulis amiſſis <emph type="italics"></emph>pq<emph.end type="italics"></emph.end>reſpon<lb></lb>dentium, uſQ.E.D.m velocitas illa in nihilum diminuta eva<lb></lb>nuerit; hoc eſt, ſector totus <emph type="italics"></emph>ADt<emph.end type="italics"></emph.end>eſt ut aſcenſus totius futuri <lb></lb>tempus. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/258.jpg" pagenum="230"></pb><arrow.to.target n="note206"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note206"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Agatur <emph type="italics"></emph>DQV<emph.end type="italics"></emph.end>abſcindens tum ſectoris <emph type="italics"></emph>DAV,<emph.end type="italics"></emph.end>tum tri<lb></lb>anguli <emph type="italics"></emph>DAQ<emph.end type="italics"></emph.end>particulas quam minimas <emph type="italics"></emph>TDV<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PDQ<emph.end type="italics"></emph.end>; & e<lb></lb>runt hæ particulæ ad invicem ut <emph type="italics"></emph>DTQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="DPq.">DPque</expan><emph.end type="italics"></emph.end>id eſt (ſi <emph type="italics"></emph>TX<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>parallelæ ſint) ut <emph type="italics"></emph><expan abbr="DXq.">DXque</expan><emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="DAq.">DAque</expan><emph.end type="italics"></emph.end>vel <emph type="italics"></emph><expan abbr="TXq.">TXque</expan><emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="APq.">APque</expan><emph.end type="italics"></emph.end>& <lb></lb>diviſim ut <emph type="italics"></emph>DXq-TXq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="DAq-APq.">DAq-APque</expan><emph.end type="italics"></emph.end>Sed ex natura <lb></lb>Hyperbolæ <emph type="italics"></emph>DXq-TXq<emph.end type="italics"></emph.end>eſt <emph type="italics"></emph>ADq,<emph.end type="italics"></emph.end>& per Hypotheſin <emph type="italics"></emph>APq<emph.end type="italics"></emph.end><lb></lb>eſt <emph type="italics"></emph>ADXAK.<emph.end type="italics"></emph.end>Ergo particulæ ſunt ad invicem ut <emph type="italics"></emph>ADq<emph.end type="italics"></emph.end>ad <lb></lb><figure id="id.039.01.258.1.jpg" xlink:href="039/01/258/1.jpg"></figure><lb></lb><emph type="italics"></emph>ADq-ADXAK<emph.end type="italics"></emph.end>; id eſt, ut <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AD-AK<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CK:<emph.end type="italics"></emph.end><lb></lb>ideoque ſectoris particula <emph type="italics"></emph>TDV<emph.end type="italics"></emph.end>eſt (<emph type="italics"></emph>PDQXAC/CK<emph.end type="italics"></emph.end>), atque adeo ob <lb></lb>datas <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AD,<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>PQ/CK<emph.end type="italics"></emph.end>), id eſt, ut incrementum velocitatis <lb></lb>directe utque vis generans incrementum inverſe, atque adeo ut par<lb></lb>ticula temporis incremento reſpondens. </s> <s>Et componendo ſit ſum <lb></lb>ma particularum temporis, quibus omnes velocitatis <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>particulæ <pb xlink:href="039/01/259.jpg" pagenum="231"></pb><emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>generantur, ut ſumma particularum ſectoris <emph type="italics"></emph>ATD,<emph.end type="italics"></emph.end>id eſt, </s></p> <p type="main"> <s><arrow.to.target n="note207"></arrow.to.target>tempus totum ut ſector totus. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note207"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>æquetur quartæ parti ipſius <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>ſpatium <lb></lb>quod corpus tempore quovis cadendo deſcribit, erit ad ſpatium <lb></lb>quod corpus velocitate maxima <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>eodem tempore uniformiter <lb></lb>progrediendo deſcribere poteſt, ut area <emph type="italics"></emph>ABNK,<emph.end type="italics"></emph.end>qua ſpatium <lb></lb>cadendo deſcriptum exponitur, ad aream <emph type="italics"></emph>ATD<emph.end type="italics"></emph.end>qua tempus ex<lb></lb>ponitur. </s> <s>Nam cum ſit <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AK,<emph.end type="italics"></emph.end>erit (per <lb></lb>Corol. </s> <s>1, Lem. </s> <s>11 hujus) <emph type="italics"></emph>LK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ut 2<emph type="italics"></emph>AK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AP,<emph.end type="italics"></emph.end>hoc eſt, <lb></lb>ut 2<emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>& inde <emph type="italics"></emph>LK<emph.end type="italics"></emph.end>ad 1/2<emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad (1/4<emph type="italics"></emph>AC<emph.end type="italics"></emph.end>vel) <lb></lb><emph type="italics"></emph>AB<emph.end type="italics"></emph.end>; eſt & <emph type="italics"></emph>KN<emph.end type="italics"></emph.end>ad (<emph type="italics"></emph>AC<emph.end type="italics"></emph.end>vel) <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>; itaque ex <lb></lb>æquo <emph type="italics"></emph>LKN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CK.<emph.end type="italics"></emph.end>Sed erat <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC.<emph.end type="italics"></emph.end>Ergo rurſus ex æquo <emph type="italics"></emph>LKN<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>; hoc eſt, ut velocitas corporis cadentis ad veloci<lb></lb>tatem maximam quam corpus cadendo poteſt acquirere. </s> <s>Cum <lb></lb>igitur arearum <emph type="italics"></emph>ABNK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ATD<emph.end type="italics"></emph.end>momenta <emph type="italics"></emph>LKN<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end><lb></lb>ſunt ut velocitates, erunt arearum illarum partes omnes ſimul <lb></lb>genitæ ut ſpatia ſimul deſcripta, ideoque areæ totæ ab initio <lb></lb>genitæ <emph type="italics"></emph>ABNK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ATD<emph.end type="italics"></emph.end>ut ſpatia tota ab initio deſcenſus de<lb></lb>ſcripta. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Idem conſequitur etiam de ſpatio quod in aſcenſu de<lb></lb>ſcribitur. </s> <s>Nimirum quod ſpatium illud omne ſit ad ſpatium, uNI<lb></lb>formi cum velocitate <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>eodem tempore deſcriptum, ut eſt area <lb></lb><emph type="italics"></emph>ABnk<emph.end type="italics"></emph.end>ad ſectorem <emph type="italics"></emph>ADt.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Velocitas corporis tempore <emph type="italics"></emph>ATD<emph.end type="italics"></emph.end>cadentis eſt ad ve<lb></lb>locitatem, quam eodem tempore in ſpatio non reſiſtente acquire<lb></lb>ret, ut triangulum <emph type="italics"></emph>APD<emph.end type="italics"></emph.end>ad ſectorem Hyperbolicum <emph type="italics"></emph>ATD.<emph.end type="italics"></emph.end><lb></lb>Nam velocitas in Medio non reſiſtente foret ut tempus <emph type="italics"></emph>ATD,<emph.end type="italics"></emph.end>& <lb></lb>in Medio reſiſtente eſt ut <emph type="italics"></emph>AP,<emph.end type="italics"></emph.end>id eſt, ut triangulum <emph type="italics"></emph>APD.<emph.end type="italics"></emph.end>Et <lb></lb>velocitates illæ initio deſcenſus æquantur inter ſe, perinde ut areæ <lb></lb>illæ <emph type="italics"></emph>ATD, APD.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Eodem argumento velocitas in aſcenſu eſt ad velocita<lb></lb>tem, qua corpus eodem tempore in ſpatio non reſiſtente omnem <lb></lb>ſuum aſcendendi motum amittere poſſet, ut triangulum <emph type="italics"></emph>ApD<emph.end type="italics"></emph.end>ad <lb></lb>ſectorem Circularem <emph type="italics"></emph>AtD<emph.end type="italics"></emph.end>; ſive ut recta <emph type="italics"></emph>Ap<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>At.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Eſt igitur tempus quo corpus in Medio reſiſtente caden<lb></lb>do velocitatem <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>acquirit, ad tempus quo velocitatem maximam <lb></lb><emph type="italics"></emph>AC<emph.end type="italics"></emph.end>in ſpatio non reſiſtente cadendo acquirere poſſet, ut ſector <lb></lb><emph type="italics"></emph>ADT<emph.end type="italics"></emph.end>ad triangulum <emph type="italics"></emph>ADC<emph.end type="italics"></emph.end>: & tempus, quo velocitatem <emph type="italics"></emph>Ap<emph.end type="italics"></emph.end>in <pb xlink:href="039/01/260.jpg" pagenum="232"></pb><arrow.to.target n="note208"></arrow.to.target>Medio reſiſtente aſcendendo poſſit amittere, ad tempus quo velo<lb></lb>citatem eandem in ſpatio non reſiſtente aſcendendo poſſet amit<lb></lb>tere, ut arcus <emph type="italics"></emph>At<emph.end type="italics"></emph.end>ad ejus tangentem <emph type="italics"></emph>Ap.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note208"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Hinc ex dato tempore datur ſpatium aſcenſu vel de<lb></lb>ſcenſu deſcriptum. </s> <s>Nam corporis in infinitum deſcendentis datur <lb></lb>velocitas maxima, per Corol. </s> <s>2, & 3, Theor. </s> <s>VI, Lib. </s> <s>11; indeque <lb></lb>datur tempus quo corpus velocitatem illam in ſpatio non reſiſtente <lb></lb>cadendo poſſet acquirere. </s> <s>Et ſumendo Sectorem <emph type="italics"></emph>ADT<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>ADt<emph.end type="italics"></emph.end><lb></lb>ad triangulum <emph type="italics"></emph>ADC<emph.end type="italics"></emph.end>in ratione temporis dati ad tempus modo <lb></lb>inventum; dabitur tum velocitas <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Ap,<emph.end type="italics"></emph.end>tum area <emph type="italics"></emph>ABNK<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>ABnk,<emph.end type="italics"></emph.end>quæ eſt ad ſectorem <emph type="italics"></emph>ADT<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>ADt<emph.end type="italics"></emph.end>ut ſpatium quæ<lb></lb>ſitum ad ſpatium quod tempore dato, cum velocitate illa maxima <lb></lb>jam ante inventa, uniformiter deſcribi poteſt. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Et regrediendo, ex dato aſcenſus vel deſcenſus ſpatio <lb></lb><emph type="italics"></emph>ABnk<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>ABNK,<emph.end type="italics"></emph.end>dabitur tempus <emph type="italics"></emph>ADt<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>ADT.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO X. PROBLEMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Tendat uniformis vis gravitatis directe ad planum Horizontis, <lb></lb>ſitque reſiſtentia ut Medii denſitas & quadratum velocitatis <lb></lb>conjunctim: requiritur tum Medii denſitas in locis ſingulis, <lb></lb>quæ faciat ut corpus in data quavis linea curva moveatur, <lb></lb>tum corporis velocitas & Medii reſiſtentia in locis ſingulis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>planum illud pla<lb></lb><figure id="id.039.01.260.1.jpg" xlink:href="039/01/260/1.jpg"></figure><lb></lb>no Schematis perpendicu<lb></lb>lare; <emph type="italics"></emph>PFHQ<emph.end type="italics"></emph.end>linea curva <lb></lb>plano huic occurrens in <lb></lb>punctis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph><expan abbr="q;">que</expan> G, H, I, K<emph.end type="italics"></emph.end><lb></lb>loca quatuor corporis in hac <lb></lb>curva ab <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>pergentis; <lb></lb>& <emph type="italics"></emph>GB, HC, ID, KE<emph.end type="italics"></emph.end>or<lb></lb>dinatæ quatuor parallelæ ab <lb></lb>his punctis ad horizontem <lb></lb>demiſſæ & lineæ horizontali <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad puncta <emph type="italics"></emph>B, C, D, E<emph.end type="italics"></emph.end>inſiſten<lb></lb>tes; & ſint <emph type="italics"></emph>BC, CD, DE<emph.end type="italics"></emph.end>diſtantiæ Ordinatarum inter ſe æqua<lb></lb>les. </s> <s>A punctis <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ducantur rectæ <emph type="italics"></emph>GL, HN<emph.end type="italics"></emph.end>curvam tan<lb></lb>gentes in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>& Ordinatis <emph type="italics"></emph>CH, DI<emph.end type="italics"></emph.end>ſurſum productis occur<lb></lb>rentes in <emph type="italics"></emph>L<emph.end type="italics"></emph.end>& <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>& compleatur parallelogrammum <emph type="italics"></emph>HCDM.<emph.end type="italics"></emph.end><pb xlink:href="039/01/261.jpg" pagenum="233"></pb>Et tempora quibus corpus deſcribit arcus <emph type="italics"></emph>GH, HI,<emph.end type="italics"></emph.end>erunt in <lb></lb><arrow.to.target n="note209"></arrow.to.target>ſubduplicata ratione altitudinum <emph type="italics"></emph>LH, NI<emph.end type="italics"></emph.end>quas corpus tempo<lb></lb>ribus illis deſcribere poſſet, a tangentibus cadendo: & velocitates <lb></lb>erunt ut longitudines deſcriptæ <emph type="italics"></emph>GH, HI<emph.end type="italics"></emph.end>directe & tempora in<lb></lb>verſe. </s> <s>Exponantur tempora per T & <emph type="italics"></emph>t,<emph.end type="italics"></emph.end>& velocitates per <lb></lb>(<emph type="italics"></emph>GH<emph.end type="italics"></emph.end>/T) & (<emph type="italics"></emph>HI/t<emph.end type="italics"></emph.end>): & decrementum velocitatis tempore <emph type="italics"></emph>t<emph.end type="italics"></emph.end>factum ex<lb></lb>ponetur per (<emph type="italics"></emph>GH<emph.end type="italics"></emph.end>/T)-(<emph type="italics"></emph>HI/t<emph.end type="italics"></emph.end>). Hoc decrementum oritur a reſiſtentia <lb></lb>corpus retardante & gravitate corpus accelerante. </s> <s>Gravitas in <lb></lb>corpore cadente & ſpatium <emph type="italics"></emph>NI<emph.end type="italics"></emph.end>cadendo deſcribente, generat ve<lb></lb>locitatem qua duplum illud ſpatium eodem tempore deſcribi po<lb></lb>tuiſſet (ut <emph type="italics"></emph>Galilæus<emph.end type="italics"></emph.end>demonſtravit) id eſt, velocitatem (2<emph type="italics"></emph>NI/t<emph.end type="italics"></emph.end>): at <lb></lb>in corpore arcum <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>deſcribente, auget arcum illum ſola longi<lb></lb>tudine <emph type="italics"></emph>HI-HN<emph.end type="italics"></emph.end>ſeu (<emph type="italics"></emph>MIXNI/HI<emph.end type="italics"></emph.end>), ideoque generat tantum velo<lb></lb>citatem (2<emph type="italics"></emph>MIXNI/tXHI<emph.end type="italics"></emph.end>). Addatur hæc velocitas ad decrementum <lb></lb>prædictum, & habebitur decrementum velocitatis ex reſiſtentia <lb></lb>ſola oriundum, nempe (<emph type="italics"></emph>GH<emph.end type="italics"></emph.end>/T)-<emph type="italics"></emph>(HI/t)+(2MIXNI/tXHI).<emph.end type="italics"></emph.end>Proindeque <lb></lb>cum gravitas eodem tempore in corpore cadente generet velocitatem <lb></lb>(2<emph type="italics"></emph>NI/t<emph.end type="italics"></emph.end>); Reſiſtentia erit ad Gravitatem ut (<emph type="italics"></emph>GH<emph.end type="italics"></emph.end>/T)-<emph type="italics"></emph>(HI/t)+(2MIXNI/tXHI)<emph.end type="italics"></emph.end><lb></lb>ad (<emph type="italics"></emph>2NI/t<emph.end type="italics"></emph.end>), ſive ut (<emph type="italics"></emph>tXGH<emph.end type="italics"></emph.end>/T)-<emph type="italics"></emph>HI+(2MIXNI/HI)<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>NI.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note209"></margin.target>LIBER <lb></lb>SECUNDUS</s></p> <p type="main"> <s>Jam pro abſciſſis <emph type="italics"></emph>CB, CD, CE<emph.end type="italics"></emph.end>ſcribantur -<emph type="italics"></emph>o, o,<emph.end type="italics"></emph.end>20. Pro <lb></lb>Ordinata <emph type="italics"></emph>CH<emph.end type="italics"></emph.end>ſcribatur P, & pro <emph type="italics"></emph>MI<emph.end type="italics"></emph.end>ſcribatur ſeries quælibet <lb></lb>Q<emph type="italics"></emph>o<emph.end type="italics"></emph.end>+R<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>+S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>+&c. </s> <s>Et ſeriei termini omnes poſt primum, <lb></lb>nempe R<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>+S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>+&c. </s> <s>erunt <emph type="italics"></emph>NI,<emph.end type="italics"></emph.end>& Ordinatæ <emph type="italics"></emph>DI, EK,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BG<emph.end type="italics"></emph.end><lb></lb>erunt P-Q<emph type="italics"></emph>o<emph.end type="italics"></emph.end>-R<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>-S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>-&c, P-2Q<emph type="italics"></emph>o<emph.end type="italics"></emph.end>-4R<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>-8S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>-&c, <lb></lb>& P+Q<emph type="italics"></emph>o<emph.end type="italics"></emph.end>-R<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>+S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>-&c. </s> <s>reſpective. </s> <s>Et quadrando diffe<lb></lb>rentias Ordinatarum <emph type="italics"></emph>BG-CH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CH-DI,<emph.end type="italics"></emph.end>& ad quadrata pro<lb></lb>deuntia addendo quadrata ipſarum <emph type="italics"></emph>BC, CD,<emph.end type="italics"></emph.end>habebuntur arcuum <lb></lb><emph type="italics"></emph>GH, HI<emph.end type="italics"></emph.end>quadrata <emph type="italics"></emph>oo<emph.end type="italics"></emph.end>+QQ<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>+2QR<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>+&c, & <emph type="italics"></emph>oo<emph.end type="italics"></emph.end>+QQ<emph type="italics"></emph>oo<emph.end type="italics"></emph.end><lb></lb>+2QR<emph type="italics"></emph>o<emph.end type="italics"></emph.end>+&c. </s> <s>Quorum radices <emph type="italics"></emph>o<emph.end type="italics"></emph.end>√1+QQ-(QR<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>/√1+QQ), & <pb xlink:href="039/01/262.jpg" pagenum="234"></pb><arrow.to.target n="note210"></arrow.to.target><emph type="italics"></emph>o<emph.end type="italics"></emph.end>√1+QQ+(QR<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>/√1+QQ) ſunt arcus <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>HI.<emph.end type="italics"></emph.end>Præterea ſi ab <lb></lb>Ordinata <emph type="italics"></emph>CH<emph.end type="italics"></emph.end>ſubducatur ſemiſumma Ordinatarum <emph type="italics"></emph>BG<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>DI,<emph.end type="italics"></emph.end><lb></lb>& ab Ordinata <emph type="italics"></emph>DI<emph.end type="italics"></emph.end>ſubducatur ſemiſumma Ordinatarum <emph type="italics"></emph>CH<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>EK,<emph.end type="italics"></emph.end>manebunt arcuum <emph type="italics"></emph>GI<emph.end type="italics"></emph.end>& <emph type="italics"></emph>HK<emph.end type="italics"></emph.end>ſagittæ R<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>& R<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>+3S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>. </s> <s><lb></lb>Et hæ ſunt lineolis <emph type="italics"></emph>LH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NI<emph.end type="italics"></emph.end>proportionales, adeoQ.E.I. du<lb></lb>plicata ratione temporum infinite parvorum T & <emph type="italics"></emph>t,<emph.end type="italics"></emph.end>& inde ratio <lb></lb><emph type="italics"></emph>t<emph.end type="italics"></emph.end>/T eſt √(R+3S<emph type="italics"></emph>o<emph.end type="italics"></emph.end>/R) ſeu (R+3/2S<emph type="italics"></emph>o<emph.end type="italics"></emph.end>/R): & (<emph type="italics"></emph>tXGH<emph.end type="italics"></emph.end>/T)-<emph type="italics"></emph>HI+(2MIXNI/HI),<emph.end type="italics"></emph.end><lb></lb>ſubſtituendo ipſorum <emph type="italics"></emph>t<emph.end type="italics"></emph.end>/T, <emph type="italics"></emph>GH, HI, MI<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NI<emph.end type="italics"></emph.end>valores jam in<lb></lb>ventos, evadit (3S<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>/2R)√1+Qq. </s> <s>Et cum 2<emph type="italics"></emph>NI<emph.end type="italics"></emph.end>ſit 2R<emph type="italics"></emph>oo,<emph.end type="italics"></emph.end>Re<lb></lb>ſiſtentia jam erit ad Gravitatem ut (3S<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>/2R)√1+QQ ad 2R<emph type="italics"></emph>oo,<emph.end type="italics"></emph.end><lb></lb>id eſt, ut 3S√1+QQ ad 4RR. </s></p> <p type="margin"> <s><margin.target id="note210"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Velocitas autem ea eſt quacum corpus de loco quovis <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>ſe<lb></lb>cundum tangentem <emph type="italics"></emph>HN<emph.end type="italics"></emph.end>egrediens, in Parabola diametrum <emph type="italics"></emph>HC<emph.end type="italics"></emph.end><lb></lb>& latus rectum (<emph type="italics"></emph>HNq/NI<emph.end type="italics"></emph.end>) ſeu (1+QQ/R) habente, deinceps in vacuo <lb></lb>moveri poteſt. </s></p> <p type="main"> <s>Et reſiſtentia eſt ut Medii denſitas & quadratum velocitatis <lb></lb>conjunctim, & propterea Medii denſitas eſt ut reſiſtentia directe <lb></lb>& quadratum velocitatis inverſe, id eſt, ut (3S√1+QQ/4RR) directe <lb></lb>& (1+QQ/R) inverſe, hoc eſt, ut (S/R√1+QQ). <emph type="italics"></emph>q.EI.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Si tangens <emph type="italics"></emph>HN<emph.end type="italics"></emph.end>producatur utrinQ.E.D.nec occurrat <lb></lb>Ordinatæ cuilibet <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>: erit (<emph type="italics"></emph>HT/AC<emph.end type="italics"></emph.end>) æqualis √1+QQ, adeo<lb></lb>Q.E.I. ſuperioribus pro √1+QQ ſcribi poteſt. </s> <s>Qua ratione <lb></lb>Reſiſtentia erit ad Gravitatem ut 3SX<emph type="italics"></emph>HT<emph.end type="italics"></emph.end>ad 4RRX<emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>Velo<lb></lb>citas erit ut (<emph type="italics"></emph>HT/AC<emph.end type="italics"></emph.end>√R), & Medii denſitas erit ut (SX<emph type="italics"></emph>AC<emph.end type="italics"></emph.end>/RX<emph type="italics"></emph>HT<emph.end type="italics"></emph.end>). </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et hinc, ſi Curva linea <emph type="italics"></emph>PFHQ<emph.end type="italics"></emph.end>definiatur per rela<lb></lb>tionem inter baſem ſeu abſciſſam <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>& ordinatim applicatam <pb xlink:href="039/01/263.jpg" pagenum="235"></pb><emph type="italics"></emph>CH,<emph.end type="italics"></emph.end>(ut moris eſt) & valor ordinatim applicatæ reſolvatur in ſe<lb></lb><arrow.to.target n="note211"></arrow.to.target>riem convergentem: Problema per primos ſeriei terminos expe<lb></lb>dite ſolvetur, ut in exemplis ſequentibus. </s></p> <p type="margin"> <s><margin.target id="note211"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>1. Sit Linea <emph type="italics"></emph>PFHQ<emph.end type="italics"></emph.end>Semicirculus ſuper diametro <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end><lb></lb>deſcriptus, & requiratur Medii denſitas quæ faciat ut Projectile <lb></lb>in hac linea moveatur. </s></p> <p type="main"> <s>Biſecetur diameter <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>dic <emph type="italics"></emph>AQ n, AC a, CH e,<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>CD o<emph.end type="italics"></emph.end>: & erit <emph type="italics"></emph>DIq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>AQq-ADq=nn-aa-2ao-oo,<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>ee-2ao-oo,<emph.end type="italics"></emph.end>& radice per methodum noſtram extracta, fiet <lb></lb><emph type="italics"></emph>DI=e-(ao/e)-(oo/2e)-(aaoo/2e<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)-(ao<emph type="sup"></emph>3<emph.end type="sup"></emph.end>/2e<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)-(a<emph type="sup"></emph>3<emph.end type="sup"></emph.end>o<emph type="sup"></emph>3<emph.end type="sup"></emph.end>/2e<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)<emph.end type="italics"></emph.end>-&c. </s> <s>Hic ſcribatur <emph type="italics"></emph>nn<emph.end type="italics"></emph.end><lb></lb>pro <emph type="italics"></emph>ee+aa,<emph.end type="italics"></emph.end>& evadet <emph type="italics"></emph>DI=e-(ao/e)-(nnoo/2e<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)-(anno<emph type="sup"></emph>3<emph.end type="sup"></emph.end>/2e<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)<emph.end type="italics"></emph.end>-&c. </s></p> <p type="main"> <s>Hujuſmodi ſeries diſtinguo in terminos ſucceſſivos in hunc mo<lb></lb>dum. </s> <s>Terminum primum appello in quo quantitas infinite par<lb></lb>va <emph type="italics"></emph>o<emph.end type="italics"></emph.end>non extat; ſecundum in quo quantitas illa eſt unius dimen<lb></lb>ſionis, tertium in quo extat <lb></lb><figure id="id.039.01.263.1.jpg" xlink:href="039/01/263/1.jpg"></figure><lb></lb>duarum, quartum in quo <lb></lb>trium eſt, & ſic in infiNI<lb></lb>tum. </s> <s>Et primus terminus <lb></lb>qui hic eſt <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>denotabit ſem<lb></lb>per longitudinem Ordinatæ <lb></lb><emph type="italics"></emph>CH<emph.end type="italics"></emph.end>inſiſtentis ad initium <lb></lb>indefinitæ quantitatis <emph type="italics"></emph>o<emph.end type="italics"></emph.end>; ſe<lb></lb>cundus terminus qui hic eſt <lb></lb>(<emph type="italics"></emph>ao/e<emph.end type="italics"></emph.end>), denotabit differentiam <lb></lb>inter <emph type="italics"></emph>CH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DN,<emph.end type="italics"></emph.end>id eſt, lineolam <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>quæ abſcinditur com<lb></lb>plendo parallelogrammum <emph type="italics"></emph>HCDM,<emph.end type="italics"></emph.end>atque adeo poſitionem tan<lb></lb>gentis <emph type="italics"></emph>HN<emph.end type="italics"></emph.end>ſemper determinat: ut in hoc caſu capiendo <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>HM<emph.end type="italics"></emph.end>ut eſt (<emph type="italics"></emph>ao/e<emph.end type="italics"></emph.end>) ad <emph type="italics"></emph>o,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>a<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>e.<emph.end type="italics"></emph.end>Terminus tertius qui hic eſt <lb></lb>(<emph type="italics"></emph>nnoo/2e<emph type="sup"></emph>3<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>) deſignabit lineolam <emph type="italics"></emph>IN<emph.end type="italics"></emph.end>quæ jacet inter tangentem & cur<lb></lb>vam, adeoQ.E.D.terminat angulum contactus <emph type="italics"></emph>IHN<emph.end type="italics"></emph.end>ſeu curvatu<lb></lb>ram quam curva linea habet in <emph type="italics"></emph>H.<emph.end type="italics"></emph.end>Si lineola illa <emph type="italics"></emph>IN<emph.end type="italics"></emph.end>finitæ eſt <lb></lb>magnitudinis, deſignabitur per terminum tertium una cum ſe<lb></lb>quentibus in infinitum. </s> <s>At ſi lineola illa minuatur in infinitum, <pb xlink:href="039/01/264.jpg" pagenum="236"></pb><arrow.to.target n="note212"></arrow.to.target>termini ſubſequentes evadent infinite minores tertio, ideoque neg<lb></lb>ligi poſſunt. </s> <s>Terminus quartus determinat variationem curva<lb></lb>turæ, quintus variationem variationis, & ſic deinceps. </s> <s>Unde obi<lb></lb>ter patet uſus non contemnendus harum Serierum in ſolutione <lb></lb>Problematum quæ pendent a tangentibus & curvatura curvarum. </s></p> <p type="margin"> <s><margin.target id="note212"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Conferatur jam ſeries <emph type="italics"></emph>e-(ao/e)-(nnoo/2e<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)-(anno<emph type="sup"></emph>3<emph.end type="sup"></emph.end>/2e<emph type="sup"></emph>5<emph.end type="sup"></emph.end>)<emph.end type="italics"></emph.end>-&c, cum ſerie <lb></lb>P-Q<emph type="italics"></emph>o<emph.end type="italics"></emph.end>-R<emph type="italics"></emph>oo<emph.end type="italics"></emph.end>-S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>-&c. </s> <s>& perinde pro P, Q, R & S ſcribatur <lb></lb><emph type="italics"></emph>e, (a/e), (nn/2e<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)<emph.end type="italics"></emph.end>& (<emph type="italics"></emph>ann/2e<emph type="sup"></emph>5<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>), & pro √1+QQ ſcribatur √1+(<emph type="italics"></emph>aa/ee<emph.end type="italics"></emph.end>) ſeu <emph type="italics"></emph>n/e,<emph.end type="italics"></emph.end>& <lb></lb>prodibit Medii denſitas ut (<emph type="italics"></emph>a/ne<emph.end type="italics"></emph.end>), hoc eſt, (ob datam <emph type="italics"></emph>n,<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>a/e,<emph.end type="italics"></emph.end>ſeu <lb></lb>(<emph type="italics"></emph>AC/CH<emph.end type="italics"></emph.end>), id eſt, ut tangentis longitudo illa <emph type="italics"></emph>HT<emph.end type="italics"></emph.end>quæ ad ſemidiame<lb></lb>trum <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>normaliter inſiſtentem terminatur: & reſiſten<lb></lb>tia erit ad gravitatem ut 3<emph type="italics"></emph>a<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>n,<emph.end type="italics"></emph.end>id eſt, ut 3 <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad Circuli <lb></lb>diametrum <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>: velocitas autem erit ut √<emph type="italics"></emph>CH.<emph.end type="italics"></emph.end>Quare ſi corpus <lb></lb>juſta cum velocitate ſecundum lineam ipſi <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>parallelam exeat <lb></lb>de loco <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>& Medii denſitas in ſingulis locis <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ſit ut longi<lb></lb>tudo tangentis <emph type="italics"></emph>HT,<emph.end type="italics"></emph.end>& reſiſtentia etiam in loco aliquo <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ſit ad <lb></lb>vim gravitatis ut 3 <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>corpus illud deſcribet Circuli <lb></lb>quadrantem <emph type="italics"></emph>FHQ. Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>At ſi corpus idem de loco <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ſecundum lineam ipſi <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>per<lb></lb>pendicularem egrederetur, & in arcu ſemicirculi <emph type="italics"></emph>PFQ<emph.end type="italics"></emph.end>moveri <lb></lb>inciperet, ſumenda eſſet <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>a<emph.end type="italics"></emph.end>ad contrarias partes centri <emph type="italics"></emph>A,<emph.end type="italics"></emph.end><lb></lb>& propterea ſignum ejus mutandum eſſet & ſcribendum -<emph type="italics"></emph>a<emph.end type="italics"></emph.end>pro <lb></lb>+<emph type="italics"></emph>a.<emph.end type="italics"></emph.end>Quo pacto prodiret Medii denſitas ut -<emph type="italics"></emph>a/e<emph.end type="italics"></emph.end>. </s> <s>Negativam <lb></lb>autem denſitatem, hoc eſt, quæ motus corporum accelerat, Na<lb></lb>tura non admittit: & propterea naturaliter fieri non poteſt, ut <lb></lb>corpus aſcendendo a <emph type="italics"></emph>P<emph.end type="italics"></emph.end>deſcribat Circuli quadrantem <emph type="italics"></emph>PF.<emph.end type="italics"></emph.end>Ad <lb></lb>hunc effectum deberet corpus a Medio impellente accelerari, non <lb></lb>a reſiſtente impediri. </s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>2. Sit linea <emph type="italics"></emph>PFHQ<emph.end type="italics"></emph.end>Parabola, axem habens <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>ho<lb></lb>rizonti <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>perpendicularem, & requiratur Medii denſitas quæ <lb></lb>faciat ut Projectile in ipſa moveatur. </s></p> <p type="main"> <s>Ex natura Parabolæ, rectangulum <emph type="italics"></emph>PDQ<emph.end type="italics"></emph.end>æquale eſt rectan<lb></lb>gulo ſub ordinata <emph type="italics"></emph>DI<emph.end type="italics"></emph.end>& recta aliqua data: hoc eſt, ſi dicantur <pb xlink:href="039/01/265.jpg" pagenum="237"></pb>recta illa <emph type="italics"></emph>b, PC a, PQ c, CH e<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CD o<emph.end type="italics"></emph.end>; rectangulum <emph type="italics"></emph>a+o<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note213"></arrow.to.target>in <emph type="italics"></emph>c-a-o<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>ac-aa-2ao+co-oo<emph.end type="italics"></emph.end>æquale eſt rectangulo <lb></lb><emph type="italics"></emph>b<emph.end type="italics"></emph.end>in <emph type="italics"></emph>DI,<emph.end type="italics"></emph.end>adeoque <emph type="italics"></emph>DI<emph.end type="italics"></emph.end>æquale <emph type="italics"></emph>(ac-aa/b)+(c-2a/b)o-(oo/b).<emph.end type="italics"></emph.end>Jam ſcri<lb></lb>bendus eſſet hujus ſeriei ſecundus terminus <emph type="italics"></emph>(c-2a/b)o<emph.end type="italics"></emph.end>pro Q<emph type="italics"></emph>o,<emph.end type="italics"></emph.end>ter<lb></lb>tius item terminus (<emph type="italics"></emph>oo/b<emph.end type="italics"></emph.end>) pro R<emph type="italics"></emph>oo.<emph.end type="italics"></emph.end>Cum vero plures non ſint ter<lb></lb>mini, debebit quarti coefficiens S evaneſcere, & propterea quan<lb></lb>titas (S/R√1+QQ) cui Medii denſitas proportionalis eſt, nihil <lb></lb>erit. </s> <s>Nulla igitur Medii denſitate movebitur Projectile in Para<lb></lb>bola, uti olim demonſtravit <emph type="italics"></emph>Galilæus, Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note213"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>3. Sit linea <emph type="italics"></emph>AGK<emph.end type="italics"></emph.end>Hyperbola, Aſymptoton habens <lb></lb><emph type="italics"></emph>NX<emph.end type="italics"></emph.end>plano horizontali <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>perpendicularem; & quæratur Medii <lb></lb>denſitas quæ faciat ut Projectile moveatur in hac linea. </s></p> <p type="main"> <s>Sit <emph type="italics"></emph>MX<emph.end type="italics"></emph.end>Aſymptotos altera, ordinatim applicatæ <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>productæ <lb></lb>occurrens in <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>& ex natura Hyperbolæ, rectangulum <emph type="italics"></emph>XV<emph.end type="italics"></emph.end>in <emph type="italics"></emph>VG<emph.end type="italics"></emph.end><lb></lb>dabitur. </s> <s>Datur autem ratio <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>VX,<emph.end type="italics"></emph.end>& propterea datur etiam <lb></lb>rectangulum <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>in <emph type="italics"></emph>VG.<emph.end type="italics"></emph.end>Sit illud <emph type="italics"></emph>bb<emph.end type="italics"></emph.end>; & completo parallelogrammo <lb></lb><emph type="italics"></emph>DNXZ,<emph.end type="italics"></emph.end>dicatur <emph type="italics"></emph>BN a, BD o, NX c,<emph.end type="italics"></emph.end>& ratio data <emph type="italics"></emph>VZ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ZX<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ponatur eſſe <emph type="italics"></emph>m/n<emph.end type="italics"></emph.end>. </s> <s>Et erit <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>a-o, VG<emph.end type="italics"></emph.end>æqualis <lb></lb><emph type="italics"></emph>(bb/a-o), VZ<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>m/n—a-o,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>GD<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>NX-VZ-VG<emph.end type="italics"></emph.end>æ<lb></lb>qualis <emph type="italics"></emph>c-m/n a+m/n o-(bb/a-o).<emph.end type="italics"></emph.end>Reſolvatur terminus (<emph type="italics"></emph>bb/a-o<emph.end type="italics"></emph.end>) in ſeriem <lb></lb>convergentem <emph type="italics"></emph>(bb/a)+(bb/aa)o+(bb/a<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)oo+(bb/a<emph type="sup"></emph>4<emph.end type="sup"></emph.end>)o<emph type="sup"></emph>3<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>&c. </s> <s>& ſiet <emph type="italics"></emph>GD<emph.end type="italics"></emph.end>æqua<lb></lb>lis <emph type="italics"></emph>c-m/n a-(bb/a)+m/n o-(bb/aa)o-(bb/a<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)o<emph type="sup"></emph>2<emph.end type="sup"></emph.end>-(bb/a<emph type="sup"></emph>4<emph.end type="sup"></emph.end>)o<emph type="sup"></emph>3<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>&c. </s> <s>Hujus ſeriei termi<lb></lb>nus ſecundus <emph type="italics"></emph>m/no-(bb/aa)o<emph.end type="italics"></emph.end>uſurpandus eſt pro Q<emph type="italics"></emph>o,<emph.end type="italics"></emph.end>tertius cum ſigno <lb></lb>mutato <emph type="italics"></emph>(bb/a<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)o<emph type="sup"></emph>2<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>pro R<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>2<emph.end type="sup"></emph.end>, & quartus cum ſigno etiam mutato <emph type="italics"></emph>(bb/a<emph type="sup"></emph>4<emph.end type="sup"></emph.end>)o<emph type="sup"></emph>1<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end><lb></lb>pro S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>, eorumque coefficientes <emph type="italics"></emph>m/n-(bb/aa), (bb/a<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)<emph.end type="italics"></emph.end>& (<emph type="italics"></emph>bb/a<emph type="sup"></emph>4<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>) ſcribendæ ſunt <lb></lb>in Regula ſuperiore, pro Q, R & S. </s> <s>Quo facto prodit medii denſitas <pb xlink:href="039/01/266.jpg" pagenum="238"></pb><arrow.to.target n="note214"></arrow.to.target>ut (<emph type="italics"></emph>(bb/a<emph type="sup"></emph>4<emph.end type="sup"></emph.end>)/(bb/a<emph type="sup"></emph>3<emph.end type="sup"></emph.end>)√1+(mm/nn)-(2mbb/naa)+(b<emph type="sup"></emph>4<emph.end type="sup"></emph.end>/a<emph type="sup"></emph>4<emph.end type="sup"></emph.end>)<emph.end type="italics"></emph.end>) ſeu (1/<emph type="italics"></emph>√aa+(mm/nn)aa-(2mbb/n)+(b<emph type="sup"></emph>4<emph.end type="sup"></emph.end>/aa)<emph.end type="italics"></emph.end>) id <lb></lb>eſt, ſi in <emph type="italics"></emph>VZ<emph.end type="italics"></emph.end>ſumatur <emph type="italics"></emph>VY<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>VG,<emph.end type="italics"></emph.end>ut (1/<emph type="italics"></emph>XY<emph.end type="italics"></emph.end>). Namque <emph type="italics"></emph>aa<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>(mm/nn)aa-(2mbb/n)+(b<emph type="sup"></emph>4<emph.end type="sup"></emph.end>/aa)<emph.end type="italics"></emph.end>ſunt ipſarum <emph type="italics"></emph>XZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ZY<emph.end type="italics"></emph.end>quadrata. </s> <s>Reſiſten<lb></lb>tia autem invenitur in ratione ad gravitatem quam habet 3 <emph type="italics"></emph>XY<emph.end type="italics"></emph.end>ad <lb></lb><figure id="id.039.01.266.1.jpg" xlink:href="039/01/266/1.jpg"></figure><lb></lb>2<emph type="italics"></emph>YG<emph.end type="italics"></emph.end>& velocitas ea eſt quacum corpus in Parabola pergeret verti<lb></lb>cem <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>diametrum <emph type="italics"></emph>DG,<emph.end type="italics"></emph.end>& latus rectum (<emph type="italics"></emph>XYquad./VG<emph.end type="italics"></emph.end>) habente. </s> <s>Pona<lb></lb>tur itaque quod Medii denſitates in locis ſingulis <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ſint reciproce <lb></lb>ut diſtantiæ <emph type="italics"></emph>XY,<emph.end type="italics"></emph.end>quodque reſiſtentia in loco aliquo <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ſit ad gra<lb></lb>vitatem ut 3<emph type="italics"></emph>XY<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>YG<emph.end type="italics"></emph.end>; & corpus de loco <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>juſta cum veloci<lb></lb>tate emiſſum, deſcribet Hyperbolam illam <emph type="italics"></emph>AGK. Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note214"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Exempl.<emph.end type="italics"></emph.end>4. Ponatur indefinite, quod linea <emph type="italics"></emph>AGK<emph.end type="italics"></emph.end>Hyperbola ſit, <lb></lb>centro <emph type="italics"></emph>X<emph.end type="italics"></emph.end>Aſymptotis <emph type="italics"></emph>MX, NX<emph.end type="italics"></emph.end>ea lege deſcripta, ut conſtructo <lb></lb>rectangulo <emph type="italics"></emph>XZDN<emph.end type="italics"></emph.end>cujus latus <emph type="italics"></emph>ZD<emph.end type="italics"></emph.end>ſecet Hyperbolam in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <pb xlink:href="039/01/267.jpg" pagenum="239"></pb>Aſymptoton ejus in <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>fuerit <emph type="italics"></emph>VG<emph.end type="italics"></emph.end>reciproce ut ipſius <emph type="italics"></emph>ZX<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>DN<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note215"></arrow.to.target>dignitas aliqua <emph type="italics"></emph>DN<emph type="sup"></emph>n<emph.end type="sup"></emph.end>,<emph.end type="italics"></emph.end>cujus index eſt numerus <emph type="italics"></emph>n<emph.end type="italics"></emph.end>: & quæratur <lb></lb>Medii denſitas, qua Projectile progrediatur in hac curva. </s></p> <p type="margin"> <s><margin.target id="note215"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Pro <emph type="italics"></emph>BN, BD, NX<emph.end type="italics"></emph.end>ſcribantur A, O, C reſpective, ſitque <emph type="italics"></emph>VZ<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>XZ<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>d<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>VG<emph.end type="italics"></emph.end>æqualis (<emph type="italics"></emph>bb/DN<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>), & erit <emph type="italics"></emph>DN<emph.end type="italics"></emph.end>æqua<lb></lb>lis A-O, <emph type="italics"></emph>VG<emph.end type="italics"></emph.end>=(<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>/—<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>A-O), <emph type="italics"></emph>VZ<emph.end type="italics"></emph.end>=<emph type="italics"></emph>d/e<emph.end type="italics"></emph.end>—A-O, & <emph type="italics"></emph>GD<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>NX-VZ <lb></lb>-VG<emph.end type="italics"></emph.end>æqualis C-<emph type="italics"></emph>d/e<emph.end type="italics"></emph.end>A+<emph type="italics"></emph>d/e<emph.end type="italics"></emph.end>O-(<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>/—<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>A-O). Reſolvatur terminus ille <lb></lb>(<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>/—<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>A-O) in ſeriem infinitam (<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>)+(<emph type="italics"></emph>nbb<emph.end type="italics"></emph.end>/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+1<emph.end type="sup"></emph.end>)O+(<emph type="italics"></emph>nn+n<emph.end type="italics"></emph.end>/2A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+2<emph.end type="sup"></emph.end>)<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>O<emph type="sup"></emph>2<emph.end type="sup"></emph.end>+ <lb></lb>(<emph type="italics"></emph>n<emph type="sup"></emph>3<emph.end type="sup"></emph.end>+3nn+2n<emph.end type="italics"></emph.end>/6A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+3<emph.end type="sup"></emph.end>)<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>O<emph type="sup"></emph>3<emph.end type="sup"></emph.end> &c. </s> <s>ac fiet <emph type="italics"></emph>GD<emph.end type="italics"></emph.end>æqualis C-<emph type="italics"></emph>d/e<emph.end type="italics"></emph.end>A-(<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>)+ <lb></lb><emph type="italics"></emph>d/e<emph.end type="italics"></emph.end>O-(<emph type="italics"></emph>nbb<emph.end type="italics"></emph.end>/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+1<emph.end type="sup"></emph.end>)O-(<emph type="italics"></emph>+nn+n<emph.end type="italics"></emph.end>/2A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+2<emph.end type="sup"></emph.end>)<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>O<emph type="sup"></emph>2<emph.end type="sup"></emph.end>-(<emph type="italics"></emph>+n<emph type="sup"></emph>3<emph.end type="sup"></emph.end>+3nn+2n<emph.end type="italics"></emph.end>/6A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+3<emph.end type="sup"></emph.end>)<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>O<emph type="sup"></emph>3<emph.end type="sup"></emph.end> &c. </s> <s>Hu<lb></lb>jus ſeriei terminus ſecundus <emph type="italics"></emph>d/e<emph.end type="italics"></emph.end>O-(<emph type="italics"></emph>nbb<emph.end type="italics"></emph.end>/A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+1<emph.end type="sup"></emph.end>)O uſurpandus eſt pro Q<emph type="italics"></emph>o,<emph.end type="italics"></emph.end><lb></lb>tertius (<emph type="italics"></emph>nn+n<emph.end type="italics"></emph.end>/2A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+2<emph.end type="sup"></emph.end>)<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>O<emph type="sup"></emph>2<emph.end type="sup"></emph.end> pro R<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>2<emph.end type="sup"></emph.end>, quartus (<emph type="italics"></emph>n<emph type="sup"></emph>3<emph.end type="sup"></emph.end>+3nn+2n<emph.end type="italics"></emph.end>/6A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+3<emph.end type="sup"></emph.end>)<emph type="italics"></emph>bb<emph.end type="italics"></emph.end>O<emph type="sup"></emph>3<emph.end type="sup"></emph.end> pro <lb></lb>S<emph type="italics"></emph>o<emph.end type="italics"></emph.end><emph type="sup"></emph>3<emph.end type="sup"></emph.end>. </s> <s>Et inde Medii denſitas (S/R√1+QQ), in loco quovis <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>fit <lb></lb>(<emph type="italics"></emph>n<emph.end type="italics"></emph.end>+2/3√A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>+(<emph type="italics"></emph>dd/ee<emph.end type="italics"></emph.end>)A<emph type="sup"></emph>2<emph.end type="sup"></emph.end>-(<emph type="italics"></emph>2dnbb/e<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>)A+(<emph type="italics"></emph>nnb<emph.end type="italics"></emph.end><emph type="sup"></emph>4<emph.end type="sup"></emph.end>/A<emph type="sup"></emph>2<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>)), adeoque ſi in <emph type="italics"></emph>VZ<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>VY<emph.end type="italics"></emph.end><lb></lb>æqualis <emph type="italics"></emph>nXVG,<emph.end type="italics"></emph.end>denſitas illa eſt reciproce ut <emph type="italics"></emph>XY.<emph.end type="italics"></emph.end>Sunt enim A<emph type="sup"></emph>2<emph.end type="sup"></emph.end><lb></lb>& (<emph type="italics"></emph>dd/ee<emph.end type="italics"></emph.end>)A<emph type="sup"></emph>3<emph.end type="sup"></emph.end>-(2<emph type="italics"></emph>dnbb/e<emph.end type="italics"></emph.end>A<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>)A+(<emph type="italics"></emph>nnb<emph.end type="italics"></emph.end><emph type="sup"></emph>4<emph.end type="sup"></emph.end>/A<emph type="sup"></emph>2<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>) ipſarum <emph type="italics"></emph>XZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ZY<emph.end type="italics"></emph.end>quadrata. </s> <s>Reſiſten<lb></lb>tia autem in eodem loco <emph type="italics"></emph>G<emph.end type="italics"></emph.end>fit ad gravitatem ut 3S in (<emph type="italics"></emph>XY<emph.end type="italics"></emph.end>/A) ad 4RR, <lb></lb>id eſt, <emph type="italics"></emph>XY<emph.end type="italics"></emph.end>ad (<emph type="italics"></emph>2nn+2n/n+2)VG.<emph.end type="italics"></emph.end>Et velocitas ibidem ea ipſa eſt qua<lb></lb>cum corpus projectum in Parabola pergeret, verticem <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>diametrum <lb></lb><emph type="italics"></emph>GD<emph.end type="italics"></emph.end>& latus rectum (1+QQ/R) ſeu (2<emph type="italics"></emph>XYquad./—nn+n<emph.end type="italics"></emph.end>in<emph type="italics"></emph>VG<emph.end type="italics"></emph.end>) habente. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end><pb xlink:href="039/01/268.jpg" pagenum="240"></pb><arrow.to.target n="note216"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note216"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Eadem ratione qua prodiit denſitas Medii ut (SX<emph type="italics"></emph>AC<emph.end type="italics"></emph.end>/RX<emph type="italics"></emph>HT<emph.end type="italics"></emph.end>) in Co<lb></lb>rollario primo, ſi reſiſtentia ponatur ut velocitatis V dignitas quæ<lb></lb>libet V<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end> prodibit denſitas Medii ut (S/R(4-<emph type="italics"></emph>n<emph.end type="italics"></emph.end>/2))X(—<emph type="italics"></emph>AC/HT<emph.end type="italics"></emph.end>|<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1.<emph.end type="sup"></emph.end>) </s></p> <p type="main"> <s>Et propterea ſi Curva inveniri poteſt ea lege ut data fuerit ratio <lb></lb>(S/R(4-<emph type="italics"></emph>n<emph.end type="italics"></emph.end>/2)) ad (—<emph type="italics"></emph>HT/AC<emph.end type="italics"></emph.end>|<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>), vel (S<emph type="sup"></emph>2<emph.end type="sup"></emph.end>/R<emph type="sup"></emph>4-<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>) ad (—1+QQ|<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>-1<emph.end type="sup"></emph.end>): corpus move<lb></lb>bitur in hac Curva in uniformi Medio cum reſiſtentia quæ ſit ut <lb></lb>velocitatis dignitas V<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>. </s> <s>Sed redeamus ad Curvas ſimpliciores. </s></p> <p type="main"> <s>Quoniam motus non fit in Parabola niſi in Medio non reſiſten<lb></lb>te, in Hyperbolis vero hic deſcriptis fit per reſiſtentiam perpetuam; <lb></lb>perſpicuum eſt quod Linea, quam projectile in Medio uniformiter <lb></lb>reſiſtente deſcribit, propius accedit ad Hyperbolas haſce quam ad <lb></lb>Parabolam. </s> <s>Eſt utique linea illa Hyperbolici generis, ſed quæ <lb></lb>circa verticem magis diſtat ab Aſymptotis; in partibus a vertice <lb></lb>remotioribus propius ad ipſas accedit quam pro ratione Hyper<lb></lb>bolarum quas hic deſcripſi. </s> <s>Tanta vero non eſt inter has & illam <lb></lb>differentia, quin illius loco poſſint hæ in rebus practicis non in<lb></lb>commode adhiberi. </s> <s>Et utiliores forſan futuræ ſunt hæ, quam <lb></lb>Hyperbola magis accurata & ſimul magis compoſita. </s> <s>Ipſæ vero <lb></lb>in uſum ſic deducentur. </s></p> <p type="main"> <s>Compleatur parallelogrammum <emph type="italics"></emph>XYGT,<emph.end type="italics"></emph.end>& recta <emph type="italics"></emph>GT<emph.end type="italics"></emph.end>tanget <lb></lb>Hyperbolam in <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>ideoQ.E.D.nſitas Medii in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>eſt reciproce ut <lb></lb>tangens <emph type="italics"></emph>GT,<emph.end type="italics"></emph.end>& velocitas ibidem ut √(<emph type="italics"></emph>GTq/GV<emph.end type="italics"></emph.end>), reſiſtentia autem ad <lb></lb>vim gravitatis ut <emph type="italics"></emph>GT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>(2nn+2n/n+2)GV.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Proinde ſi corpus de loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſecundum rectam <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>projectum <lb></lb>deſcribat Hyperbolam <emph type="italics"></emph>AGK,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>producta occurrat Aſymp<lb></lb>toto <emph type="italics"></emph>MX<emph.end type="italics"></emph.end>in <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>actaque <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>eidem parallela occurrat alteri Aſymp<lb></lb>toto <emph type="italics"></emph>MX<emph.end type="italics"></emph.end>in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>: erit Medii denſitas in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>reciproce ut <emph type="italics"></emph>AH,<emph.end type="italics"></emph.end>& cor<lb></lb>poris velocitas ut √(<emph type="italics"></emph>AHq/AI<emph.end type="italics"></emph.end>), ac reſiſtentia ibidem ad gravitatem ut <lb></lb><emph type="italics"></emph>AH<emph.end type="italics"></emph.end>ad (<emph type="italics"></emph>2nn+2n/n+2<emph.end type="italics"></emph.end>) in <emph type="italics"></emph>AI.<emph.end type="italics"></emph.end>Unde prodeunt ſequentes Regulæ. </s></p><pb xlink:href="039/01/269.jpg" pagenum="241"></pb> <p type="main"> <s><emph type="italics"></emph>Reg.<emph.end type="italics"></emph.end>1. Si ſervetur tum Medii denſitas in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>tum velocitas qua<lb></lb><arrow.to.target n="note217"></arrow.to.target>cum corpus projicitur, & mutetur angulus <emph type="italics"></emph>NAH<emph.end type="italics"></emph.end>; manebunt lon<lb></lb>gitudines <emph type="italics"></emph>AH, AI, HX.<emph.end type="italics"></emph.end>Ideoque ſi longitudines illæ in aliquo <lb></lb>caſu inveniantur, Hyperbola deinceps ex dato quovis angulo <emph type="italics"></emph>NAH<emph.end type="italics"></emph.end><lb></lb>expedite determinari poteſt. </s></p> <p type="margin"> <s><margin.target id="note217"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Reg.<emph.end type="italics"></emph.end>2. Si ſervetur tum angulus <emph type="italics"></emph>NAH,<emph.end type="italics"></emph.end>tum Medii denſitas <lb></lb>in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>& mutetur velocitas quacum corpus projicitur; ſervabitur <lb></lb>longitudo <emph type="italics"></emph>AH,<emph.end type="italics"></emph.end>& mutabitur <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>in duplicata ratione velocitatis <lb></lb>reciproce. </s></p> <p type="main"> <s><emph type="italics"></emph>Reg.<emph.end type="italics"></emph.end>3. Si tam angulus <emph type="italics"></emph>NAH<emph.end type="italics"></emph.end>quam corporis velocitas in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end><lb></lb>gravitaſque acceleratrix ſervetur, & proportio reſiſtentiæ in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <lb></lb><figure id="id.039.01.269.1.jpg" xlink:href="039/01/269/1.jpg"></figure><lb></lb>gravitatem motricem augeatur in ratione quacunque: augebitur <lb></lb>proportio <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>in eadem ratione, manente Parabolæ late<lb></lb>re recto, eique proportionali longitudine (<emph type="italics"></emph>AHq/AI<emph.end type="italics"></emph.end>); & propterea mi<lb></lb>nuetur <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>in eadem ratione, & <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>minuetur in ratione illa du<lb></lb>plicata. </s> <s>Augetur vero proportio reſiſtentiæ ad pondus, ubi vel gra<lb></lb>vitas ſpecifica ſub æquali magnitudine fit minor, vel Medii denſi<lb></lb>tas major, vel reſiſtentia, ex magnitudine diminuta, diminuitur in <lb></lb>minore ratione quam pondus. <pb xlink:href="039/01/270.jpg" pagenum="242"></pb><arrow.to.target n="note218"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note218"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Reg.<emph.end type="italics"></emph.end>4. Quoniam denſitas Medii prope verticem Hyperbolæ <lb></lb>major eſt quam in loco <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut habeatur denſitas mediocris, debet <lb></lb>ratio minimæ tangentium <emph type="italics"></emph>GT<emph.end type="italics"></emph.end>ad tangentem <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>inveniri, & <lb></lb>denſitas in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>angeri in ratione paudo majore quam ſemiſummæ <lb></lb>harum tangentium ad minimam tangentium <emph type="italics"></emph>GT.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Reg.<emph.end type="italics"></emph.end>5. Si dantur longitudines <emph type="italics"></emph>AH, AI,<emph.end type="italics"></emph.end>& deſcribenda ſit Figu<lb></lb>ra <emph type="italics"></emph>AGK:<emph.end type="italics"></emph.end>produc <emph type="italics"></emph>HN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>X,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>HX<emph.end type="italics"></emph.end>æqualis facto ſub <emph type="italics"></emph>n<emph.end type="italics"></emph.end>+1 & <lb></lb><emph type="italics"></emph>AI<emph.end type="italics"></emph.end>; centroque <emph type="italics"></emph>X<emph.end type="italics"></emph.end>& Aſymptotis <emph type="italics"></emph>MX, NX<emph.end type="italics"></emph.end>per punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>deſcriba<lb></lb>tur Hyperbola, ea lege, ut ſit <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>ad quamvis <emph type="italics"></emph>VG<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>XV<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>ad <emph type="italics"></emph>XI<emph type="sup"></emph>n<emph.end type="sup"></emph.end>.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Reg.<emph.end type="italics"></emph.end>6. Quo major eſt numerus <emph type="italics"></emph>n,<emph.end type="italics"></emph.end>eo magis accuratæ ſunt hæ <lb></lb>Hyperbolæ in aſcenſu corporis ab <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>& minus accuratæ in ejus de<lb></lb>ſcenſu ad <emph type="italics"></emph>K<emph.end type="italics"></emph.end>; & contra. </s> <s>Hyperbola Conica mediocrem rationem <lb></lb>tenet, eſt que cæteris ſimplicior. </s> <s>Igitur ſi Hyperbola ſit hujus generis, <lb></lb>& punctum <emph type="italics"></emph>K,<emph.end type="italics"></emph.end>ubi corpus projectum incidet in rectam quamvis <emph type="italics"></emph>AN<emph.end type="italics"></emph.end><lb></lb>per punctum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>tranſeuntem, quæratur: occurrat producta <emph type="italics"></emph>AN<emph.end type="italics"></emph.end><lb></lb>Aſymptotis <emph type="italics"></emph>MX, NX<emph.end type="italics"></emph.end>in <emph type="italics"></emph>M<emph.end type="italics"></emph.end>& <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>& ſumatur <emph type="italics"></emph>NK<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>AM<emph.end type="italics"></emph.end>æqualis. </s></p> <p type="main"> <s><emph type="italics"></emph>Reg.<emph.end type="italics"></emph.end>7. Et hinc liquet methodus expedita determinandi hanc <lb></lb>Hyperbolam ex Phænomenis. </s> <s>Projiciantur corpora duo ſimilia & <lb></lb>æqualia, eadem velocitate, in angulis diverſis <emph type="italics"></emph>HAK, hAk,<emph.end type="italics"></emph.end>inci<lb></lb>dantQ.E.I. planum Horizontis in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>k<emph.end type="italics"></emph.end>; & notetur proportio <emph type="italics"></emph>AK<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>Ak.<emph.end type="italics"></emph.end>Sit ea <emph type="italics"></emph>d<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>e.<emph.end type="italics"></emph.end>Tum erecto cujuſvis longitudinis perpen<lb></lb>diculo <emph type="italics"></emph>AI,<emph.end type="italics"></emph.end>aſſume utcunque longitudinem <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Ah,<emph.end type="italics"></emph.end>& inde <lb></lb>collige graphice longitudines <emph type="italics"></emph>AK, Ak,<emph.end type="italics"></emph.end>per Reg. </s> <s>6. Si ratio <emph type="italics"></emph>AK<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>Ak<emph.end type="italics"></emph.end>ſit eadem cum ratione <emph type="italics"></emph>d<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>longitudo <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>recte aſſump<lb></lb>ta fuit. </s> <s>Sin minus cape in recta infinita <emph type="italics"></emph>SM<emph.end type="italics"></emph.end>longitudinem <emph type="italics"></emph>SM<emph.end type="italics"></emph.end><lb></lb>æqualem aſſumptæ <emph type="italics"></emph>AH,<emph.end type="italics"></emph.end>& erige perpendiculum <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>æquale ra<lb></lb>tionum differentiæ <emph type="italics"></emph>(AK/Ak)-d/e<emph.end type="italics"></emph.end>ductæ in rectam quamvis datam. </s> <s>Si<lb></lb>mili methodo ex aſſumptis pluribus longitudinibus <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>invenien<lb></lb>da ſunt plura puncta <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>& per omnia a<lb></lb><figure id="id.039.01.270.1.jpg" xlink:href="039/01/270/1.jpg"></figure><lb></lb>genda Curva linea regularis <emph type="italics"></emph>NNXN,<emph.end type="italics"></emph.end>ſe<lb></lb>cans rectam <emph type="italics"></emph>SMMM<emph.end type="italics"></emph.end>in <emph type="italics"></emph>X.<emph.end type="italics"></emph.end>Aſſumatur <lb></lb>demum <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>æqualie abſciſſæ <emph type="italics"></emph>SX<emph.end type="italics"></emph.end>& inde <lb></lb>denuo inveniatur longitudo <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>; & lon<lb></lb>gitudines, quæ ſint ad aſſumptam longitu<lb></lb>dinem <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>& hanc ultimam <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>ut longitudo <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>per experi<lb></lb>mentum cognita ad ultimo inventam longitudinem <emph type="italics"></emph>AK,<emph.end type="italics"></emph.end>erunt veræ <lb></lb>illæ longitudines <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AH,<emph.end type="italics"></emph.end>quas invenire oportuit. </s> <s>Hiſce vero <lb></lb>datis dabitur & reſiſtentia Medii in loco <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>quippe quæ ſit ad vim <lb></lb>gravitatis ut <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>AI.<emph.end type="italics"></emph.end>Augenda eſt autem denſitas. </s> <s>Medii per <lb></lb>Reg. </s> <s>4; & reſiſtentia modo inventa, ſi in eadem ratione augeatur, fiet <lb></lb>accuratior. </s></p><pb xlink:href="039/01/271.jpg" pagenum="243"></pb> <p type="main"> <s><emph type="italics"></emph>Reg.<emph.end type="italics"></emph.end>8. Inventis longitudinibus <emph type="italics"></emph>AH, HX<emph.end type="italics"></emph.end>; ſi jam deſideretur </s></p> <p type="main"> <s><arrow.to.target n="note219"></arrow.to.target>poſitio rectæ <emph type="italics"></emph>AH,<emph.end type="italics"></emph.end>ſecundum quam Projectile, data illa cum veloci<lb></lb>tate emiſſum, incidit in punctum quodvis <emph type="italics"></emph>K:<emph.end type="italics"></emph.end>ad puncta <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end><lb></lb>erigantur rectæ <emph type="italics"></emph>AC, KF<emph.end type="italics"></emph.end>horizonti perpendiculares, quarum <emph type="italics"></emph>AC<emph.end type="italics"></emph.end><lb></lb>deorſum tendat, & æquetur ipſi <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>ſeu 1/2<emph type="italics"></emph>HX.<emph.end type="italics"></emph.end>Aſymptotis <emph type="italics"></emph>AK, <lb></lb>KF<emph.end type="italics"></emph.end>deſcribatur Hyperbola, cujus conjugata tranſeat per punctum <lb></lb><emph type="italics"></emph>C,<emph.end type="italics"></emph.end>centroque <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& intervallo <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>deſcribatur Circulus ſecans Hy<lb></lb>perbolam illam in puncto <emph type="italics"></emph>H;<emph.end type="italics"></emph.end>& Projectile ſecundum rectam <emph type="italics"></emph>AH<emph.end type="italics"></emph.end><lb></lb>emiſſum incidet in punctum <emph type="italics"></emph>K. Q.E.I.<emph.end type="italics"></emph.end>Nam punctum <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>ob <lb></lb>datam longitudinem <emph type="italics"></emph>AH,<emph.end type="italics"></emph.end>locatur alicubi in Circulo deſcripto. </s> <s>A<lb></lb>gatur <emph type="italics"></emph>CH<emph.end type="italics"></emph.end>occurrens ipſis <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>KF,<emph.end type="italics"></emph.end>illi in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>huic in <emph type="italics"></emph>F;<emph.end type="italics"></emph.end>& ob <lb></lb><figure id="id.039.01.271.1.jpg" xlink:href="039/01/271/1.jpg"></figure><lb></lb>parallelas <emph type="italics"></emph>CH, MX<emph.end type="italics"></emph.end>& æquales <emph type="italics"></emph>AC, AI,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>AM,<emph.end type="italics"></emph.end><lb></lb>& propterea etiam æqualis <emph type="italics"></emph>KN.<emph.end type="italics"></emph.end>Sed <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>FH<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>KN,<emph.end type="italics"></emph.end>& propterea <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FH<emph.end type="italics"></emph.end>æquantur. </s> <s>Incidit ergo punctum <lb></lb><emph type="italics"></emph>H<emph.end type="italics"></emph.end>in Hyperbolam Aſymptotis <emph type="italics"></emph>AK, KF<emph.end type="italics"></emph.end>deſcriptam, cujus conju<lb></lb>gata tranſit per punctum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>atque adeo reperitur in communi in<lb></lb>terſectione Hyperbolæ hujus & Circuli deſcripti. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end>No<lb></lb>tandum eſt autem quod hæc operatio perinde ſe habet, ſive recta <lb></lb><emph type="italics"></emph>AKN<emph.end type="italics"></emph.end>horizonti parallela ſit, ſive ad horizontem in angulo quo<lb></lb>vis inclinata: quodque ex duabus interſectionibus <emph type="italics"></emph>H, H<emph.end type="italics"></emph.end>duo pro<lb></lb>deunt anguli <emph type="italics"></emph>NAH, NAH<emph.end type="italics"></emph.end>; & quod in Praxi mechanica ſufficit <pb xlink:href="039/01/272.jpg" pagenum="244"></pb><arrow.to.target n="note220"></arrow.to.target>Circulum ſemel deſcribere, deinde regulam interminatam <emph type="italics"></emph>CH<emph.end type="italics"></emph.end>ita ap<lb></lb>plicare ad punctum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>ut ejus pars <emph type="italics"></emph>FH,<emph.end type="italics"></emph.end>Circulo & rectæ <emph type="italics"></emph>FK<emph.end type="italics"></emph.end>interje<lb></lb>cta, æqualis ſit ejus parti <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>inter punctum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& rectam <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>ſitæ. </s></p> <p type="margin"> <s><margin.target id="note219"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="margin"> <s><margin.target id="note220"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Quæ de Hyperbolis dicta ſunt fa<lb></lb><figure id="id.039.01.272.1.jpg" xlink:href="039/01/272/1.jpg"></figure><lb></lb>cile applicantur ad Parabolas. </s> <s>Nam <lb></lb>ſi <emph type="italics"></emph>XAGK<emph.end type="italics"></emph.end>Parabolam deſignet quam <lb></lb>recta <emph type="italics"></emph>XV<emph.end type="italics"></emph.end>tangat in vertice <emph type="italics"></emph>X,<emph.end type="italics"></emph.end>ſintque <lb></lb>ordinatim applicatæ <emph type="italics"></emph>IA, VG<emph.end type="italics"></emph.end>ut quæ<lb></lb>libet abſciſſarum <emph type="italics"></emph>XI, XV<emph.end type="italics"></emph.end>dignitates <lb></lb><emph type="italics"></emph>XI<emph type="sup"></emph>n<emph.end type="sup"></emph.end>, XV<emph type="sup"></emph>n<emph.end type="sup"></emph.end>;<emph.end type="italics"></emph.end>agantur <emph type="italics"></emph>XT, GT, AH,<emph.end type="italics"></emph.end><lb></lb>quarum <emph type="italics"></emph>XT<emph.end type="italics"></emph.end>parallela ſit <emph type="italics"></emph>VG,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>GT, <lb></lb>AH<emph.end type="italics"></emph.end>Parabolam tangant in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>A:<emph.end type="italics"></emph.end>& <lb></lb>corpus de loco quovis <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ſecundum <lb></lb>rectam <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>productam, juſta cum <lb></lb>velocitate projectum, deſcribet hanc <lb></lb>Parabolam, ſi modo denſitas Medii, <lb></lb>in locis ſingulis <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>ſit reciproce ut <lb></lb>tangens <emph type="italics"></emph>GT.<emph.end type="italics"></emph.end>Velocitas autem in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ea erit quacum Projectile per<lb></lb>geret, in ſpatio non reſiſtente, in Parabola Conica verticem <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>dia<lb></lb>metrum <emph type="italics"></emph>VG<emph.end type="italics"></emph.end>deorſum productam, & latus rectum (<emph type="italics"></emph>2GTq./nn-nXVG<emph.end type="italics"></emph.end>) <lb></lb>habente. </s> <s>Et reſiſtentia in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>erit ad vim gravitatis ut <emph type="italics"></emph>GT<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>(2nn-2n/n-2) VG.<emph.end type="italics"></emph.end>Unde ſi <emph type="italics"></emph>NAK<emph.end type="italics"></emph.end>lineam horizontalem deſignet, & <lb></lb>manente tum denſitate Medii in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>tum velocitate quacum corpus <lb></lb>projicitur, mutetur utcunque angulus <emph type="italics"></emph>NAH;<emph.end type="italics"></emph.end>manebunt longitu<lb></lb>dines <emph type="italics"></emph>AH, AI, HX,<emph.end type="italics"></emph.end>& inde datur Parabolæ vertex <emph type="italics"></emph>X,<emph.end type="italics"></emph.end>& poſitio <lb></lb>rectæ <emph type="italics"></emph>XI,<emph.end type="italics"></emph.end>& ſumendo <emph type="italics"></emph>VG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>IA<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>XV<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>ad <emph type="italics"></emph>XI<emph type="sup"></emph>n<emph.end type="sup"></emph.end>,<emph.end type="italics"></emph.end>dantur om<lb></lb>nia Parabolæ puncta <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>per quæ Projectile tranſibit. <pb xlink:href="039/01/273.jpg" pagenum="245"></pb><arrow.to.target n="note221"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note221"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Motu Corporum quibus reſiſtitur partim in ratione <lb></lb>velocitatis, partim in ejuſdem ratione duplicata.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XI. THEOREMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Corpori reſiſtitur partim in ratione velocitatis, partim in ve<lb></lb>locitatis ratione duplicata, & idem ſola vi inſita in Medio ſi<lb></lb>milari movetur, ſumantur autem tempora in progreſſione Arith<lb></lb>metica: quantitates velocitatibus reciproce proportionales, datâ <lb></lb>quadam quantitate auctæ, erunt in progreſſione Geometrica.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Centro <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>Aſymptotis rectan<lb></lb><figure id="id.039.01.273.1.jpg" xlink:href="039/01/273/1.jpg"></figure><lb></lb>gulis <emph type="italics"></emph>CADd<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CH,<emph.end type="italics"></emph.end>deſcribatur <lb></lb>Hyperbola <emph type="italics"></emph>BEeS,<emph.end type="italics"></emph.end>& Aſympto<lb></lb>to <emph type="italics"></emph>CH<emph.end type="italics"></emph.end>parallelæ ſint <emph type="italics"></emph>AB, DE, <lb></lb>de.<emph.end type="italics"></emph.end>In Aſymptoto <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>dentur <lb></lb>puncta <emph type="italics"></emph>A, G:<emph.end type="italics"></emph.end>Et ſi tempus ex<lb></lb>ponatur per aream Hyperbolicam <lb></lb><emph type="italics"></emph>ABED<emph.end type="italics"></emph.end>uniformiter creſcentem; <lb></lb>dico quod velocitas exponi poteſt <lb></lb>per longitudinem <emph type="italics"></emph>DF,<emph.end type="italics"></emph.end>cujus reci<lb></lb>proca <emph type="italics"></emph>GD<emph.end type="italics"></emph.end>una cum data <emph type="italics"></emph>CG<emph.end type="italics"></emph.end>com<lb></lb>ponat longitudinem <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>in progreſſione Geometrica creſcentem. </s></p> <p type="main"> <s>Sit enim areola <emph type="italics"></emph>DEed<emph.end type="italics"></emph.end>datum temporis incrementum quam <lb></lb>minimum, & erit <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>reciproce ut <emph type="italics"></emph>DE,<emph.end type="italics"></emph.end>adeoQ.E.D.recte ut <lb></lb><emph type="italics"></emph>CD.<emph.end type="italics"></emph.end>Ipſius autem (1/<emph type="italics"></emph>G-D<emph.end type="italics"></emph.end>) decrementum, quod (per hujus Lem. </s> <s>11) <lb></lb>eſt (<emph type="italics"></emph>Dd/GDq<emph.end type="italics"></emph.end>), erit ut (<emph type="italics"></emph>CD/GDq<emph.end type="italics"></emph.end>) ſeu (<emph type="italics"></emph>CG+GD/GDq<emph.end type="italics"></emph.end>), id eſt, ut (1/<emph type="italics"></emph>GD<emph.end type="italics"></emph.end>)+(<emph type="italics"></emph>CG/GDq<emph.end type="italics"></emph.end>). <lb></lb>Igitur tempore <emph type="italics"></emph>ABED<emph.end type="italics"></emph.end>peradditionem datarum particularum <emph type="italics"></emph>ED de<emph.end type="italics"></emph.end><lb></lb>uniformiter creſcente, decreſcit (1/<emph type="italics"></emph>GD<emph.end type="italics"></emph.end>) in eadem ratione cum veloci<lb></lb>tate. </s> <s>Nam decrementum velocitatis eſt ut reſiſtentia, hoc eſt (per <lb></lb>Hypotheſin) ut ſumma duarum quantitatum, quarum una eſt ut <pb xlink:href="039/01/274.jpg" pagenum="246"></pb><arrow.to.target n="note222"></arrow.to.target>velocitas, altera ut quadratum velocitatis: & ipſius (1/<emph type="italics"></emph>GD<emph.end type="italics"></emph.end>) decremen<lb></lb>tum eſt ut ſumma quantitatum (1/<emph type="italics"></emph>GD<emph.end type="italics"></emph.end>) & (<emph type="italics"></emph>CG/GDq<emph.end type="italics"></emph.end>), quarum prior eſt <lb></lb>ipſa (1/<emph type="italics"></emph>GD<emph.end type="italics"></emph.end>), & poſterior (<emph type="italics"></emph>CG/GDq<emph.end type="italics"></emph.end>) eſt ut (1/<emph type="italics"></emph>GDq<emph.end type="italics"></emph.end>). Proinde (1/<emph type="italics"></emph>GD<emph.end type="italics"></emph.end>), ob an<lb></lb>alogum decrementum, eſt ut velocitas. </s> <s>Et ſi quantitas <emph type="italics"></emph>GD,<emph.end type="italics"></emph.end>ipſi (1/<emph type="italics"></emph>GD<emph.end type="italics"></emph.end>) <lb></lb>reciproce proportionalis, quantitate data <emph type="italics"></emph>CG<emph.end type="italics"></emph.end>augeatur; ſumma <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end><lb></lb>tempore <emph type="italics"></emph>ABED<emph.end type="italics"></emph.end>uniformiter creſcente, creſcet in progreſſione <lb></lb>Geometrica. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note222"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur. </s> <s>ſi, datis punctis <emph type="italics"></emph>A, G,<emph.end type="italics"></emph.end>exponatur tempus per <lb></lb>aream Hyperbolicam <emph type="italics"></emph>ABED,<emph.end type="italics"></emph.end>exponi poteſt velocitas per ipſius <lb></lb><emph type="italics"></emph>GD<emph.end type="italics"></emph.end>reciprocam (1/<emph type="italics"></emph>GD<emph.end type="italics"></emph.end>). </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Sumendo autem <emph type="italics"></emph>GA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GD<emph.end type="italics"></emph.end>ut velocitatis reciproca ſub <lb></lb>initio, ad velocitatis reciprocam in fine temporis cujuſvis <emph type="italics"></emph>ABED,<emph.end type="italics"></emph.end><lb></lb>invenietur punctum <emph type="italics"></emph>G.<emph.end type="italics"></emph.end>Eo autem invento, velocitas ex dato quo<lb></lb>vis alio tempore inveniri poteſt. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XII. THEOREMA IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, dico quod ſi ſpatia deſcripta ſumantur in progreſſio<lb></lb>ne Arithmetica, velocitates data quadam quantitate auctæ e<lb></lb>runt in progreſſione Geometrica.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>In Aſymptoto <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>detur pun<lb></lb><figure id="id.039.01.274.1.jpg" xlink:href="039/01/274/1.jpg"></figure><lb></lb>ctum <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>& erecto perpendiculo <emph type="italics"></emph>RS,<emph.end type="italics"></emph.end><lb></lb>quod occurrat Hyperbolæ in <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>ex<lb></lb>ponatur deſcriptum ſpatium per a<lb></lb>ream Hyperbolicam <emph type="italics"></emph>RSED<emph.end type="italics"></emph.end>; & <lb></lb>velocitas erit ut longitudo <emph type="italics"></emph>GD,<emph.end type="italics"></emph.end><lb></lb>quæ cum data <emph type="italics"></emph>CG<emph.end type="italics"></emph.end>componit longi<lb></lb>tudinem <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end>in progreſſione Geo<lb></lb>metrica decreſcentem, interea dum <lb></lb>ſpatium <emph type="italics"></emph>RSED<emph.end type="italics"></emph.end>augetur in Arith<lb></lb>metica. </s></p> <p type="main"> <s>Etenim ob datum ſpatii incrementum <emph type="italics"></emph>EDde,<emph.end type="italics"></emph.end>lineola <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>quæ <pb xlink:href="039/01/275.jpg" pagenum="247"></pb>decrementum eſt ipſius <emph type="italics"></emph>GD,<emph.end type="italics"></emph.end>erit reciproce ut <emph type="italics"></emph>ED,<emph.end type="italics"></emph.end>adeoQ.E.D.<lb></lb><arrow.to.target n="note223"></arrow.to.target>recte ut <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end>hoc eſt, ut ſumma ejuſdom <emph type="italics"></emph>GD<emph.end type="italics"></emph.end>& longitudinis datæ <lb></lb><emph type="italics"></emph>CG.<emph.end type="italics"></emph.end>Sed velocitatis decrementum, tempore ſibi reciproce pro<lb></lb>portionali, quo data ſpatii particula <emph type="italics"></emph>D de E<emph.end type="italics"></emph.end>deſcribitur, eſt ut re<lb></lb>ſiſtentia & tempus conjunctim, id eſt, directe ut ſumma duarum <lb></lb>quantitatum, quarum una eſt ut velocitas, altera ut velocitatis qua<lb></lb>dratum, & inverſe ut velocitas; adeoQ.E.D.recte ut ſumma duarum <lb></lb>quantitatum, quarum una datur, altera eſt ut velocitas. </s> <s>Igitur de<lb></lb>crementum tam velocitatis quam lineæ <emph type="italics"></emph>GD,<emph.end type="italics"></emph.end>eſt ut quantitas data <lb></lb>& quantitas decreſcens conjunctim, & propter analoga decremen<lb></lb>ta, analogæ ſemper crunt quantitates decreſcentes: nimirum veloci<lb></lb>tas & linea <emph type="italics"></emph>G.D. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note223"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur ſi velocitas exponatur per longitudinem <emph type="italics"></emph>GD,<emph.end type="italics"></emph.end>ſpa<lb></lb>tium deſcriptum erit ut area Hyperbolica <emph type="italics"></emph>DESR.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et ſi utcunque aſſumatur punctum <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>invenietur pun<lb></lb>ctum <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>capiendo <emph type="italics"></emph>GR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GD,<emph.end type="italics"></emph.end>ut eſt velocitas ſub initio ad ve<lb></lb>locitatem poſt ſpatium quodvis <emph type="italics"></emph>RSED<emph.end type="italics"></emph.end>deſcriptum. </s> <s>Invento au<lb></lb>tem puncto <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>datur ſpatium ex data velocitate, & contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Unde cum, per Prop. </s> <s>XI. detur velocitas ex dato tem<lb></lb>pore, & per hanc Propoſitionem detur ſpatium ex data velocitate; <lb></lb>dabitur ſpatium ex dato tempore: & contra. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XIII. THEOREMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſito quod Corpus ab uniformi gravitate deorſum attractum recta: <lb></lb>aſcendit vel deſcendit, & quod eidem reſiſtitur partim in ra<lb></lb>tione velocitatis, partim in ejuſdem ratione duplicata: dico quod <lb></lb>ſi Circuli & Hyperbolæ diametris parallelæ rectæ per conjuga<lb></lb>tarum diametrorum terminos ducantur, & velocitates ſint ut <lb></lb>ſegmenta quædam parallelarum a dato puncto ducta, Tempora <lb></lb>erunt ut arearum Sectores, rectis a centro ad ſegmentorum ter<lb></lb>minos ductis abſciſſi: & contra.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>1. Ponamus primo quod corpus aſcendit, centroque <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <lb></lb>ſemidiametro quovis <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>deſcribatur Circuli quadrans <emph type="italics"></emph>BETF,<emph.end type="italics"></emph.end>& <lb></lb>per ſemidiametri <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>terminum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>agatur infinita <emph type="italics"></emph>BAP,<emph.end type="italics"></emph.end>ſemidia<lb></lb>metro <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>parallela. </s> <s>In ea detur punctum <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>& capiatur ſegmen<lb></lb>tum <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>velocitati proportionale. </s> <s>Et cum reſiſtentiæ pars aliqua ſit <pb xlink:href="039/01/276.jpg" pagenum="248"></pb><arrow.to.target n="note224"></arrow.to.target>ut velocitas & pars altera ut <lb></lb><figure id="id.039.01.276.1.jpg" xlink:href="039/01/276/1.jpg"></figure><lb></lb>velocitatis quadratum, fit re<lb></lb>ſiſtentia tota in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AP quad<emph.end type="italics"></emph.end><lb></lb>+2 <emph type="italics"></emph>BAP.<emph.end type="italics"></emph.end>Jungantur <emph type="italics"></emph>DA, <lb></lb>DP<emph.end type="italics"></emph.end>Circulum ſecantes in <emph type="italics"></emph>E<emph.end type="italics"></emph.end><lb></lb>ac <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>& exponatur gravitas per <lb></lb><emph type="italics"></emph>DA quad,<emph.end type="italics"></emph.end>ita ut ſit gravitas ad <lb></lb>reſiſtentiam in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DAq<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>APq<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>BAP:<emph.end type="italics"></emph.end>& tempus <lb></lb>aſcenſus omnis ſuturi erit ut <lb></lb>Circuli ſector <emph type="italics"></emph>EDTE.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note224"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Agatur enim <emph type="italics"></emph>DVQ,<emph.end type="italics"></emph.end>ab<lb></lb>ſcindens & velocitatis <emph type="italics"></emph>AP<emph.end type="italics"></emph.end><lb></lb>momentum <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>& Sectoris <lb></lb><emph type="italics"></emph>DET<emph.end type="italics"></emph.end>momentum <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>da<lb></lb>to temporis momento reſpondens: & velocitatis decrementum il<lb></lb>lud <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>erit ut ſumma virium gravitatis <emph type="italics"></emph>DAq<emph.end type="italics"></emph.end>& reſiſtentiæ <lb></lb><emph type="italics"></emph>APq<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>BAP,<emph.end type="italics"></emph.end>id eſt (per Prop. </s> <s>12, Lib. </s> <s>2. Elem.) ut <emph type="italics"></emph>DPquad.<emph.end type="italics"></emph.end><lb></lb>Proinde area <emph type="italics"></emph>DPQ,<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>proportionalis, eſt ut <emph type="italics"></emph>DP quad<emph.end type="italics"></emph.end>; <lb></lb>& area <emph type="italics"></emph>DTV,<emph.end type="italics"></emph.end>(quæ eſt ad aream <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DTq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DPq<emph.end type="italics"></emph.end>) <lb></lb>eſt ut datum <emph type="italics"></emph>DTQ<emph.end type="italics"></emph.end>Decreſcit igitur area <emph type="italics"></emph>EDT<emph.end type="italics"></emph.end>uniformiter ad mo<lb></lb>dum temporis futuri, per ſubductionem datarum particularum <emph type="italics"></emph>DTV,<emph.end type="italics"></emph.end><lb></lb>& propterea tempori aſcenſus futuri proportionalis eſt. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>2. Si veloci<lb></lb><figure id="id.039.01.276.2.jpg" xlink:href="039/01/276/2.jpg"></figure><lb></lb>tas in aſcenſu cor<lb></lb>poris exponatur per <lb></lb>longitudinem <emph type="italics"></emph>AP<emph.end type="italics"></emph.end><lb></lb>ut prius, & reſiſten<lb></lb>tia ponatur eſſe ut <lb></lb><emph type="italics"></emph>APq<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>BAP,<emph.end type="italics"></emph.end>& <lb></lb>ſi vis gravitatis mi<lb></lb>nor ſit quam quæ per <lb></lb><emph type="italics"></emph>DAq<emph.end type="italics"></emph.end>exponi poſ<lb></lb>ſit; capiatur <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>e<lb></lb>jus longitudinis, ut <lb></lb>ſit <emph type="italics"></emph>ABq-BDq<emph.end type="italics"></emph.end><lb></lb>gravitati proportio<lb></lb>nale, ſitque <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>ipſi <lb></lb><emph type="italics"></emph>DB<emph.end type="italics"></emph.end>perpendicularis & æqualis, & per verticem <emph type="italics"></emph>F<emph.end type="italics"></emph.end>deſcribatur Hy<lb></lb>perbola <emph type="italics"></emph>FTVE<emph.end type="italics"></emph.end>cujus ſemidiametri conjugatæ ſint <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DF,<emph.end type="italics"></emph.end><lb></lb>quæque ſecet <emph type="italics"></emph>DA<emph.end type="italics"></emph.end>in <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DP, DQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>V<emph.end type="italics"></emph.end>; & crit tempus <lb></lb>aſcenſus futuri ut Hyperbolæ ſector <emph type="italics"></emph>TDE.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/277.jpg" pagenum="249"></pb> <p type="main"> <s>Nam velocitatis decrementum <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>in data temporis particula <lb></lb><arrow.to.target n="note225"></arrow.to.target>factum, eſt ut ſumma reſiſtentiæ <emph type="italics"></emph>APq<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>BAP<emph.end type="italics"></emph.end>& gravitatis <lb></lb><emph type="italics"></emph>ABq-BDq,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>BPq-BDq.<emph.end type="italics"></emph.end>Eſt autem area <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end><lb></lb>ad aream <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DTq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DPq<emph.end type="italics"></emph.end>adeoque, ſi ad <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>demitta<lb></lb>tur perpendiculum <emph type="italics"></emph>GT,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>GTq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>GDq-DFq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BDq<emph.end type="italics"></emph.end><lb></lb>utque <emph type="italics"></emph>GDq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BPq<emph.end type="italics"></emph.end>& diviſim ut <emph type="italics"></emph>DFq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BPq-BDq.<emph.end type="italics"></emph.end><lb></lb>Quare cum area <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ſit ut <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>BPq-BDq<emph.end type="italics"></emph.end>; erit <lb></lb>area <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>ut datum <emph type="italics"></emph>DFq.<emph.end type="italics"></emph.end>Decreſcit igitur area <emph type="italics"></emph>EDT<emph.end type="italics"></emph.end>unifor<lb></lb>miter ſingulis temporis particulis æqualibus, per ſubductionem par<lb></lb>ticularum totidem datarum <emph type="italics"></emph>DTV,<emph.end type="italics"></emph.end>& propterea tempori propor<lb></lb>tionalis eſt. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note225"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>3. Sit <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>velocitas in deſcenſu corporis, & <emph type="italics"></emph>APq+2BAP<emph.end type="italics"></emph.end><lb></lb>reſiſtentia, & <emph type="italics"></emph>BDq-ABq<emph.end type="italics"></emph.end>vis gravitatis, exiſtente angulo <emph type="italics"></emph>DBA<emph.end type="italics"></emph.end><lb></lb>recto. </s> <s>Et ſi centro <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>vertice <lb></lb><figure id="id.039.01.277.1.jpg" xlink:href="039/01/277/1.jpg"></figure><lb></lb>principali <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>deſcribatur Hy<lb></lb>perbola rectangula <emph type="italics"></emph>BETV<emph.end type="italics"></emph.end><lb></lb>ſecans productas <emph type="italics"></emph>DA, DP<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>DQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>E, T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>V<emph.end type="italics"></emph.end>; erit Hy<lb></lb>perbolæ hujus ſector <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>ut <lb></lb>tempus deſcenſus. </s></p> <p type="main"> <s>Nam velocitatis <expan abbr="incremẽtum">incrementum</expan> <lb></lb><emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>eique proportionalis area <lb></lb><emph type="italics"></emph>DPQ,<emph.end type="italics"></emph.end>eſt ut exceſſus gravita<lb></lb>tis ſupra reſiſtentiam, id eſt, ut <lb></lb><emph type="italics"></emph>BDq-ABq-2BAP-APq<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>BDq-BPq.<emph.end type="italics"></emph.end>Et area <lb></lb><emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>eſt ad aream <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>DTq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DPq,<emph.end type="italics"></emph.end>adeoque ut <lb></lb><emph type="italics"></emph>GTq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>GDq-BDq<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>BPq<emph.end type="italics"></emph.end>utque <emph type="italics"></emph>GDq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BDq<emph.end type="italics"></emph.end><lb></lb>& diviſim ut <emph type="italics"></emph>BDq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BDq-BPq.<emph.end type="italics"></emph.end>Quare cum area <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end><lb></lb>ſit ut <emph type="italics"></emph>BDq-BPq,<emph.end type="italics"></emph.end>erit area <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>ut datum <emph type="italics"></emph>BDq.<emph.end type="italics"></emph.end>Creſcit <lb></lb>igitur area <emph type="italics"></emph>EDT<emph.end type="italics"></emph.end>uniformiter ſingulis temporis particulis æquali<lb></lb>bus, per additionem totidem datarum particularum <emph type="italics"></emph>DTV,<emph.end type="italics"></emph.end>& prop<lb></lb>terea tempori deſcenſus proportionalis eſt. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Igitur velocitas <emph type="italics"></emph>AP<emph.end type="italics"></emph.end>eſt ad velocitatem quam corpus tem<lb></lb>pore <emph type="italics"></emph>EDT,<emph.end type="italics"></emph.end>in ſpatio non reſiſtente, aſcendendo amittere vel de<lb></lb>ſcendendo acquirere poſſet, ut area trianguli <emph type="italics"></emph>DAP<emph.end type="italics"></emph.end>ad aream ſe<lb></lb>ctoris centro <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>DA,<emph.end type="italics"></emph.end>angulo <emph type="italics"></emph>ADT<emph.end type="italics"></emph.end>deſcripti; ideoque ex <lb></lb>dato tempore datur. </s> <s>Nam velocitas, in Medio non reſiſtente, tem-<pb xlink:href="039/01/278.jpg" pagenum="250"></pb><arrow.to.target n="note226"></arrow.to.target>pori atque adeo ſectori huic proportionalis eſt; in Medio reſiſten<lb></lb>te eſt ut triangulum; & in Medio utroque, ubi quam minima eſt, ac<lb></lb>cedit ad rationem æqualitatis, pro more ſectoris & trianguli. </s></p> <p type="margin"> <s><margin.target id="note226"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XIV. THEOREMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, dico quod ſpatium aſcenſu vel deſcenſu deſcriptum, <lb></lb>eſt ut differentia areæ per quam tempus exponitur, & areæ cu<lb></lb>juſdam alterius quæ augetur vel diminuitur in progreſſione A<lb></lb>rithmetica; ſi vires ex reſiſtentia & gravitate compoſitæ ſu<lb></lb>mantur in progreſſione Geometrica.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Capiatur <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>(in Fig. </s> <s>tribus ultimis,) gravitati, & <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>reſi<lb></lb>ſtentiæ proportionalis. </s> <s>Capiantur autem ad eaſdem partes pun<lb></lb>cti <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſi corpus deſcendit, aliter ad contrarias. </s> <s>Erigatur <emph type="italics"></emph>Ab<emph.end type="italics"></emph.end>quæ <lb></lb>ſit ad <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DBq<emph.end type="italics"></emph.end>ad 4 <emph type="italics"></emph>BAC:<emph.end type="italics"></emph.end>& area <emph type="italics"></emph>AbNK<emph.end type="italics"></emph.end>augebitur vel <lb></lb>diminuetur in progreſſione Arithmetica, dum vires <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>in pro<lb></lb>greſſione Geometrica ſumuntur. </s> <s>Dico igitur quod diſtantia cor<lb></lb>poris ab ejus altitudine maxima ſit ut exceſſus areæ <emph type="italics"></emph>AbNK<emph.end type="italics"></emph.end>ſupra <lb></lb>aream <emph type="italics"></emph>DET.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam cum <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>ſit ut reſiſtentia, id eſt, ut <emph type="italics"></emph>APq+2BAP<emph.end type="italics"></emph.end>: <lb></lb>aſſumatur data quævis quantitas Z, & ponatur <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>æqualis <lb></lb>(<emph type="italics"></emph>APq+2BAP<emph.end type="italics"></emph.end>/Z); & (per hujus Lemma 11.) erit ipſius <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>mo<lb></lb>mentum <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>æquale (2<emph type="italics"></emph>APQ+2BAXPQ<emph.end type="italics"></emph.end>/Z) ſeu (2<emph type="italics"></emph>BPQ<emph.end type="italics"></emph.end>/Z), & <lb></lb>areæ <emph type="italics"></emph>AbNK<emph.end type="italics"></emph.end>momentum <emph type="italics"></emph>KLON<emph.end type="italics"></emph.end>æquale (2<emph type="italics"></emph>BPQXLO<emph.end type="italics"></emph.end>/Z) ſeu <lb></lb>(<emph type="italics"></emph>BPQXBD cub.<emph.end type="italics"></emph.end>/2ZX<emph type="italics"></emph>CRXAB<emph.end type="italics"></emph.end>). </s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>1. Jam ſi corpus aſcendit, ſitque gravitas ut <emph type="italics"></emph>ABq+BDq<emph.end type="italics"></emph.end><lb></lb>exiſtente <emph type="italics"></emph>BET<emph.end type="italics"></emph.end>Circulo, (in Fig. </s> <s>Caſ. </s> <s>1. Prop. </s> <s>XIII.) linea <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end><lb></lb>quæ gravitati proportionalis eſt, erit (<emph type="italics"></emph>ABq+BDq<emph.end type="italics"></emph.end>/Z), & <emph type="italics"></emph>DPq<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>APq+2BAP+ABq+BDq<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>XZ+<emph type="italics"></emph>AC<emph.end type="italics"></emph.end>XZ ſeu <lb></lb><emph type="italics"></emph>CK<emph.end type="italics"></emph.end>XZ; ideoque area <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>erit ad aream <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DTq<emph.end type="italics"></emph.end>vel <lb></lb><emph type="italics"></emph>DBq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>XZ. </s></p><pb xlink:href="039/01/279.jpg" pagenum="251"></pb> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>2. Sin corpus aſcendit, & gravitas ſit ut <emph type="italics"></emph>ABq-BDq<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note227"></arrow.to.target>linea <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>(Fig. </s> <s>Caſ. </s> <s>2. Prop. </s> <s>XIII) erit (<emph type="italics"></emph>ABq-BDq<emph.end type="italics"></emph.end>/Z), & <emph type="italics"></emph>DTq<emph.end type="italics"></emph.end><lb></lb>erit ad <emph type="italics"></emph>DPq<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DFq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>DBq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BPq-BDq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>APq+ <lb></lb>2BAP+ABq-BDq,<emph.end type="italics"></emph.end>id eſt, ad <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>XZ+<emph type="italics"></emph>AC<emph.end type="italics"></emph.end>XZ ſeu <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>XZ. </s> <s><lb></lb>Ideoque area <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>erit ad aream <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DBq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>XZ. </s></p> <p type="margin"> <s><margin.target id="note227"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>3. Et eodem argumento, ſi corpus deſcendit, & propterea <lb></lb>gravitas ſit ut <emph type="italics"></emph>BDq-ABq,<emph.end type="italics"></emph.end>& linea <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>(Fig. </s> <s>Caſ.3. Prop. </s> <s>præced.) <lb></lb>æquetur (<emph type="italics"></emph>BDq-ABq<emph.end type="italics"></emph.end>/Z) erit area <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>ad aream <emph type="italics"></emph>DPQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DBq<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>XZ: ut ſupra. </s></p> <p type="main"> <s>Cum igitur areæ illæ ſemper ſint in hac ratione; ſi pro area <lb></lb><emph type="italics"></emph>DTV,<emph.end type="italics"></emph.end>qua momentum temporis ſibimet ipſi ſemper æquale ex<lb></lb>ponitur, ſcribatur determinatum quodvis rectangulum, puta <lb></lb><emph type="italics"></emph>BDXm,<emph.end type="italics"></emph.end>erit area <emph type="italics"></emph>DPQ,<emph.end type="italics"></emph.end>id eſt, 1/2<emph type="italics"></emph>BDXPQ<emph.end type="italics"></emph.end>; ad <emph type="italics"></emph>BDXm<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>CK<emph.end type="italics"></emph.end>XZ ad <emph type="italics"></emph><expan abbr="BDq.">BDque</expan><emph.end type="italics"></emph.end>AtQ.E.I.de fit <emph type="italics"></emph>PQXBD cub.<emph.end type="italics"></emph.end>æquale <lb></lb>2<emph type="italics"></emph>BDXmXCK<emph.end type="italics"></emph.end>XZ, & areæ <emph type="italics"></emph>AbNK<emph.end type="italics"></emph.end>momentum <emph type="italics"></emph>KLON<emph.end type="italics"></emph.end>ſu<lb></lb>perius inventum, fit (<emph type="italics"></emph>BPXBDXm/AB<emph.end type="italics"></emph.end>). Auferatur areæ <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>mo<lb></lb>mentum <emph type="italics"></emph>DTV<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>BDXm,<emph.end type="italics"></emph.end>& reſtabit (<emph type="italics"></emph>APXBDXm/AB<emph.end type="italics"></emph.end>). Eſt igi<lb></lb>tur differentia momentorum, id eſt, momentum differentiæ area<lb></lb>rum, æqualis (<emph type="italics"></emph>APXBDXm/AB<emph.end type="italics"></emph.end>); & propterea (ob datum (<emph type="italics"></emph>BDXm/AB<emph.end type="italics"></emph.end>)) <lb></lb>ut velocitas <emph type="italics"></emph>AP,<emph.end type="italics"></emph.end>id eſt, ut momentum ſpatii quod corpus aſcen<lb></lb>dendo vel deſcendendo deſcribit. </s> <s>IdeoQ.E.D.fferentia arearum <lb></lb>& ſpatium illud, proportionalibus momentis creſcentia vel decre<lb></lb>ſcentia & ſimul incipientia vel ſimul evaneſcentia, ſunt proportio<lb></lb>nalia. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Igitur ſi longitudo aliqua V ſumatur in ea ratione ad du<lb></lb>plum longitudinis M, quæ oritur applicando aream <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BD,<emph.end type="italics"></emph.end><lb></lb>quam habet linea <emph type="italics"></emph>DA<emph.end type="italics"></emph.end>ad lineam <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>; ſpatium quod corpus aſcen<lb></lb>ſu vel deſcenſu toto in Medio reſiſtente deſcribit, erit ad ſpatium <lb></lb>quod in Medio non reſiſtente eodem tempore deſcribere poſſet, <lb></lb>ut arearum illarum differentia ad (<emph type="italics"></emph>BD<emph.end type="italics"></emph.end>XV<emph type="sup"></emph>2<emph.end type="sup"></emph.end>/4<emph type="italics"></emph>AB<emph.end type="italics"></emph.end>), ideoque ex dato tem<lb></lb>pore datur. </s> <s>Nam ſpatium in Medio non reſiſtente eſt in dupli<lb></lb>cata ratione temporis, ſive ut V<emph type="sup"></emph>2<emph.end type="sup"></emph.end>, & ob datas <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>ut <pb xlink:href="039/01/280.jpg" pagenum="252"></pb><arrow.to.target n="note228"></arrow.to.target>(<emph type="italics"></emph>BD<emph.end type="italics"></emph.end>XV<emph type="sup"></emph>2<emph.end type="sup"></emph.end>/4<emph type="italics"></emph>AB<emph.end type="italics"></emph.end>). Momentum hujus areæ ſive huic æqualis (<emph type="italics"></emph>DAqXBD<emph.end type="italics"></emph.end>XM<emph type="sup"></emph>2<emph.end type="sup"></emph.end>/<emph type="italics"></emph>DEqXAB<emph.end type="italics"></emph.end>) <lb></lb>eſt ad momentum differentiæ arearum <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AbNK,<emph.end type="italics"></emph.end>ut <lb></lb>(<emph type="italics"></emph>DAqXBD<emph.end type="italics"></emph.end>X2MX<emph type="italics"></emph>m<emph.end type="italics"></emph.end>/<emph type="italics"></emph>DEqXAB<emph.end type="italics"></emph.end>) ad (<emph type="italics"></emph>APXBDXm/AB<emph.end type="italics"></emph.end>), hoc eſt, ut (<emph type="italics"></emph>DAqXBD<emph.end type="italics"></emph.end>XM/<emph type="italics"></emph>DEq<emph.end type="italics"></emph.end>) <lb></lb>ad 1/2<emph type="italics"></emph>BDXAP,<emph.end type="italics"></emph.end>ſive ut (<emph type="italics"></emph>DAq/DEq<emph.end type="italics"></emph.end>) in <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DAP<emph.end type="italics"></emph.end>; adeoque ubi <lb></lb>areæ <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DAP<emph.end type="italics"></emph.end>quam minimæ ſunt, in ratione æqualitatis. <lb></lb></s> <s>Æqualis igitur eſt area quam minima (<emph type="italics"></emph>BD<emph.end type="italics"></emph.end>XV<emph type="sup"></emph>2<emph.end type="sup"></emph.end>/4<emph type="italics"></emph>AB<emph.end type="italics"></emph.end>) differentiæ quam <lb></lb>minimæ arearum <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AbNK.<emph.end type="italics"></emph.end>Unde cum ſpatia in Me<lb></lb>dio utroque, in principio deſcenſus vel fine aſcenſus ſimul deſcrip<lb></lb>ta accedunt ad æqualitatem, adeoque tunc ſunt ad invicem ut area <lb></lb>(<emph type="italics"></emph>BD<emph.end type="italics"></emph.end>XV<emph type="sup"></emph>2<emph.end type="sup"></emph.end>/4<emph type="italics"></emph>AB<emph.end type="italics"></emph.end>) & arearum <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AbNK<emph.end type="italics"></emph.end>differentia; ob eorum ana<lb></lb>loga incrementa neceſſe eſt ut in æqualibus quibuſcunque tempo<lb></lb>ribus ſint ad invicem ut area illa (<emph type="italics"></emph>BD<emph.end type="italics"></emph.end>XV<emph type="sup"></emph>2<emph.end type="sup"></emph.end>/4<emph type="italics"></emph>AB<emph.end type="italics"></emph.end>) & arearum <emph type="italics"></emph>DET<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>AbNK<emph.end type="italics"></emph.end>differentia. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/281.jpg" pagenum="253"></pb></subchap2><subchap2> <p type="margin"> <s><margin.target id="note228"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO IV.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note229"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note229"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Corporum Circulari Motu in Mediis reſiſtentibus.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Sit<emph.end type="italics"></emph.end>PQRr <emph type="italics"></emph>Spiralis quæ ſecet radios omnes<emph.end type="italics"></emph.end>SP, SQ, SR, <emph type="italics"></emph>&c. </s> <s><lb></lb>in æqualibus angulis. </s> <s>Agatur recta<emph.end type="italics"></emph.end>PT <emph type="italics"></emph>quæ tangat eandem in <lb></lb>puncto quovis<emph.end type="italics"></emph.end>P, <emph type="italics"></emph>ſecetque radium<emph.end type="italics"></emph.end>SQ <emph type="italics"></emph>in<emph.end type="italics"></emph.end>T; <emph type="italics"></emph>& ad Spiralem <lb></lb>erectis perpendiculis<emph.end type="italics"></emph.end>PO, QO <emph type="italics"></emph>concurrentibus in<emph.end type="italics"></emph.end>O, <emph type="italics"></emph>jungatur<emph.end type="italics"></emph.end><lb></lb>SO. <emph type="italics"></emph>Dico quod ſi puncta<emph.end type="italics"></emph.end>P <emph type="italics"></emph>&<emph.end type="italics"></emph.end>Q <emph type="italics"></emph>accedant ad invicem & co<lb></lb>eant, angulus<emph.end type="italics"></emph.end>PSO <emph type="italics"></emph>evadet rectus, & ultima ratio rectanguli<emph.end type="italics"></emph.end><lb></lb>TQX2PS <emph type="italics"></emph>ad<emph.end type="italics"></emph.end>PQ<emph type="italics"></emph>quad. </s> <s>erit ratio æqualitatis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Etenim de angulis rectis <emph type="italics"></emph>OPQ, OQR<emph.end type="italics"></emph.end>ſubducantur anguli <lb></lb>æquales <emph type="italics"></emph>SPQ, SQR,<emph.end type="italics"></emph.end>& manebunt anguli æquales <emph type="italics"></emph>OPS, OQS.<emph.end type="italics"></emph.end><lb></lb>Ergo Circulus qui tranſit <lb></lb><figure id="id.039.01.281.1.jpg" xlink:href="039/01/281/1.jpg"></figure><lb></lb>per puncta <emph type="italics"></emph>O, S, P<emph.end type="italics"></emph.end>tranſ<lb></lb>ibit etiam per punctum <emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end><lb></lb>Coeant puncta <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q,<emph.end type="italics"></emph.end><lb></lb>& hic Circulus in loco co<lb></lb>itus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>tanget Spiralem, <lb></lb>adeoque perpendiculariter <lb></lb>ſecabit rectam <emph type="italics"></emph>OP.<emph.end type="italics"></emph.end>Fiet <lb></lb>igitur <emph type="italics"></emph>OP<emph.end type="italics"></emph.end>diameter Cir<lb></lb>culi hujus, & angulus <lb></lb><emph type="italics"></emph>OSP<emph.end type="italics"></emph.end>in ſemicirculo re<lb></lb>ctus. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Ad <emph type="italics"></emph>OP<emph.end type="italics"></emph.end>demittantur perpendicula <emph type="italics"></emph>QD, SE,<emph.end type="italics"></emph.end>& linearum ratio<lb></lb>nes ultimæ erunt hujuſmodi: <emph type="italics"></emph>TQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>TS<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PE,<emph.end type="italics"></emph.end><lb></lb>ſeu 2<emph type="italics"></emph>PO<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>PS.<emph.end type="italics"></emph.end>Item <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>PO.<emph.end type="italics"></emph.end>Et ex <lb></lb>æquo perturbate <emph type="italics"></emph>TQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>PS.<emph.end type="italics"></emph.end>Unde fit <emph type="italics"></emph>PQq<emph.end type="italics"></emph.end><lb></lb>æquale <emph type="italics"></emph>TQX2PS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/282.jpg" pagenum="254"></pb><arrow.to.target n="note230"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note230"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XV. THEOREMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Medii denſitas in locis ſingulis ſit reciproce ut diſtantia loeorum <lb></lb>a centro immobili, ſitque vis centripeta in duplicata ratione den<lb></lb>ſitatis: dico quod corpus gyrari potest in Spirali, quæ radios <lb></lb>omnes a centro illo ductos interſecat in angulo dato.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Ponantur quæ in ſuperiore Lemmate, & producatur <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>V,<emph.end type="italics"></emph.end><lb></lb>ut ſit <emph type="italics"></emph>SV<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>SP.<emph.end type="italics"></emph.end>Tempore quovis, in Medio reſiſtente, de<lb></lb>ſcribat corpus arcum quam minimum <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>& tempore duplo ar<lb></lb>cum quam minimum <emph type="italics"></emph>PR<emph.end type="italics"></emph.end>; & decrementa horum arcuum ex reſi<lb></lb>ſtentia oriunda, ſive defe<lb></lb><figure id="id.039.01.282.1.jpg" xlink:href="039/01/282/1.jpg"></figure><lb></lb>ctus ab arcubus qui in Me<lb></lb>dio non reſiſtente iiſdem <lb></lb>temporibus deſcriberen<lb></lb>tur, erunt ad invicem ut <lb></lb>quadrata temporum in <lb></lb>quibus generantur: Eſt <lb></lb>itaQ.E.D.crementum arcus <lb></lb><emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>pars quarta decre<lb></lb>menti arcus <emph type="italics"></emph>PR.<emph.end type="italics"></emph.end>Unde <lb></lb>etiam, ſi areæ <emph type="italics"></emph>PSQ<emph.end type="italics"></emph.end>æ<lb></lb>qualis capiatur area <emph type="italics"></emph>QSr,<emph.end type="italics"></emph.end><lb></lb>erit decrementum arcus <lb></lb><emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>æquale dimidio lineolæ <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>; adeoque vis reſiſtentiæ & vis cen<lb></lb>tripeta ſunt ad invicem ut lineolæ 1/2<emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>& <emph type="italics"></emph>TQ<emph.end type="italics"></emph.end>quas ſimul generant. </s> <s><lb></lb>Quoniam vis centripeta, qua corpus urgetur in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>eſt reciproce ut <lb></lb><emph type="italics"></emph>SPq,<emph.end type="italics"></emph.end>& (per Lem. </s> <s>X. Lib. </s> <s>1,) lineola <emph type="italics"></emph>TQ,<emph.end type="italics"></emph.end>quæ vi illa generatur, eſt <lb></lb>in ratione compoſita ex ratione hujus vis & ratione duplicata tem<lb></lb>poris quo arcus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>deſcribitur, (Nam reſiſtentiam in hoc caſu, <lb></lb>ut infinite minorem quam vis centripeta, negligo) erit <emph type="italics"></emph>TQXSPq<emph.end type="italics"></emph.end><lb></lb>id eſt (per Lemma noviſſimum) 1/2<emph type="italics"></emph>PQqXSP,<emph.end type="italics"></emph.end>in ratione duplicata <lb></lb>temporis, adeoque tempus eſt ut <emph type="italics"></emph>PQX√SP<emph.end type="italics"></emph.end>; & corporis veloci<lb></lb>tas, qua arcus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>illo tempore deſcribitur, ut (<emph type="italics"></emph>PQ/PQX√SP<emph.end type="italics"></emph.end>) ſeu <lb></lb>(1/√<emph type="italics"></emph>SP<emph.end type="italics"></emph.end>), hoc eſt, in ſubduplicata ratione ipſius <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>reciproce. </s> <s>Et ſi<lb></lb>mili argumento, velocitas qua arcus <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>deſcribitur, eſt in ſub-<pb xlink:href="039/01/283.jpg" pagenum="255"></pb>duplicata ratione ipſius <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end>reciproce. </s> <s>Sunt autem arcus illi <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note231"></arrow.to.target>& <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>ut velocitates deſcriptrices ad invicem, id eſt, in ſubdupli<lb></lb>cata ratione <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>ſive ut <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end>ad √<emph type="italics"></emph>SPXSQ<emph.end type="italics"></emph.end>; & ob æqua<lb></lb>les angulos <emph type="italics"></emph>SPQ, SQr<emph.end type="italics"></emph.end>& æquales areas <emph type="italics"></emph>PSQ, QSr,<emph.end type="italics"></emph.end>eſt ar<lb></lb>cus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>Qr<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SP.<emph.end type="italics"></emph.end>Sumantur proportionalium <lb></lb>conſequentium differentiæ, & fiet arcus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>SP-√SPXSQ,<emph.end type="italics"></emph.end>ſeu 1/2<emph type="italics"></emph>VQ<emph.end type="italics"></emph.end>; nam punctis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>coeunti<lb></lb>bus, ratio ultima <emph type="italics"></emph>SP-√SPXSQ<emph.end type="italics"></emph.end>ad 1/2<emph type="italics"></emph>VQ<emph.end type="italics"></emph.end>ſit æqualitatis. </s> <s><lb></lb>Quoniam decrementum arcus <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>ex reſiſtentia oriundum, ſive <lb></lb>hujus duplum <emph type="italics"></emph>Rr,<emph.end type="italics"></emph.end>eſt ut reſiſtentia & quadratum temporis con<lb></lb>junctim; erit reſiſtentia ut (<emph type="italics"></emph>Rr/PQqXSP<emph.end type="italics"></emph.end>). Erat autem <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Rr,<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end>ad 1/2<emph type="italics"></emph>VQ,<emph.end type="italics"></emph.end>& inde (<emph type="italics"></emph>Rr/PQqXSP<emph.end type="italics"></emph.end>) fit ut (1/2<emph type="italics"></emph>VQ/PQXSPXSQ<emph.end type="italics"></emph.end>) ſive <lb></lb>ut (1/2<emph type="italics"></emph>OS/OPXSPq<emph.end type="italics"></emph.end>). Namque punctis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>coeuntibus, <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end><lb></lb>coincidunt, & angulus <emph type="italics"></emph>PVQ<emph.end type="italics"></emph.end>fit rectus; & ob ſimilia triangula <lb></lb><emph type="italics"></emph>PVQ, PSO,<emph.end type="italics"></emph.end>fit <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad 1/2<emph type="italics"></emph>VQ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>OP<emph.end type="italics"></emph.end>ad 1/2<emph type="italics"></emph>OS.<emph.end type="italics"></emph.end>Eſt igitur <lb></lb>(<emph type="italics"></emph>OS/OPXSPq<emph.end type="italics"></emph.end>) ut reſiſtentia, id eſt, in ratione denſitatis Medii in <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>& ratione duplicata velocitatis conjunctim. </s> <s>Auferatur duplicata <lb></lb>ratio velocitatis, nempe ratio (1/<emph type="italics"></emph>SP<emph.end type="italics"></emph.end>), & manebit Medii denſitas in <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>OS/OPXSP<emph.end type="italics"></emph.end>). Detur Spiralis, & ob datam rationem <emph type="italics"></emph>OS<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>OP,<emph.end type="italics"></emph.end>denſitas Medii in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erit ut (1/<emph type="italics"></emph>SP<emph.end type="italics"></emph.end>). In Medio igitur cujus <lb></lb>denſitas eſt reciproce ut diſtantia a centro <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>corpus gyrari po<lb></lb>teſt in hac Spirali. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note231"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Velocitas in loco quovis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ea ſemper eſt quacum cor<lb></lb>pus in Medio non reſiſtente gyrari poteſt in Circulo, ad eandem a <lb></lb>centro diſtantiam <emph type="italics"></emph>SP.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Medii denſitas, ſi datur diſtantia <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>eſt ut (<emph type="italics"></emph>OS/OP<emph.end type="italics"></emph.end>), ſin <lb></lb>diſtantia illa non datur, ut (<emph type="italics"></emph>OS/OPXSP<emph.end type="italics"></emph.end>). Et inde Spiralis ad quam<lb></lb>libet Medii denſitatem aptari poteſt. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Vis reſiſtentiæ in loco quovis <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>eſt ad vim centripe-<pb xlink:href="039/01/284.jpg" pagenum="256"></pb><arrow.to.target n="note232"></arrow.to.target>tam in eodem loco ut 1/2<emph type="italics"></emph>OS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OP.<emph.end type="italics"></emph.end>Nam vires illæ ſunt ad invi<lb></lb>vicem ut 1/4<emph type="italics"></emph>Rr<emph.end type="italics"></emph.end>& <emph type="italics"></emph>TQ<emph.end type="italics"></emph.end>ſive ut (1/4<emph type="italics"></emph>VQXPQ/SQ<emph.end type="italics"></emph.end>) & (1/2<emph type="italics"></emph>PQq/SP<emph.end type="italics"></emph.end>), hoc eſt, ut 1/2<emph type="italics"></emph>VQ<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>ſeu 1/2<emph type="italics"></emph>OS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>OP.<emph.end type="italics"></emph.end>Data igitur Spirali datur proportio re<lb></lb>ſiſtentiæ ad vim centripetam, & viceverſa ex data illa proportione <lb></lb>datur Spiralis. </s></p> <p type="margin"> <s><margin.target id="note232"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Corpus itaque gyrari nequit in hac Spirali, niſi ubi vis <lb></lb>reſiſtentiæ minor eſt quam dimidium vis centripetæ. </s> <s>Fiat reſiſten<lb></lb>tia æqualis dimidio vis centripetæ & Spiralis conveniet cum linea <lb></lb>recta <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end>inque hac recta corpus deſcendet ad centrum, ea cum <lb></lb>velocitate quæ ſit ad velocitatem qua probavimus in ſuperioribus <lb></lb>in caſu Parabolæ (Theor. </s> <s>X, Lib. </s> <s>I,) deſcenſum in Medio non reſi<lb></lb>ſtente fieri, in ſubduplicata ratione unitatis ad numerum binarium. </s> <s><lb></lb>Et tempora deſcenſus hic erunt reciproce ut velocitates, atque <lb></lb>adeo dantur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et quoniam in æqualibus a centro diſtantiis velocitas <lb></lb>eadem eſt in Spirali <emph type="italics"></emph>PQR<emph.end type="italics"></emph.end>atQ.E.I. recta <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>& longitudo Spi<lb></lb>ralis ad longitudinem rectæ <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>eſt in data ratione, nempe in <lb></lb>ratione <emph type="italics"></emph>OP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OS<emph.end type="italics"></emph.end>; tempus deſcenſus in Spirali erit ad tem<lb></lb>pus deſcenſus in recta <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>in eadem illa data ratione, proinde<lb></lb>Q.E.D.tur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Si centro <emph type="italics"></emph>S<emph.end type="italics"></emph.end>intervallis duobus quibuſcunQ.E.D.tis deſcri<lb></lb>bantur duo Circuli; & manentibus hiſce Circulis, mutetur utcun<lb></lb>que angulus quem Spiralis continet cum radio <emph type="italics"></emph>PS:<emph.end type="italics"></emph.end>numerus revo<lb></lb>lutionum quas corpus intra Circulorum circumferentias, pergendo <lb></lb>in Spirali a circumferentia ad circumferentiam, complere poteſt, eſt <lb></lb>ut (<emph type="italics"></emph>PS/OS<emph.end type="italics"></emph.end>), ſive ut Tangens anguli illius quem Spiralis continet cum <lb></lb>radio <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>; tempus vero revolutionum earundem ut (<emph type="italics"></emph>OP/OS<emph.end type="italics"></emph.end>), id eſt, ut <lb></lb>Secans anguli ejuſdem, vel etiam reciproce ut Medii denſitas. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Si corpus, in Medio cujus denſitas eſt reciproce ut di<lb></lb>ſtantia loeorum a centro, revolutionem in Curva quacunque <emph type="italics"></emph>AEB<emph.end type="italics"></emph.end><lb></lb>circa centrum illud fecerit, & Radium primum <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>in eodem an<lb></lb>gulo ſecuerit in <emph type="italics"></emph>B<emph.end type="italics"></emph.end>quo prius in <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>idque cum velocitate quæ fue<lb></lb>rit ad velocitatem ſuam primam in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>reciproce in ſubduplica<lb></lb>ta ratione diſtantiarum a centro (id eſt, ut <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>ad mediam pro<lb></lb>portionalem inter <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BS<emph.end type="italics"></emph.end>) corpus illud perget innume<lb></lb>ras conſimiles revolutiones <emph type="italics"></emph>BFC, CGD<emph.end type="italics"></emph.end>&c. </s> <s>facere, & interſe-<pb xlink:href="039/01/285.jpg" pagenum="257"></pb>ctionibus diſtinguet Radium <emph type="italics"></emph>AS<emph.end type="italics"></emph.end>in partes <emph type="italics"></emph>AS, BS, CS, DS,<emph.end type="italics"></emph.end>&c. <lb></lb><arrow.to.target n="note233"></arrow.to.target>continue proportionales. </s> <s>Revolutionum vero tempora erunt ut <lb></lb><figure id="id.039.01.285.1.jpg" xlink:href="039/01/285/1.jpg"></figure><lb></lb>perimetri Orbitarum <emph type="italics"></emph>AEB, BFC, CGD,<emph.end type="italics"></emph.end>&c. </s> <s>directe, & veloci<lb></lb>tates in principiis <emph type="italics"></emph>A, B, C,<emph.end type="italics"></emph.end>inverſe; id eſt, ut <emph type="italics"></emph>AS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>, <emph type="italics"></emph>BS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>, <emph type="italics"></emph>CS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>. </s> <s>At<lb></lb>que tempus totum, quo corpus perveniet ad centrum, erit ad tem<lb></lb>pus revolutionis primæ, ut ſumma omnium continue proportiona<lb></lb>lium <emph type="italics"></emph>AS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>, <emph type="italics"></emph>BS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>, <emph type="italics"></emph>CS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end> pergentium in infinitum, ad terminum pri<lb></lb>mum <emph type="italics"></emph>AS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>; id eſt, ut terminus ille primus <emph type="italics"></emph>AS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end> ad differentiam du<lb></lb>orum primorum <emph type="italics"></emph>AS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>-<emph type="italics"></emph>BS<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph.end type="sup"></emph.end>, ſive ut 2/3<emph type="italics"></emph>AS<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>quam proxime. </s> <s><lb></lb>Unde tempus illud totum expedite invenitur. </s></p> <p type="margin"> <s><margin.target id="note233"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Ex his etiam præter propter colligere licet motus cor<lb></lb>porum in Mediis, quorum denſitas aut uniformis eſt, aut aliam <lb></lb>quamcunque legem aſſignatam obſervat. </s> <s>Centro <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>intervallis con<lb></lb>tinue proportionalibus <emph type="italics"></emph>SA, SB, SC,<emph.end type="italics"></emph.end>&c. </s> <s>deſcribe Circulos quot<lb></lb>cunque, & ſtatue tempus revolutionum inter perimetros duorum <lb></lb>quorumvis ex his Circulis, in Medio de quo egimus, eſſe ad tempus <lb></lb>revolutionum inter eoſdem in Medio propoſito, ut Medii propo<lb></lb>ſiti denſitas mediocris inter hos Circulos ad Medii, de quo egimus, <lb></lb>denſitatem mediocrem inter eoſdem quam proxime: Sed & in ea<lb></lb>dem quoque ratione eſſe Secantem anguli quo Spiralis præfinita, <lb></lb>in Medio de quo egimus, ſecat radium <emph type="italics"></emph>AS,<emph.end type="italics"></emph.end>ad Secantem anguli <pb xlink:href="039/01/286.jpg" pagenum="258"></pb><arrow.to.target n="note234"></arrow.to.target>quo Spiralis nova ſecat radium eundem in Medio propoſito: At<lb></lb>que etiam ut ſunt eorundem angulorum Tangentes ita eſſe numeros <lb></lb>revolutionum omnium inter Circulos eoſdem duos quam proxime. </s> <s><lb></lb>Si hæc fiant paſſim inter Circulos binos, continuabitur motus per <lb></lb>Circulos omnes. </s> <s>Atque hoc pacto haud difficulter imaginari poſſi<lb></lb>mus quibus modis ac temporibus corpora in Medio quocunque re<lb></lb>gulari gyrari debebunt. </s></p> <p type="margin"> <s><margin.target id="note234"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>9. Et quamvis motus excentrici in Spiralibus ad formam <lb></lb>Ovalium accedentibus peragantur; tamen concipiendo Spiralium <lb></lb>illarum ſingulas revolutiones iiſdem ab invicem intervallis diſtare, <lb></lb>iiſdemque gradibus ad centrum accedere cum Spirali ſuperius de<lb></lb>ſcripta, intelligemus etiam quomodo motus corporum in hujuſmo<lb></lb>di Spiralibus peragantur. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XVI. THEOREMA XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Medii denſitas in locis ſingulis ſit reciproce ut diſtantia loco<lb></lb>rum a centro immobili, ſitque vis centripeta reciproce ut dig<lb></lb>nitas quælibet ejuſdem diſtantiæ: dico quod corpus gyrari potest <lb></lb>in Spirali quæ radios omnes a centro illo ductos interſecat in <lb></lb>angulo dato.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Demonſtratur eadem methodo cum Propoſitione ſuperiore. </s> <s><lb></lb>Nam ſi vis centripeta in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ſit reciproce ut diſtantiæ <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>dignitas <lb></lb>quælibet <emph type="italics"></emph>SP<emph type="sup"></emph>n<emph.end type="italics"></emph.end>+1<emph.end type="sup"></emph.end> cujus index eſt <emph type="italics"></emph>n<emph.end type="italics"></emph.end>+1; colligetur ut ſupra, <lb></lb>quod tempus quo corpus deſcribit arcum quemvis <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>erit ut <lb></lb><emph type="italics"></emph>PQXSP<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, & reſiſtentia in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>Rr/PQqXSP<emph type="sup"></emph>n<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>), ſive ut (—1-1/2<emph type="italics"></emph>nXVQ/PQXSP<emph type="sup"></emph>n<emph.end type="sup"></emph.end>XSQ<emph.end type="italics"></emph.end>), <lb></lb>adeoque ut (—1-1/2<emph type="italics"></emph>nXOS/OPXSP<emph type="sup"></emph>n+1<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>), hoc eſt, ob datum (—1-1/2<emph type="italics"></emph>nXOS/OP<emph.end type="italics"></emph.end>), recipro<lb></lb>ce ut <emph type="italics"></emph>SP<emph type="sup"></emph>n+1<emph.end type="sup"></emph.end>.<emph.end type="italics"></emph.end>Et propterea, cum velocitas ſit reciproce ut <emph type="italics"></emph>SP<emph.end type="italics"></emph.end><emph type="sup"></emph>1/2<emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, <lb></lb>denſitas in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erit reciproce ut <emph type="italics"></emph>SP.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Reſiſtentia eſt ad vim centripetam, ut —1-1/2<emph type="italics"></emph>nXOS<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>OP.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si vis centripeta ſit reciproce ut <emph type="italics"></emph>SPcub,<emph.end type="italics"></emph.end>erit 1-1/2<emph type="italics"></emph>n=o<emph.end type="italics"></emph.end>; <lb></lb>adeoque reſiſtentia & denſitas Medii nulla erit, ut in Propoſitione <lb></lb>nona Libri primi. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si vis centripeta ſit reciproce ut dignitas aliqua radii <lb></lb><emph type="italics"></emph>SP<emph.end type="italics"></emph.end>cujus index eſt major numero 3, reſiſtentia affirmativa in nega<lb></lb>tivam mutabitur. </s></p><pb xlink:href="039/01/287.jpg" pagenum="259"></pb> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note235"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note235"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Cæterum hæc Propoſitio & ſuperiores, quæ ad Media inæquali<lb></lb>ter denſa ſpectant, intelligendæ ſunt de motu corporum adeo par<lb></lb>vorum, ut Medii ex uno corporis latere major denſitas quam ex al<lb></lb>tero non conſideranda veniat. </s> <s>Reſiſtentiam quoque cæteris paribus <lb></lb>denſitati proportionalem eſſe ſuppono. </s> <s>Unde in Mediis quorum <lb></lb>vis reſiſtendi non eſt ut denſitas, debet denſitas eo uſque augeri vel <lb></lb>diminui, ut reſiſtentiæ vel tollatur exceſſus vel defectus ſuppleatur. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XVII. PROBLEMA IV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Invenire & vim centripetam & Medii reſiſtentiam qua corpus <lb></lb>in data Spirali, data velocitatis Lege, revolvi potest.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit Spiralis illa <emph type="italics"></emph>PQR.<emph.end type="italics"></emph.end>Ex velocitate qua corpus percurrit ar<lb></lb>cum quam minimum <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>dabitur tempus, & ex altitudine <emph type="italics"></emph>TQ,<emph.end type="italics"></emph.end><lb></lb>quæ eſt ut vis centripeta & quadratum temporis, dabitur vis. </s> <s>De<lb></lb>inde ex arearum, æqualibus temporum particulis confectarum <emph type="italics"></emph>PSQ<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>QSR,<emph.end type="italics"></emph.end>differentia <emph type="italics"></emph>RSr,<emph.end type="italics"></emph.end>dabitur corporis retardatio, & ex re<lb></lb>tardatione invenietur reſiſtentia ac denſitas Medii. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XVIII. PROBLEMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Data Lege vis centripetæ, invenire Medii denſitatem in locis ſin<lb></lb>gulis, qua corpus datam Spiralem deſcribet.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ex vi centripeta invenienda eſt velocitas in locis ſingulis, de<lb></lb>inde ex velocitatis retardatione quærenda Medii denſitas: ut in <lb></lb>Propoſitione ſuperiore. </s></p> <p type="main"> <s>Methodum vero tractandi hæc Problemata aperui in hujus Pro<lb></lb>poſitione decima, & Lemmate ſecundo; & Lectorem in hujuſmodi <lb></lb>perplexis diſquiſitionibus diutius detinere nolo. </s> <s>Addenda jam <lb></lb>ſunt aliqua de viribus corporum ad progrediendum, deQ.E.D.nſi<lb></lb>tate & reſiſtentia Mediorum, in quibus motus hactenus expoſiti & <lb></lb>his affines peraguntur. </s></p><pb xlink:href="039/01/288.jpg" pagenum="260"></pb></subchap2><subchap2> <p type="main"> <s><arrow.to.target n="note236"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note236"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Denſitate & Compreſſione Fluidorum, deque <lb></lb>Hydroſtatica.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>Definitio Fluidi.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Fluidum eſt corpus omne cujus partes cedunt vi cuicunQ.E.I.latæ, <lb></lb>& cedendo facile moventur inter ſe.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XIX. THEOREMA XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Fluidi homogenei & immoti quod in vaſe quocunQ.E.I.moto clau<lb></lb>ditur & undique comprimitur, partes omnes (ſepoſita conden<lb></lb>ſationis, gravitatis & virium omnium centripetarum conſide<lb></lb>ratione) æqualiter premuntur undique, & abſque omni motu a <lb></lb>preſſione illa orto permanent in locis ſuis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>1. In vaſe ſphærico <emph type="italics"></emph>ABC<emph.end type="italics"></emph.end>claudatur & uniformiter com<lb></lb>primatur fluidum undique: dico quod ejuſdem pars nulla ex illa <lb></lb>preſſione movebitur. </s> <s>Nam ſi pars aliqua <emph type="italics"></emph>D<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.288.1.jpg" xlink:href="039/01/288/1.jpg"></figure><lb></lb>moveatur, neceſſe eſt ut omnes hujuſmodi <lb></lb>partes, ad eandem a centro diſtantiam un<lb></lb>dique conſiſtentes, ſimili motu ſimul move<lb></lb>antur; atque hoc adeo quia ſimilis & æ<lb></lb>qualis eſt omnium preſſio, & motus omnis <lb></lb>excluſus ſupponitur, niſi qui a preſſione il<lb></lb>la oriatur. </s> <s>Atqui non poſſunt omnes ad <lb></lb>centrum propius accedere, niſi fluidum ad <lb></lb>centrum condenſetur; contra Hypotheſin. </s> <s><lb></lb>Non poſſunt longius ab eo recedere, niſi <lb></lb>fluidum ad circumferentiam condenſetur; <lb></lb>etiam contra Hypotheſin. </s> <s>Non poſſunt ſervata ſua a centro di<lb></lb>ſtantia moveri in plagam quamcunque, quia pari ratione movebun<lb></lb>tur in plagam contrariam; in plagas autem contrarias non poteſt <pb xlink:href="039/01/289.jpg" pagenum="261"></pb>pars eadem, eodem tempore, moveri. </s> <s>Ergo fluidi pars nulla de lo<lb></lb><arrow.to.target n="note237"></arrow.to.target>co ſuo movebitur. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note237"></margin.target>LIBER <lb></lb>SECUNDUS</s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>2. Dico jam quod fluidi hujus partes omnes ſphæricæ æqua<lb></lb>liter premuntur undique: ſit enim <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>pars ſphærica fluidi, & ſi <lb></lb>hæc undique non premitur æqualiter, augeatur preſſio minor, uſ<lb></lb>Q.E.D.m ipſa undique prematur æqualiter; & partes ejus, per <lb></lb>Caſum primum, permanebunt in locis ſuis. </s> <s>Sed ante auctam preſ<lb></lb>ſionem permanebunt in locis ſuis, per Caſum eundum primum, & <lb></lb>additione preſſionis novæ movebuntur de locis ſuis, per definitio<lb></lb>nem Fluidi. </s> <s>Quæ duo repugnant. </s> <s>Ergo falſo dicebatur quod Sphæ<lb></lb>ra <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>non undique premebatur æqualiter. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>3. Dico præterea quod diverſarum partium ſphæricarum æ<lb></lb>qualis ſit preſſio. </s> <s>Nam partes ſphæricæ contiguæ ſe mutuo pre<lb></lb>munt æqualiter in puncto contactus, per motus Legem III. </s> <s>Sed &, <lb></lb>per Caſum ſecundum, undique premuntur eadem vi. </s> <s>Partes igitur <lb></lb>duæ quævis ſphæricæ non contiguæ, quia pars ſphærica intermedia <lb></lb>tangere poteſt utramque, prementur eadem vi. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>4. Dico jam quod fluidi partes omnes ubique premuntur <lb></lb>æqualiter. </s> <s>Nam partes duæ quævis tangi poſſunt a partibus Sphæ<lb></lb>ricis in punctis quibuſcunque, & ibi partes illas Sphæricas æquali<lb></lb>ter premunt, per Caſum 3. & viciſſim ab illis æqualiter premuntur, <lb></lb>per Motus Legem tertiam. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>5. Cum igitur fluidi pars quælibet <emph type="italics"></emph>GHI<emph.end type="italics"></emph.end>in fluido reliquo <lb></lb>tanquam in vaſe claudatur, & undique prematur æqualiter, partes <lb></lb>autem ejus ſe mutuo æqualiter premant & quieſcant inter ſe; ma<lb></lb>nifeſtum eſt quod Fluidi cujuſcunque <emph type="italics"></emph>GHI,<emph.end type="italics"></emph.end>quod undique premi<lb></lb>tur æqualiter, partes omnes ſe mutuo premunt æqualiter, & qui<lb></lb>eſcunt inter ſe. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>6. Igitur ſi Fluidum illud in vaſe non rigido claudatur, & <lb></lb>undique non prematur æqualiter, cedet idem preſſioni fortiori, per <lb></lb>Definitionem Fluiditatis. </s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>7. IdeoQ.E.I. vaſe rigido Fluidum non ſuſtinebit preſſio<lb></lb>nem fortiorem ex uno latere quam ex alio, ſed eidem cedet, idque <lb></lb>in momento temporis, quia latus vaſis rigidum non perſequitur li<lb></lb>quorem cedentem. </s> <s>Cedendo autem urgebit latus oppoſitum, & <lb></lb>ſic preſſio undique ad æqualitatem verget. </s> <s>Et quoniam Fluidum, <lb></lb>quam primum a parte magis preſſa recedere conatur, inhibetur per <lb></lb>reſiſtentiam vaſis ad latus oppoſitum; reducetur preſſio undique <lb></lb>ad æqualitatem, in momento temporis, abſque motu locali: & ſub<lb></lb>inde partes fluidi, per Caſum quintum, ſe mutuo prement æqua<lb></lb>liter, & quieſcent inter ſe. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/290.jpg" pagenum="262"></pb><arrow.to.target n="note238"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note238"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Unde nec motus partium fluidi inter ſe, per preſſionem <lb></lb>fluido ubivis in externa ſuperficie illatam, mutari poſſunt, niſi qua<lb></lb>tenus aut figura ſuperficiei alicubi mutatur, aut omnes fluidi partes <lb></lb>intenſius vel remiſſius ſeſe premendo difficilius vel facilius labun<lb></lb>tur inter ſe. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XX. THEOREMA XV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Fluidi Sphærici, & in æqualibus a centro diſtantiis homogenei, <lb></lb>fundo Sphærico concentrico incumbentis partes ſingulæ verſus <lb></lb>centrum totius gravitent; ſuſtinet fundum pondus Cylindri, cu<lb></lb>jus bafis æqualis est ſuperficiei fundi, & altitudo eadem quæ <lb></lb>Fluidi incumbentis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>DHM<emph.end type="italics"></emph.end>ſuperficies ſundi, & <emph type="italics"></emph>AEI<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.290.1.jpg" xlink:href="039/01/290/1.jpg"></figure><lb></lb>ſuperficies ſuperior fluidi. </s> <s>Superficiebus <lb></lb>ſphæricis innumeris <emph type="italics"></emph>BFK, CGL<emph.end type="italics"></emph.end>diſtin<lb></lb>guatur fluidum in Orbes concentricos æ<lb></lb>qualiter craſſos; & concipe vim gravita<lb></lb>tis agere ſolummodo in ſuperficiem ſupe<lb></lb>riorem Orbis cujuſque, & æquales eſſe a<lb></lb>ctiones in æquales partes ſuperficierum om<lb></lb>nium. </s> <s>Premitur ergo ſuperficies ſuprema <lb></lb><emph type="italics"></emph>AE<emph.end type="italics"></emph.end>vi ſimplici gravitatis propriæ, qua & <lb></lb>omnes Orbis ſupremi partes & ſuperficies <lb></lb>ſecunda <emph type="italics"></emph>BFK<emph.end type="italics"></emph.end>(per Prop. </s> <s>XIX.) pro menſura ſua æqualiter pre<lb></lb>muntur. </s> <s>Premitur præterea ſuperficies ſecunda <emph type="italics"></emph>BFK<emph.end type="italics"></emph.end>vi propriæ <lb></lb>gravitatis, quæ addita vi priori facit preſſionem duplam. </s> <s>Hac <lb></lb>preſſione, pro menſura ſua, & inſuper vi propriæ gravitatis, id eſt <lb></lb>preſſione tripla, urgetur ſuperficies tertia <emph type="italics"></emph>CGL.<emph.end type="italics"></emph.end>Et ſimiliter preſ<lb></lb>ſione quadrupla urgetur ſuperficies quarta, quintupla quinta, & <lb></lb>ſic deinceps. </s> <s>Preſſio igitur qua ſuperficies unaquæque urgetur, <lb></lb>non eſt ut quantitas ſolida fluidi incumbentis, ſed ut numerus Or<lb></lb>bium ad uſque ſummitatem fluidi; & æquatur gravitati Orbis infi<lb></lb>mi multiplicatæ per numerum Orbium: hoc eſt, gravitati ſolidi cu<lb></lb>jus ultima ratio ad Cylindrum præfinitum, (ſi modo Orbium au<lb></lb>geatur numerus & minuatur craſſitudo in infinitum, ſic ut actio <lb></lb>gravitatis a ſuperficie infima ad ſupremam continua reddatur) fiet <lb></lb>ratio æqualitatis. </s> <s>Suſtinet ergo ſuperficies infima pondus Cylindri <pb xlink:href="039/01/291.jpg" pagenum="263"></pb>præfiniti. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end>Et ſimili argumentatione patet Propoſitio, </s></p> <p type="main"> <s><arrow.to.target n="note239"></arrow.to.target>ubi gravitas decreſcit in ratione quavis aſſignata diſtantiæ a centro, <lb></lb>ut & ubi Fluidum ſurſum rarius eſt, deorſum denſius. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note239"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur fundum non urgetur a toto fluidi incumbentis <lb></lb>pondere, ſed eam ſolummodo ponderis partem ſuſtinet quæ in <lb></lb>propoſitione deſcribitur; pondere reliquo a fluidi figura fornicata <lb></lb>ſuſtentato. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. In æqualibus autem a centro diſtantiis eadem ſemper eſt <lb></lb>preſſionis quantitas, ſive ſuperficies preſſa ſit Horizonti parallela <lb></lb>vel perpendicularis vel obliqua; ſive fluidum, a ſuperficie preſſa ſur<lb></lb>ſum continuatum, ſurgat perpendiculariter ſecundum lineam rectam, <lb></lb>vel ſerpit oblique per tortas cavitates & canales, eaſque regulares <lb></lb>vel maxime irregulares, amplas vel anguſtiſſimas. </s> <s>Hiſce circum<lb></lb>ſtantiis preſſionem nil mutari colligitur, applicando demonſtratio<lb></lb>nem Theorematis hujus ad Caſus ſingulos Fluidorum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Eadem Demonſtratione colligitur etiam (per Prop. </s> <s>XIX) <lb></lb>quod fluidi gravis partes nullum, ex preſſione ponderis incumben<lb></lb>tis, acquirunt motum inter ſe, ſi modo excludatur motus qui ex <lb></lb>condenſatione oriatur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Et propterea ſi aliud ejuſdem gravitatis ſpecificæ cor<lb></lb>pus, quod ſit condenſationis expers, ſubmergatur in hoc fluido, id <lb></lb>ex preſſione ponderis incumbentis nullum acquiret motum: non <lb></lb>deſcendet, non aſcendet, non cogetur figuram ſuam mutare. </s> <s>Si <lb></lb>ſphæricum eſt manebit ſphæricum, non obſtante preſſione; ſi qua<lb></lb>dratum eſt manebit quadratum: idque ſive molle ſit, ſive fluidiſſi<lb></lb>mum; ſive fluido libere innatet, ſive fundo incumbat. </s> <s>Habet e<lb></lb>nim fluidi pars quælibet interna rationem corporis ſubmerſi, & par <lb></lb>eſt ratio omnium ejuſdem magnitudinis, figuræ & gravitatis ſpeci<lb></lb>ficæ ſubmerſorum corporum. </s> <s>Si corpus ſubmerſum ſervato pon<lb></lb>dere liqueſceret & indueret formam fluidi; hoc, ſi prius aſcende<lb></lb>ret vel deſcenderet vel ex preſſione figuram novam indueret, etiam <lb></lb>nunc aſcenderet vel deſcenderet vel figuram novam induere coge<lb></lb>retur: id adeo quia gravitas ejus cæteræque motuum cauſæ per<lb></lb>manent. </s> <s>Atqui, per Caſ. </s> <s>5. Prop. </s> <s>XIX, jam quieſceret & figuram <lb></lb>retineret. </s> <s>Ergo & prius. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Proinde corpus quod ſpecifice gravius eſt quam Flui<lb></lb>dum ſibi contiguum ſubſidebit, & quod ſpecifice levius eſt aſcen<lb></lb>det, motumque & figuræ mutationem conſequetur, quantum ex<lb></lb>ceſſus ille vel defectus gravitatis efficere poſſit. </s> <s>Namque exceſſus <lb></lb>ille vel deſectus rationem habet impulſus, quo corpus, alias in <pb xlink:href="039/01/292.jpg" pagenum="264"></pb><arrow.to.target n="note240"></arrow.to.target>æquilibrio cum fluidi partibus conſtitutum, urgetur; & comparari <lb></lb>poteſt cum exceſſu vel defectu ponderis in lance alterutra libræ. </s></p> <p type="margin"> <s><margin.target id="note240"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Corporum igitur in fluidis conſtitutorum duplex eſt Gra<lb></lb>vitas: altera vera & abſoluta, altera apparens, vulgaris & compa<lb></lb>rativa. </s> <s>Gravitas abſoluta eſt vis tota qua corpus deorſum tendit: <lb></lb>relativa & vulgaris eſt exceſſus gravitatis quo corpus magis tendit <lb></lb>deorſum quam fluidum ambiens. </s> <s>Prioris generis Gravitate partes <lb></lb>fluidorum & corporum omnium gravitant in locis ſuis: ideoque <lb></lb>conjunctis ponderibus componunt pondus totius. </s> <s>Nam totum <lb></lb>omne grave eſt, ut in vaſis liquorum plenis experiri licet; & pon<lb></lb>dus totius æquale eſt ponderibus omnium partium, ideoque ex iiſ<lb></lb>dem componitur. </s> <s>Alterius generis Gravitate corpora non gravi<lb></lb>tant in locis ſuis, id eſt, inter ſe collata non prægravant, ſed mu<lb></lb>tuos ad deſcendendum conatus impedientia permanent in locis <lb></lb>ſuis, perinde ac ſi gravia non eſſent. </s> <s>Quæ in Aere ſunt & non <lb></lb>prægravant, vulgus gravia non judicat. </s> <s>Quæ prægravant vulgus <lb></lb>gravia judicat, quatenus ab Aeris pondere non ſuſtinentur. </s> <s>Pon<lb></lb>dera vulgi nihil aliud ſunt quam exceſſus verorum ponderum ſu<lb></lb>pra pondus Aeris. </s> <s>Unde & vulgo dicuntur levia, quæ ſunt mi<lb></lb>nus gravia, Aerique prægravanti cedendo ſuperiora petunt. </s> <s>Com<lb></lb>parative levia ſunt, non vere, quia deſcendunt in vacuo. </s> <s>Sic & <lb></lb>in Aqua, corpora, quæ ob majorem vel minorem gravitatem de<lb></lb>ſcendunt vel aſcendunt, ſunt comparative & apparenter gravia vel <lb></lb>levia, & eorum gravitas vel levitas comparativa & apparens eſt ex<lb></lb>ceſſus vel defectus quo vera eorum gravitas vel ſuperat gravita<lb></lb>tem aque vel ab ea ſuperatur. </s> <s>Quæ vero nec prægravando de<lb></lb>ſcendunt, nec prægravanti cedendo aſcendunt, etiamſi veris ſuis <lb></lb>ponderibus adaugeant pondus totius, comparative tamen & in ſen<lb></lb>ſu vulgi non gravitant in aqua. </s> <s>Nam ſimilis eſt horum Caſuum <lb></lb>Demonſtratio. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Quæ de gravitate demonſtrantur, obtinent in aliis qui<lb></lb>buſcunque viribus centripetis. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Proinde ſi Medium, in quo corpus aliquod movetur, <lb></lb>urgeatur vel a gravitate propria, vel ab alia quacunque vi centri<lb></lb>peta, & corpus ab eadem vi urgeatur fortius: differentia virium <lb></lb>eſt vis illa motrix, quam in præcedentibus Propoſitionibus ut vim <lb></lb>centripetam conſideravimus. </s> <s>Sin corpus a vi illa urgeatur levius, <lb></lb>differentia virium pro vi centrifuga haberi debet. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>9. Cum autem fluida premendo corpora incluſa non <lb></lb>mutent eorum Figuras externas, patet inſuper, per Corollarium <pb xlink:href="039/01/293.jpg" pagenum="265"></pb>Prop. </s> <s>XIX, quod non mutabunt ſitum partium internarum inter <lb></lb><arrow.to.target n="note241"></arrow.to.target>ſe: proindeque, ſi Animalia immergantur, & ſenſatio omnis a mo<lb></lb>tu partium oriatur; nec lædent corpora immerſa, nec ſenſatio<lb></lb>nem ullam excitabunt, niſi quatenus hæc corpora a compreſſione <lb></lb>condenſari poſſunt. </s> <s>Et par eſt ratio cujuſcunque corporum Sy<lb></lb>ſtematis fluido comprimente circundati. </s> <s>Syſtematis partes omnes <lb></lb>iiſdem agitabuntur motibus, ac ſi in vacuo conſtituerentur, ac ſo<lb></lb>lam retinerent gravitatem ſuam comparativam, niſi quatenus flui<lb></lb>dum vel motibus earum nonnihil reſiſtat, vel ad eaſdem compreſſi<lb></lb>one conglutinandas requiratur. </s></p> <p type="margin"> <s><margin.target id="note241"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXI. THEOREMA XVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Sit Fluidi cujuſdam denſitas compreſſioni proportionalis, & partes <lb></lb>ejus a vi centripeta diſtantiis ſuis a centro reciproce proportio<lb></lb>nali deorſum trabantur: dico quod, fi diſtantiæ illæ ſumantur <lb></lb>continue proportionales, denſitates Fluidi in iiſdem diſtantiis e<lb></lb>runt etiam continue proportionales.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Deſignet <emph type="italics"></emph>ATV<emph.end type="italics"></emph.end>fundum Sphæricum cui fluidum incumbit, <emph type="italics"></emph>S<emph.end type="italics"></emph.end><lb></lb>centrum, <emph type="italics"></emph>SA, SB, SC, SD, SE,<emph.end type="italics"></emph.end>&c. </s> <s>diſtantias continue propor<lb></lb>tionales. </s> <s>Erigantur perpendicula <emph type="italics"></emph>AH, BI, CK, DL, EM, &c.<emph.end type="italics"></emph.end><lb></lb>quæ ſint ut denſitates Medii in locis <emph type="italics"></emph>A, B, C, D, E<emph.end type="italics"></emph.end>; & ſpecificæ <lb></lb>gravitates in iiſdem locis erunt ut <emph type="italics"></emph>(AH/AS), (BI/BS), (CK/CS),<emph.end type="italics"></emph.end>&c. </s> <s>vel, quod <lb></lb>perinde eſt, ut <emph type="italics"></emph>(AH/AB), (BI/BC), (CK/CD),<emph.end type="italics"></emph.end>&c. </s> <s>Finge pri<lb></lb><figure id="id.039.01.293.1.jpg" xlink:href="039/01/293/1.jpg"></figure><lb></lb>mum has gravitates uniformiter continuari ab <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>a <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>&c. </s> <s>factis per <lb></lb>gradus decrementis in punctis <emph type="italics"></emph>B, C, D,<emph.end type="italics"></emph.end>&c. </s> <s>Et <lb></lb>hæ gravitates ductæ in altitudines <emph type="italics"></emph>AB, BC, <lb></lb>CD,<emph.end type="italics"></emph.end>&c. </s> <s>conficient preſſiones <emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end><lb></lb>quibus fundum <emph type="italics"></emph>ATV<emph.end type="italics"></emph.end>(juxta Theorema XV.) <lb></lb>urgetur. </s> <s>Suſtinet ergo particula <emph type="italics"></emph>A<emph.end type="italics"></emph.end>preſſiones <lb></lb>omnes <emph type="italics"></emph>AH, BI, CK, DL,<emph.end type="italics"></emph.end>pergendo in <lb></lb>infinitum; & particula <emph type="italics"></emph>B<emph.end type="italics"></emph.end>preſſiones omnes <lb></lb>præter primam <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>; & particula <emph type="italics"></emph>C<emph.end type="italics"></emph.end>omnes <lb></lb>præter duas primas <emph type="italics"></emph>AH, BI<emph.end type="italics"></emph.end>; & ſic deinceps: adeoque parti<lb></lb>culæ primæ <emph type="italics"></emph>A<emph.end type="italics"></emph.end>denſitas <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>eſt ad particulæ ſecundæ <emph type="italics"></emph>B<emph.end type="italics"></emph.end>denſi-<pb xlink:href="039/01/294.jpg" pagenum="266"></pb><arrow.to.target n="note242"></arrow.to.target>tatem <emph type="italics"></emph>BI<emph.end type="italics"></emph.end>ut ſumma omnium <emph type="italics"></emph>AH+BI+CK+DL,<emph.end type="italics"></emph.end>in infiNI<lb></lb>tum, ad ſummam omnium <emph type="italics"></emph>BI+CK+DL,<emph.end type="italics"></emph.end>&c. </s> <s>Et <emph type="italics"></emph>BI<emph.end type="italics"></emph.end>den<lb></lb>ſitas ſecundæ <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>denſitatem tertiæ <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>ut ſumma om<lb></lb>nium <emph type="italics"></emph>BI+CK+DL,<emph.end type="italics"></emph.end>&c. </s> <s>ad ſummam omnium <emph type="italics"></emph>CK+DL,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>Sunt igitur ſummæ illæ differentiis ſuis <emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end>&c. </s> <s>pro<lb></lb>portionales, atque adeo continue proportionales, per hujus Lem. </s> <s>I. <lb></lb>proindeQ.E.D.fferentiæ <emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end>&c. </s> <s>ſummis proportionales, <lb></lb>ſunt etiam continue proportionales. </s> <s>Quare cum denſitates in locis <emph type="italics"></emph>A, <lb></lb>B, C,<emph.end type="italics"></emph.end>&c. </s> <s>ſint ut <emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end>&c. </s> <s>erunt etiam hæ continue propor<lb></lb>tionales. </s> <s>Pergatur per ſaltum, & (ex æquo) in diſtantiis <emph type="italics"></emph>SA, SC, <lb></lb>SE<emph.end type="italics"></emph.end>continue proportionalibus, erunt denſitates <emph type="italics"></emph>AH, CK, EM<emph.end type="italics"></emph.end><lb></lb>continue proportionales. </s> <s>Et eodem argumento, in diſtantiis qui<lb></lb>buſvis continue proportionalibus <emph type="italics"></emph>SA, SD, SG,<emph.end type="italics"></emph.end>denſitates <emph type="italics"></emph>AH, DL, <lb></lb>GO<emph.end type="italics"></emph.end>erunt continue proportionales. </s> <s>Coeant jam puncta <emph type="italics"></emph>A, B, C, <lb></lb>D, E,<emph.end type="italics"></emph.end>&c. </s> <s>eo ut progreſſio gravitatum ſpecificarum a fundo <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <lb></lb>ſummitatem Fluidi continua reddatur, & in diſtantiis quibuſvis con<lb></lb>tinue proportionalibus <emph type="italics"></emph>SA, SD, SG,<emph.end type="italics"></emph.end>denſitates <emph type="italics"></emph>AH, DL, GO,<emph.end type="italics"></emph.end><lb></lb>ſemper exiſtentes continue proportionales, manebunt etiamnum <lb></lb>continue proportionales. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note242"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc ſi detur denſitas Fluidi in duobus locis, puta <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>E,<emph.end type="italics"></emph.end>colligi poteſt ejus denſitas <lb></lb><figure id="id.039.01.294.1.jpg" xlink:href="039/01/294/1.jpg"></figure><lb></lb>in alio quovis loco <emph type="italics"></emph><expan abbr="q.">que</expan><emph.end type="italics"></emph.end>Centro <lb></lb><emph type="italics"></emph>S,<emph.end type="italics"></emph.end>Aſymptotis rectangulis <emph type="italics"></emph>SQ, <lb></lb>SX,<emph.end type="italics"></emph.end>deſcribatur Hyperbola ſe<lb></lb>cans perpendicula <emph type="italics"></emph>AH, EM, <lb></lb>QT<emph.end type="italics"></emph.end>in <emph type="italics"></emph>a, e, q,<emph.end type="italics"></emph.end>ut & perpendicu<lb></lb>la <emph type="italics"></emph>HX, MY, TZ,<emph.end type="italics"></emph.end>ad Aſymp<lb></lb>toton <emph type="italics"></emph>SX<emph.end type="italics"></emph.end>demiſſa, in <emph type="italics"></emph>h, m<emph.end type="italics"></emph.end>& <emph type="italics"></emph>t.<emph.end type="italics"></emph.end><lb></lb>Fiat area <emph type="italics"></emph>ZYmtZ<emph.end type="italics"></emph.end>ad aream da<lb></lb>tam <emph type="italics"></emph>YmhX<emph.end type="italics"></emph.end>ut area data <emph type="italics"></emph>EeqQ<emph.end type="italics"></emph.end><lb></lb>ad aream datam <emph type="italics"></emph>EeaA<emph.end type="italics"></emph.end>; & li<lb></lb>nea <emph type="italics"></emph>Zt<emph.end type="italics"></emph.end>producta abſcindet li<lb></lb>neam <emph type="italics"></emph>QT<emph.end type="italics"></emph.end>denſitati proportio<lb></lb>nalem. </s> <s>Namque ſi lineæ <emph type="italics"></emph>SA, SE, SQ<emph.end type="italics"></emph.end>ſunt continue proportiona<lb></lb>les, erunt areæ <emph type="italics"></emph>EeqQ, EeaA<emph.end type="italics"></emph.end>æquales, & inde areæ his propor<lb></lb>tionales <emph type="italics"></emph>YmtZ, XhmY<emph.end type="italics"></emph.end>etiam æquales, & lineæ <emph type="italics"></emph>SX, SY, SZ,<emph.end type="italics"></emph.end>id eſt <lb></lb><emph type="italics"></emph>AH, EM, QT<emph.end type="italics"></emph.end>continue proportionales, ut oportet. </s> <s>Et ſi lineæ <lb></lb><emph type="italics"></emph>SA, SE, SQ<emph.end type="italics"></emph.end>obtinent alium quemvis ordinem in ſerie continue <lb></lb>proportionalium, lineæ <emph type="italics"></emph>AH, EM, QT,<emph.end type="italics"></emph.end>ob proportionales areas <lb></lb>Hyperbolicas, obtinebunt eundem ordinem in alia ſerie quantita<lb></lb>tum continue proportionalium. <pb xlink:href="039/01/295.jpg" pagenum="267"></pb><arrow.to.target n="note243"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note243"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXII. THEOREMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Sit Fluidi cujuſdam denſitas compreſſioni proportionalis, & partes <lb></lb>ejus a gravitate quadratis diſtantiarum ſuarum a centro reci<lb></lb>proce proportionali deorſum trabantur: dico quod, ſi diſtantiæ <lb></lb>ſumantur in progreſſione Muſica, denſitates Fluidi in bis di<lb></lb>ſtantiis erunt in progreſſione Geometrica.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Deſignet <emph type="italics"></emph>S<emph.end type="italics"></emph.end>centrum, & <emph type="italics"></emph>SA, SB, SC, SD, SE<emph.end type="italics"></emph.end>diſtantias in pro<lb></lb>greſſione Geometrica. </s> <s>Erigantur perpendicula <emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>quæ ſint ut Fluidi denſitates in locis <emph type="italics"></emph>A, B, C, D, E,<emph.end type="italics"></emph.end>&c. </s> <s>& ipſius <lb></lb><figure id="id.039.01.295.1.jpg" xlink:href="039/01/295/1.jpg"></figure><lb></lb>gravitates ſpecificæ in iiſdem locis erunt <emph type="italics"></emph>(AH/SAq), (BI/SBq), (CK/SCq),<emph.end type="italics"></emph.end>&c. </s> <s>Fin<lb></lb>ge has gravitates uniformiter continuari, primam ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>ſe<lb></lb>cundam a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>tertiam a <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>&c. </s> <s>Et hæ ductæ in altitu<lb></lb>dines <emph type="italics"></emph>AB, BC, CD, DE,<emph.end type="italics"></emph.end>&c. </s> <s>vel, quod perinde eſt, in diſtantias <lb></lb><emph type="italics"></emph>SA, SB, SC,<emph.end type="italics"></emph.end>&c. </s> <s>altitudinibus illis proportionales, conficient ex<lb></lb>ponentes preſſionum <emph type="italics"></emph>(AH/SA), (BI/SB), (CK/SC),<emph.end type="italics"></emph.end>&c. </s> <s>Quare cum denſitates <lb></lb>ſint ut harum preſſionum ſummæ, differentiæ denſitatum <emph type="italics"></emph>AH-BI, <lb></lb>BI-CK,<emph.end type="italics"></emph.end>&c. </s> <s>erunt ut ſummarum differentiæ <emph type="italics"></emph>(AH/SA), (BI/SB), (CK/SC),<emph.end type="italics"></emph.end>&c. <pb xlink:href="039/01/296.jpg" pagenum="268"></pb><arrow.to.target n="note244"></arrow.to.target>Centro <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>Aſymptotis <emph type="italics"></emph>SA, Sx,<emph.end type="italics"></emph.end>deſcribatur Hyperbola quæ<lb></lb>vis, quæ ſecet perpendicula <emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end>&c. </s> <s>in <emph type="italics"></emph>a, b, c,<emph.end type="italics"></emph.end>&c. </s> <s>ut & <lb></lb>perpendicula ad Aſymptoton <emph type="italics"></emph>Sx<emph.end type="italics"></emph.end>demiſſa <emph type="italics"></emph>Ht, Iu, Kw<emph.end type="italics"></emph.end>in <emph type="italics"></emph>h, i, k<emph.end type="italics"></emph.end>; <lb></lb>& denſitatum differentiæ <emph type="italics"></emph>tu, uw,<emph.end type="italics"></emph.end>&c. </s> <s>erunt üt <emph type="italics"></emph>(AH/SA), (BI/SB),<emph.end type="italics"></emph.end>&c. </s> <s>Et <lb></lb>rectangula <emph type="italics"></emph>tuXth, uwXui,<emph.end type="italics"></emph.end>&c. </s> <s>ſeu <emph type="italics"></emph>tp, uq,<emph.end type="italics"></emph.end>&c. </s> <s>ut <emph type="italics"></emph>(AHXtb/SA), <lb></lb>(BIXui/SB),<emph.end type="italics"></emph.end>&c. </s> <s>id eſt, ut <emph type="italics"></emph>Aa, Bb,<emph.end type="italics"></emph.end>&c. </s> <s>Eſt enim, ex natura Hyperbolæ, <lb></lb><emph type="italics"></emph>SA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>St,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>th<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end>adeoque (<emph type="italics"></emph>AHXth/SA<emph.end type="italics"></emph.end>) æquale <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.296.1.jpg" xlink:href="039/01/296/1.jpg"></figure><lb></lb>Et ſimili argumento eſt (<emph type="italics"></emph>BIXui/SB<emph.end type="italics"></emph.end>) æquale <emph type="italics"></emph>Bb,<emph.end type="italics"></emph.end>&c. </s> <s>Sunt autem <emph type="italics"></emph>Aa, <lb></lb>Bb, Cc,<emph.end type="italics"></emph.end>&c. </s> <s>continue proportionales, & propterea differentiis ſu<lb></lb>is <emph type="italics"></emph>Aa-Bb, Bb-Cc,<emph.end type="italics"></emph.end>&c. </s> <s>proportionales; ideoQ.E.D.fferentiis <lb></lb>hiſce proportionalia ſunt rectangula <emph type="italics"></emph>tp, uq,<emph.end type="italics"></emph.end>&c. </s> <s>ut & ſummis diffe<lb></lb>rentiarum <emph type="italics"></emph>Aa-Cc<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Aa-Dd<emph.end type="italics"></emph.end>ſummæ rectangulorum <emph type="italics"></emph>tp+uq<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>tp+uq+wr.<emph.end type="italics"></emph.end>Sunto ejuſmodi termini quam plurimi, & ſum<lb></lb>ma omnium differentiarum, puta <emph type="italics"></emph>Aa-Ff,<emph.end type="italics"></emph.end>erit ſummæ omnium <lb></lb>rectangulorum, puta <emph type="italics"></emph>zthn,<emph.end type="italics"></emph.end>proportionalis. </s> <s>Augeatur numerus <lb></lb>terminorum & minuantur diſtantiæ punctorum <emph type="italics"></emph>A, B, C,<emph.end type="italics"></emph.end>&c. </s> <s>in in<lb></lb>nitum, & rectangula illa evadent æqualia areæ Hyperbolicæ <emph type="italics"></emph>zthn,<emph.end type="italics"></emph.end><lb></lb>adeoque huic areæ proportionalis eſt differentia <emph type="italics"></emph>Aa-Ff.<emph.end type="italics"></emph.end>Suman-<pb xlink:href="039/01/297.jpg" pagenum="269"></pb>tur jam diſtantiæ quælibet, puta <emph type="italics"></emph>SA, SD, SF<emph.end type="italics"></emph.end>in progreſſione Mu<lb></lb><arrow.to.target n="note245"></arrow.to.target>ſica, & differentiæ <emph type="italics"></emph>Aa-Dd, Dd-Ff<emph.end type="italics"></emph.end>erunt æquales; & propter<lb></lb>ea differentiis hiſce proportionales areæ <emph type="italics"></emph>thlx, xlnz<emph.end type="italics"></emph.end>æquales erunt <lb></lb>inter ſe, & denſitates <emph type="italics"></emph>St, Sx, Sz,<emph.end type="italics"></emph.end>id eſt, <emph type="italics"></emph>AH, DL, FN,<emph.end type="italics"></emph.end>conti<lb></lb>nue proportionales. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note244"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="margin"> <s><margin.target id="note245"></margin.target>LIBER <lb></lb>SECUNDUS</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc ſi dentur Fluidi denſitates duæ quævis, puta <emph type="italics"></emph>AH<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>CK,<emph.end type="italics"></emph.end>dabitur area <emph type="italics"></emph>thkw<emph.end type="italics"></emph.end>harum differentiæ <emph type="italics"></emph>tw<emph.end type="italics"></emph.end>reſpondens; & <lb></lb>inde invenietur denſitas <emph type="italics"></emph>FN<emph.end type="italics"></emph.end>in altitudine quacunque <emph type="italics"></emph>SF,<emph.end type="italics"></emph.end>ſumen<lb></lb>do aream <emph type="italics"></emph>thnz<emph.end type="italics"></emph.end>ad aream illam datam <emph type="italics"></emph>thkw<emph.end type="italics"></emph.end>ut eſt differentia <lb></lb><emph type="italics"></emph>Aa-Ff<emph.end type="italics"></emph.end>ad differentiam <emph type="italics"></emph>Aa-Cc.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Simili argumentatione probari poteſt, quod ſi gravitas particu<lb></lb>larum Fluidi diminuatur in triplicata ratione diſtantiarum a centro; <lb></lb>& quadratorum diſtantiarum <emph type="italics"></emph>SA, SB, SC,<emph.end type="italics"></emph.end>&c. </s> <s>reciproca (nem<lb></lb>pe <emph type="italics"></emph>(SAcub./SAq), (SAcub./SBq), (SAcub./SCq)<emph.end type="italics"></emph.end>) ſumantur in progreſſione Arithme<lb></lb>tica; denſitates <emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end>&c. </s> <s>erunt in progreſſione Geome<lb></lb>trica. </s> <s>Et ſi gravitas diminuatur in quadruplicata ratione diſtan<lb></lb>tiarum, & cuborum diſtantiarum reciproca (puta <emph type="italics"></emph>(SAqq/SAcub), (SAqq/SBcub), <lb></lb>(SAqq/SCcub.),<emph.end type="italics"></emph.end>&c.) ſumantur in progreſſione Arithmetica; denſitates <lb></lb><emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end>&c. </s> <s>erunt in progreſſione Geometrica. </s> <s>Et ſic in <lb></lb>infinitum. </s> <s>Rurſus. </s> <s>ſi gravitas particularum Fluidi in omnibus di<lb></lb>ſtantiis eadem ſit, & diſtantiæ ſint in progreſſione Arithmetica, <lb></lb>denſitates erunt in progreſſione Geometrica, uti Vir Cl. <emph type="italics"></emph>Edmundus <lb></lb>Hælleius<emph.end type="italics"></emph.end>invenit. </s> <s>Si gravitas ſit ut diſtantia, & quadrata diſtantia<lb></lb>rum ſint in progreſſione Arithmetica, denſitates erunt in progreſ<lb></lb>ſione Geometrica. </s> <s>Et ſic in infinitum. </s> <s>Hæc ita ſe habent ubi Fluidi <lb></lb>compreſſione condenſati denſitas eſt ut vis compreſſionis, vel, quod <lb></lb>perinde eſt, ſpatium a Fluido occupatum reciproce ut hæc vis. </s> <s><lb></lb>Fingi poſſunt aliæ condenſationis Leges, ut quod cubus vis com<lb></lb>primentis ſit ut quadrato-quadratum denſitatis, feu triplicata ra<lb></lb>tio Vis æqualis quadruplicatæ rationi denſitatis. </s> <s>Quo in caſu, ſi gra<lb></lb>vitas eſt reciproce ut quadratum diſtantiæ a centro, denſitas erit <lb></lb>reciproce ut cubus diſtantiæ. </s> <s>Fingatur quod cubus vis compri<lb></lb>mentis ſit ut quadrato-cubus denſitatis, & ſi gravitas eſt reciproce <lb></lb>ut quadratum diſtantiæ, denſitas erit reciproce in ſuſquiplicata ra-<pb xlink:href="039/01/298.jpg" pagenum="270"></pb><arrow.to.target n="note246"></arrow.to.target>tione diſtantiæ. </s> <s>Fingatur quod vis comprimens ſit in duplicata <lb></lb>ratione denſitatis, & gravitas reciproce in ratione duplicata diſtan<lb></lb>tiæ, & denſitas erit reciproce ut diſtantia. </s> <s>Caſus omnes percurre<lb></lb>re longum eſſet. </s></p> <p type="margin"> <s><margin.target id="note246"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIII. THEOREMA XVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Fluidi ex particulis ſe mutuo fugientibus compoſiti denſitas ſit <lb></lb>ut compreſſio, vires centrifugæ particularum ſunt reciproce pro<lb></lb>portionales diſtantiis centrorum ſuorum. </s> <s>Et vice verſa, par<lb></lb>ticulæ viribus quæ ſunt reciproce proportionales diſtantiis cen<lb></lb>trorum ſuorum ſe mutuo fugientes componunt Fluidum Elaſti<lb></lb>cum, cujus denſitas est compreſſioni proportionalis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Includi intelligatur Fluidum in ſpatio cubico <emph type="italics"></emph>ACE,<emph.end type="italics"></emph.end>dein com<lb></lb>preſſione redigi in ſpatium cubicum minus <emph type="italics"></emph>ace<emph.end type="italics"></emph.end>; & particularum, <lb></lb>ſimilem ſitum inter ſe in utro<lb></lb><figure id="id.039.01.298.1.jpg" xlink:href="039/01/298/1.jpg"></figure><lb></lb>que ſpatio obtinentium, diſtan<lb></lb>tiæ erunt ut cuborum latera <lb></lb><emph type="italics"></emph>AB, ab<emph.end type="italics"></emph.end>; & Medii denſitates <lb></lb>reciproce ut ſpatia continentia <lb></lb><emph type="italics"></emph>AB cub.<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ab cub.<emph.end type="italics"></emph.end>In latere <lb></lb>cubi majoris <emph type="italics"></emph>ABCD<emph.end type="italics"></emph.end>capiatur <lb></lb>quadratum <emph type="italics"></emph>DP<emph.end type="italics"></emph.end>æquale lateri <lb></lb>cubi minoris <emph type="italics"></emph>db<emph.end type="italics"></emph.end>; & ex Hypo<lb></lb>theſi, preſſio qua quadratum <emph type="italics"></emph>DP<emph.end type="italics"></emph.end>urget Fluidum incluſum, erit ad <lb></lb>preſſionem qua latus illud quadratum <emph type="italics"></emph>db<emph.end type="italics"></emph.end>urget Fluidum incluſum <lb></lb>ut Medii denſitates ad invicem, hoc eſt, ut <emph type="italics"></emph>ab cub.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ABcub.<emph.end type="italics"></emph.end>Sed <lb></lb>preſſio qua quadratum <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>urget Fluidum incluſum, eſt ad preſſi<lb></lb>onem qua quadratum <emph type="italics"></emph>DP<emph.end type="italics"></emph.end>urget idem Fluidum, ut quadratum <emph type="italics"></emph>DB<emph.end type="italics"></emph.end><lb></lb>ad quadratum <emph type="italics"></emph>DP,<emph.end type="italics"></emph.end>hoc eſt, ut <emph type="italics"></emph>AB quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ab quad.<emph.end type="italics"></emph.end>Ergo, ex <lb></lb>æquo, preſſio qua latus <emph type="italics"></emph>DB<emph.end type="italics"></emph.end>urget Fluidum, eſt ad preſſionem qua <lb></lb>latus <emph type="italics"></emph>db<emph.end type="italics"></emph.end>urget Fluidum, ut <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Planis <emph type="italics"></emph>FGH, fgh,<emph.end type="italics"></emph.end>per <lb></lb>media cuborum ductis, diſtinguatur Fluidum in duas partes, & hæ <lb></lb>ſe mutuo prement iiſdem viribus, quibus premuntur a planis <emph type="italics"></emph>AC, ac,<emph.end type="italics"></emph.end><lb></lb>hoc eſt, in proportione <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB:<emph.end type="italics"></emph.end>adeoque vires centrifugæ, qui<lb></lb>bus hæ preſſiones ſuſtinentur, ſunt in eadem ratione. </s> <s>Ob eundem <lb></lb>particularum numerum ſimilemque ſitum in utroque cubo, vires <lb></lb>quas particulæ omnes ſecundum plana <emph type="italics"></emph>FGH, fgh<emph.end type="italics"></emph.end>exercent in om-<pb xlink:href="039/01/299.jpg" pagenum="271"></pb>nes, ſunt ut vires quas ſingulæ exercent in ſingulas. </s> <s>Ergo vires, <lb></lb><arrow.to.target n="note247"></arrow.to.target>quas ſingulæ exercent in ſingulas ſecundum planum <emph type="italics"></emph>FGH<emph.end type="italics"></emph.end>in cubo <lb></lb>majore, ſunt ad vires quas ſingulæ exercent in ſingulas ſecundum <lb></lb>planum <emph type="italics"></emph>fgh<emph.end type="italics"></emph.end>in cubo minore ut <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>hoc eſt, reciproce ut <lb></lb>diſtantiæ particularum ad invicem. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note247"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Et vice verſa, ſi vires particularum ſingularum ſunt reciproce <lb></lb>ut diſtantiæ, id eſt, reciproce ut cuborum latera <emph type="italics"></emph>AB, ab<emph.end type="italics"></emph.end>; ſummæ <lb></lb>virium erunt in eadem ratione, & preſſiones laterum <emph type="italics"></emph>DB, db<emph.end type="italics"></emph.end>ut <lb></lb>ſummæ virium; & preſſio quadrati <emph type="italics"></emph>DP<emph.end type="italics"></emph.end>ad preſſionem lateris <emph type="italics"></emph>DB<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>ab quad.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB quad.<emph.end type="italics"></emph.end>Et, ex æquo, preſſio quadrati <emph type="italics"></emph>DP<emph.end type="italics"></emph.end>ad preſ<lb></lb>ſionem lateris <emph type="italics"></emph>db<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>ab cub.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB cub.<emph.end type="italics"></emph.end>id eſt, vis compreſſionis ad <lb></lb>vim compreſſionis ut denſitas ad denſitatem. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Simili argumento, ſi particularum vires centrifugæ ſint reciproce <lb></lb>in duplicata ratione diſtantiarum inter centra, cubi virium compri<lb></lb>mentium erunt ut quadrato-quadrata denſitarum. </s> <s>Si vires centri<lb></lb>fugæ ſint reciproce in triplicata vel quadruplicata ratione diſtantia<lb></lb>rum, cubi virium comprimentium erunt ut quadrato-cubi vel cubo<lb></lb>cubi denſitatum. </s> <s>Et univerſaliter, ſi D ponatur pro diſtantia, & <lb></lb>E pro denſitate Fluidi compreſſi, & vires centrifugæ ſint reciproce <lb></lb>ut diſtantiæ dignitas quælibet D<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end><emph.end type="sup"></emph.end>, cujus index eſt numerus <emph type="italics"></emph>n<emph.end type="italics"></emph.end>; vi<lb></lb>res comprimentes erunt ut latera cubica dignitatis E<emph type="sup"></emph><emph type="italics"></emph>n<emph.end type="italics"></emph.end>+2<emph.end type="sup"></emph.end>, cujus <lb></lb>index eſt numerus <emph type="italics"></emph>n<emph.end type="italics"></emph.end>+2: & contra. </s> <s>Intelligenda vero ſunt hæc <lb></lb>omnia de particularum Viribus centrifugis quæ terminantur in par<lb></lb>ticulis proximis, aut non longe ultra diffunduntur. </s> <s>Exemplum <lb></lb>habemus in corporibus Magneticis. </s> <s>Horum Virtus attractiva ter<lb></lb>minatur fere in ſui generis corporibus ſibi proximis. </s> <s>Magnetis <lb></lb>virtus per interpoſitam laminam ferri contrahitur, & in lamina fere <lb></lb>terminatur. </s> <s>Nam corpora ulteriora non tam a Magnete quam a <lb></lb>lamina trahuntur. </s> <s>Ad eundem modum ſi particulæ fugant alias ſui <lb></lb>generis particulas ſibi proximas, in particulas autem remotiores <lb></lb>virtutem nullam exerceant, ex hujuſmodi particulis componentur <lb></lb>Fluida de quibus actum eſt in hac Propoſitione. </s> <s>Quod ſi particulæ <lb></lb>cujuſque virtus in infinitum propagetur, opus erit vi majori ad æqua<lb></lb>lem condenſationem majoris quantitatis Fluidi. </s> <s>An vero Fluida <lb></lb>Elaſtica ex particulis ſe mutuo fugantibus conſtent, Quæſtio Phy<lb></lb>ſica eſt. </s> <s>Nos proprietatem Fluidorum ex ejuſmodi particulis con<lb></lb>ſtantium Mathematice demonſtravimus, ut Philoſophis anſam præ<lb></lb>beamus Quæſtionem illam tractandi. <pb xlink:href="039/01/300.jpg" pagenum="272"></pb><arrow.to.target n="note248"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note248"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Motu & Reſiſtentia Corporum Funependulorum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIV. THEOREMA XIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Quantitates materiæ in corporibus funependulis, quorum centra <lb></lb>oſcillationum a centro ſuſpenſionis æqualiter diſtant, ſunt in ra<lb></lb>tione compoſita ex ratione ponderum & ratione duplicata tem<lb></lb>porum oſcillationum in vacuo.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam velocitas, quam data vis in data materia dato tempore ge<lb></lb>nerare poteſt, eſt ut vis & tempus directe, & materia inverſe. </s> <s>Quo <lb></lb>major eſt vis vel majus tempus vel minor materia, eo major gene<lb></lb>rabitur velocitas. </s> <s>Id quod per motus Legem ſecundam manife<lb></lb>ſtum eſt. </s> <s>Jam vero ſi Pendula ejuſdem ſint longitudinis, vires mo<lb></lb>trices in locis a perpendiculo æqualiter diſtantibus ſunt ut ponde<lb></lb>ra: ideoque ſi corpora duo oſcillando deſcribant arcus æquales, & <lb></lb>arcus illi dividantur in partes æquales; cum tempora quibus cor<lb></lb>pora deſcribant ſingulas arcuum partes correſpondentes ſint ut <lb></lb>tempora oſcillationum totarum, erunt velocitates ad invicem in <lb></lb>correſpondentibus oſcillationum partibus, ut vires motrices & tota <lb></lb>oſcillationum tempora directe & quantitates materiæ reciproce: <lb></lb>adeoque quantitates materiæ ut vires & oſcillationum tempora di<lb></lb>recte & velocitates reciproce. </s> <s>Sed velocitates reciproce ſunt ut <lb></lb>tempora, atque adeo tempora directe & velocitates reciproce ſunt <lb></lb>ut quadrata temporum, & propterea quantitates materiæ ſunt ut <lb></lb>vires motrices & quadrata temporum, id eſt, ut pondera & quadra<lb></lb>ta temporum. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Ideoque ſi tempora ſunt æqualia, quantitates materiæ <lb></lb>in ſingulis corporibus erunt ut pondera. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si pondera ſunt æqualia, quantitates materiæ erunt ut <lb></lb>quadrata temporum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si quantitates materiæ æquantur, pondera erunt reci<lb></lb>proce ut quadrata temporum. </s></p><pb xlink:href="039/01/301.jpg" pagenum="273"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Unde cum quadrata temporum, cæteris paribus, ſint ut <lb></lb><arrow.to.target n="note249"></arrow.to.target>longitudines pendulorum; ſi & tempora & quantitates materiæ æ<lb></lb>qualia ſunt, pondera erunt ut longitudines pendulorum. </s></p> <p type="margin"> <s><margin.target id="note249"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et univerſaliter, quantitas materiæ pendulæ eſt ut pon<lb></lb>dus & quadratum temporis directe, & longitudo penduli inverſe. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Sed & in Medio non reſiſtente quantitas materiæ pen<lb></lb>dulæ eſt ut pondus comparativum & quadratum temporis directe <lb></lb>& longitudo penduli inverſe. </s> <s>Nam pondus comparativum eſt vis <lb></lb>motrix corporis in Medio quovis gravi, ut ſupra explicui; adeoque <lb></lb>idem præſtat in tali Medio non reſiſtente atque pondus abſolutum <lb></lb>in vacuo. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Et hinc liquet ratio tum comparandi corpora inter ſe, <lb></lb>quoad quantitatem materiæ in ſingulis; tum comparandi pondera <lb></lb>ejuſdem corporis in diverſis locis, ad cognoſcendam variationem <lb></lb>gravitatis. </s> <s>Factis autem experimentis quam accuratiſſimis inveni <lb></lb>ſemper quantitatem materiæ in corporibus ſingulis eorum ponderi <lb></lb>proportionalem eſſe. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXV. THEOREMA XX:<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corpora Funependula quibus, in Medio quovis, reſiſtitur in ratione <lb></lb>momentorum temporis, & corpora Funependula quæ in ejuſdem <lb></lb>gravitatis ſpecificæ Medio non reſiſtente moventur, oſcillatio<lb></lb>nes in Cycloide eodem tempore peragunt, & arcuum partes pro<lb></lb>portionales ſimul deſcribunt.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>Cycloidis <lb></lb><figure id="id.039.01.301.1.jpg" xlink:href="039/01/301/1.jpg"></figure><lb></lb>arcus, quem corpus <lb></lb><emph type="italics"></emph>D<emph.end type="italics"></emph.end>tempore quovis in <lb></lb>Medio non reſiſtente <lb></lb>oſcillando deſcribit. </s> <s><lb></lb>Biſecetur idem in <emph type="italics"></emph>C,<emph.end type="italics"></emph.end><lb></lb>ita ut <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ſit infimum <lb></lb>ejus punctum; & erit <lb></lb>vis acceleratrix qua <lb></lb>corpus urgetur in lo<lb></lb>co quovis <emph type="italics"></emph>D<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>d<emph.end type="italics"></emph.end>vel <lb></lb><emph type="italics"></emph>E<emph.end type="italics"></emph.end>ut longitudo arcus <lb></lb><emph type="italics"></emph>CD<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Cd<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>CE.<emph.end type="italics"></emph.end>Exponatur vis illa per eundem arcum; & <lb></lb>cum reſiſtentia ſit ut momentum temporis, adeoQ.E.D.tur, expona-<pb xlink:href="039/01/302.jpg" pagenum="274"></pb><arrow.to.target n="note250"></arrow.to.target>tur eadem per datam arcus Cycloidis partem <emph type="italics"></emph>CO,<emph.end type="italics"></emph.end>& ſumatur ar<lb></lb>cus <emph type="italics"></emph>Od<emph.end type="italics"></emph.end>in ratione ad arcum <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>quam habet arcus <emph type="italics"></emph>OB<emph.end type="italics"></emph.end>ad arcum <lb></lb><emph type="italics"></emph>CB:<emph.end type="italics"></emph.end>& vis qua corpus in <emph type="italics"></emph>d<emph.end type="italics"></emph.end>urgetur in Medio reſiſtente, cum ſit ex<lb></lb>ceſſus vis <emph type="italics"></emph>Cd<emph.end type="italics"></emph.end>ſupra reſiſtentiam <emph type="italics"></emph>CO,<emph.end type="italics"></emph.end>exponetur per arcum <emph type="italics"></emph>Od,<emph.end type="italics"></emph.end>ad<lb></lb>eoque erit ad vim qua corpus <emph type="italics"></emph>D<emph.end type="italics"></emph.end>urgetur in Medio non reſiſtente, <lb></lb>in loco <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ut arcus <emph type="italics"></emph>Od<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>; & propterea etiam in lo<lb></lb>co <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut arcus <emph type="italics"></emph>OB<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>CB.<emph.end type="italics"></emph.end>Proinde ſi corpora duo, <emph type="italics"></emph>D, d<emph.end type="italics"></emph.end><lb></lb>exeant de loco <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& his viribus urgeantur: cum vires ſub initio <lb></lb>ſint ut arcus <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>OB,<emph.end type="italics"></emph.end>erunt velocitates primæ & arcus primo <lb></lb>deſcripti in eadem ratione. </s> <s>Sunto arcus illi <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Bd,<emph.end type="italics"></emph.end>& arcus <lb></lb>reliqui <emph type="italics"></emph>CD, Od<emph.end type="italics"></emph.end>erunt in eadem ratione. </s> <s>Proinde vires, ipſis <lb></lb><emph type="italics"></emph>CD, Od<emph.end type="italics"></emph.end>proportionales, manebunt in eadem ratione ac ſub initio, <lb></lb>& propterea corpora pergent arcus in eadem ratione ſimul deſcri<lb></lb>bere. </s> <s>Igitur vires & <lb></lb><figure id="id.039.01.302.1.jpg" xlink:href="039/01/302/1.jpg"></figure><lb></lb>velocitates & arcus re<lb></lb>liqui <emph type="italics"></emph>CD, Od<emph.end type="italics"></emph.end>ſemper <lb></lb>erunt ut arcus toti <emph type="italics"></emph>CB, <lb></lb>OB,<emph.end type="italics"></emph.end>& propterea ar<lb></lb>cus illi reliqui ſimul <lb></lb>deſcribentur. </s> <s>Quare <lb></lb>corpora duo <emph type="italics"></emph>D, d<emph.end type="italics"></emph.end>ſi<lb></lb>mul pervenient ad loca <lb></lb><emph type="italics"></emph>C<emph.end type="italics"></emph.end>& <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>alterum qui<lb></lb>dem in Medio non re<lb></lb>ſiſtente ad locum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>& <lb></lb>alterum in Medio reſiſtente ad locum <emph type="italics"></emph>O.<emph.end type="italics"></emph.end>Cum autem velocitates in <lb></lb><emph type="italics"></emph>C<emph.end type="italics"></emph.end>& <emph type="italics"></emph>O<emph.end type="italics"></emph.end>ſint ut arcus <emph type="italics"></emph>CB, OB<emph.end type="italics"></emph.end>; erunt arcus quos corpora ulterius <lb></lb>pergendo ſimul deſcribunt, in eadem ratione. </s> <s>Sunto illi <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>Oe.<emph.end type="italics"></emph.end>Vis qua corpus <emph type="italics"></emph>D<emph.end type="italics"></emph.end>in Medio non reſiſtente retardatur in <emph type="italics"></emph>E<emph.end type="italics"></emph.end><lb></lb>eſt ut <emph type="italics"></emph>CE,<emph.end type="italics"></emph.end>& vis qua corpus <emph type="italics"></emph>d<emph.end type="italics"></emph.end>in Medio reſiſtente retardatur in <emph type="italics"></emph>e<emph.end type="italics"></emph.end><lb></lb>eſt ut ſumma vis <emph type="italics"></emph>Ce<emph.end type="italics"></emph.end>& reſiſtentiæ <emph type="italics"></emph>CO,<emph.end type="italics"></emph.end>id eſt ut <emph type="italics"></emph>Oe<emph.end type="italics"></emph.end>; ideoque vi<lb></lb>res, quibus corpora retardantur, ſunt ut arcubus <emph type="italics"></emph>CE, Oe<emph.end type="italics"></emph.end>propor<lb></lb>tionales arcus <emph type="italics"></emph>CB, OB<emph.end type="italics"></emph.end>; proindeque velocitates, in data illa ratio<lb></lb>ne retardatæ, manent in eadem illa data ratione. </s> <s>Velocitates igitur <lb></lb>& arcus iiſdem deſcripti ſemper ſunt ad invicem in data illa ratio<lb></lb>ne arcuum <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>OB<emph.end type="italics"></emph.end>; & propterea ſi ſumantur arcus toti <emph type="italics"></emph>AB, <lb></lb>aB<emph.end type="italics"></emph.end>in eadem ratione, corpora <emph type="italics"></emph>D, d<emph.end type="italics"></emph.end>ſimul deſcribent hos arcus, & <lb></lb>in locis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>a<emph.end type="italics"></emph.end>motum omnem ſimul amittent. </s> <s>Iſochronæ ſunt <lb></lb>igitur oſcillationes totæ, & arcubus totis <emph type="italics"></emph>BA, Ba<emph.end type="italics"></emph.end>proportionales <lb></lb>ſunt arcuum partes quælibet <emph type="italics"></emph>BD, Bd<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>BE, Be<emph.end type="italics"></emph.end>quæ ſimul de<lb></lb>ſcribuntur. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/303.jpg" pagenum="275"></pb> <p type="margin"> <s><margin.target id="note250"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Igitur motus velociſſimus in Medio reſiſtente non incidit <lb></lb><arrow.to.target n="note251"></arrow.to.target>in punctum infimum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>ſed reperitur in puncto illo <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>quo arcus <lb></lb>totus deſcriptus <emph type="italics"></emph>aB<emph.end type="italics"></emph.end>biſecatur. </s> <s>Et corpus ſubinde pergendo ad <emph type="italics"></emph>a,<emph.end type="italics"></emph.end><lb></lb>iiſdem gradibus retardatur quibus antea accelerabatur in deſcenſu <lb></lb>ſuo a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>O.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note251"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVI. THEOREMA XXI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corporum Funependulorum, quibus reſiſtitur in ratione velocitatum, <lb></lb>oſcillationes in Cycloide ſunt Iſochronæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi corpora duo, a centris ſuſpenſionum æqualiter diſtantia, <lb></lb>oſcillando deſcribant arcus inæquales, & velocitates in arcuum par<lb></lb>tibus correſpondentibus ſint ad invicem ut arcus toti: reſiſtentiæ <lb></lb>velocitatibus proportionales, erunt etiam ad invicem ut iidem ar<lb></lb>cus. </s> <s>Proinde ſi viribus motricibus a gravitate oriundis, quæ ſint <lb></lb>ut iidem arcus, auferantur vel addantur hæ reſiſtentiæ, erunt dif<lb></lb>ferentiæ vel ſummæ ad invicem in eadem arcuum ratione: cumque <lb></lb>velocitatum incrementa vel decrementa ſint ut hæ differentiæ vel <lb></lb>ſummæ, velocitates ſemper erunt ut arcus toti: Igitur velocitates, <lb></lb>ſi ſint in aliquo caſu ut arcus toti, manebunt ſemper in eadem ra<lb></lb>tione. </s> <s>Sed in principio motus, ubi corpora incipiunt deſcendere <lb></lb>& arcus illos deſcribere, vires, cum ſint arcubus proportionales, ge<lb></lb>nerabunt velocitates arcubus proportionales. </s> <s>Ergo velocitates ſem<lb></lb>per erunt ut arcus toti deſcribendi, & propterea arcus illi ſimul de<lb></lb>ſcribentur. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVII. THEOREMA XXII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Corporibus Funependulis reſiſtitur in duplicata ratione veloci<lb></lb>tatum, differentiæ inter tempora oſcillationum in Medio reſi<lb></lb>ſtente ac tempora oſcillationum in ejuſdem gravitatis ſpecificæ <lb></lb>Medio non reſiſtente, erunt arcubus oſcillando deſcriptis pro<lb></lb>portionales, quam proxime.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam pendulis æqualibus in Medio reſiſtente deſcribantur arcus <lb></lb>inæquales A, B; & reſiſtentia corporis in arcu A, erit ad reſiſten<lb></lb>tiam corporis in parte correſpondente arcus B, in duplicata ratio<lb></lb>ne velocitatum, id eſt, ut AA ad BB, quam proxime. </s> <s>Si reſi-<pb xlink:href="039/01/304.jpg" pagenum="276"></pb><arrow.to.target n="note252"></arrow.to.target>ſtentia in arcu B eſſet ad reſiſtentiam in arcu A ut AB ad AA; <lb></lb>tempora in arcubus A & B forent æqualia, per Propoſitionem ſu<lb></lb>periorem. </s> <s>Ideoque reſiſtentia AA in arcu A, vel AB in arcu B, <lb></lb>efficit exceſſum temporis in arcu A ſupra tempus in Medio non <lb></lb>reſiſtente; & reſiſtentia BB efficit exceſſum temporis in arcu B <lb></lb>ſupra tempus in Medio non reſiſtente. </s> <s>Sunt autem exceſſus illi <lb></lb>ut vires efficientes AB & BB quam proxime, id eſt, ut arcus <lb></lb>A & B. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note252"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ex oſcillationum temporibus, in Medio reſiſtente, <lb></lb>in arcubus inæqualibus factarum, cognoſci poſſunt tempora oſcilla<lb></lb>tionum in ejuſdem gravitatis ſpecificæ Medio non reſiſtente. </s> <s>Nam <lb></lb>differentia temporum erit ad exceſſum temporis in arcu minore ſu<lb></lb>pra tempus in Medio non reſiſtente, ut differentia arcuum ad ar<lb></lb>cum minorem. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Oſcillationes breviores ſunt magis Iſochronæ, & bre<lb></lb>viſſimæ iiſdem temporibus peraguntur ac in Medio non reſiſtente, <lb></lb>quam proxime. </s> <s>Earum vero quæ in majoribus arcubus fiunt, tem<lb></lb>ra ſunt paulo majora, propterea quod reſiſtentia in deſcenſu cor<lb></lb>poris qua tempus producitur, major ſit pro ratione longitudinis <lb></lb>in deſcenſu deſcriptæ, quam reſiſtentia in aſcenſu, ſubſequente qua <lb></lb>tempus contrahitur. </s> <s>Sed & tempus oſcillationum tam brevium <lb></lb>quam longarum nonnihil produci videtur per motum Medii. </s> <s>Nam <lb></lb>corporibus tardeſcentibus paulo minus reſiſtitur, pro ratione velo<lb></lb>citatis, & corporibus acceleratis paulo magis quam iis quæ unifor<lb></lb>miter progrediuntur: id adeo quia Medium, eo quem a corporibus <lb></lb>accepit motu, in eandem plagam pergendo, in priore caſu magis <lb></lb>agitatur, in poſteriore minus; ac proinde magis vel minus cum <lb></lb>corporibus motis conſpirat. </s> <s>Pendulis igitur in deſcenſu magis re<lb></lb>ſiſtit, in aſcenſu minus quam pro ratione velocitatis, & ex utraque <lb></lb>cauſa tempus producitur. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVIII. THEOREMA XXIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Corpori Funependulo in Cycloide oſcillanti reſiſtitur in ratione <lb></lb>momentorum temporis, erit ejus reſiſtentia ad vim gravitatis <lb></lb>ut exceſſus arcus deſcenſu toto deſcripti ſupra arcum aſcenſu <lb></lb>ſubſequente deſcriptum, ad penduli longitudinem duplicatam.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Deſignet <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>arcum deſcenſu deſcriptum, <emph type="italics"></emph>Ca<emph.end type="italics"></emph.end>arcum aſcenſu de<lb></lb>ſcriptum, & <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>differentiam arcuum: & ſtantibus quæ in Propo-<pb xlink:href="039/01/305.jpg" pagenum="277"></pb>ſitione XXV conſtructa & demonſtrata ſunt, erit vis qua corpus <lb></lb><arrow.to.target n="note253"></arrow.to.target>olcnlans urgetur in loco quovis <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ad vim reſiſtentiæ ut arcus <lb></lb><emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>CO,<emph.end type="italics"></emph.end>qui ſemiſſis eſt differentiæ illius <emph type="italics"></emph>Aa.<emph.end type="italics"></emph.end>Ideoque <lb></lb>vis qua corpus oſcillans urgetur in Cycloidis principio ſeu puncto <lb></lb>altiſſimo, id eſt, vis gravitatis, erit ad reſiſtentiam ut arcus Cy<lb></lb>cloidis inter punctum illud ſupremum & punctum infimum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ad <lb></lb>arcum <emph type="italics"></emph>CO<emph.end type="italics"></emph.end>; id eſt (ſi arcus duplicentur) ut Cycloidis totius arcus, <lb></lb>ſeu dupla penduli longitudo, ad arcum <emph type="italics"></emph>Aa. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note253"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIX. PROBLEMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Poſito quod Corpori in Cycloide oſcillanti reſiſtitur in duplicata ra<lb></lb>tione velocitatis: invenire reſiſtentiam in locis ſingulis.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>(Fig. </s> <s>Prop. </s> <s>XXV) arcus oſcillatione integra deſcriptus, <lb></lb>ſitque <emph type="italics"></emph>C<emph.end type="italics"></emph.end>infimum Cycloidis punctum, & <emph type="italics"></emph>CZ<emph.end type="italics"></emph.end>ſemiſſis arcus Cycloi<lb></lb>dis totius, longitudini Penduli æqualis; & quæratur reſiſtentia cor<lb></lb><figure id="id.039.01.305.1.jpg" xlink:href="039/01/305/1.jpg"></figure><lb></lb>poris in loco quovis <emph type="italics"></emph>D.<emph.end type="italics"></emph.end>Secetur recta infinita <emph type="italics"></emph>OQ<emph.end type="italics"></emph.end>in punctis <emph type="italics"></emph>O, <lb></lb>C, P, Q,<emph.end type="italics"></emph.end>ea lege, ut (ſi erigantur perpendicula <emph type="italics"></emph>OK, CT, PI, QE,<emph.end type="italics"></emph.end><lb></lb>centroque <emph type="italics"></emph>O<emph.end type="italics"></emph.end>& Aſymptotis <emph type="italics"></emph>OK, OQ<emph.end type="italics"></emph.end>deſcribatur Hyperbola <emph type="italics"></emph>TIGE<emph.end type="italics"></emph.end><lb></lb>ſecans perpendicula <emph type="italics"></emph>CT, PI, QE<emph.end type="italics"></emph.end>in <emph type="italics"></emph>T, I<emph.end type="italics"></emph.end>& <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>& per punctum <emph type="italics"></emph>I<emph.end type="italics"></emph.end><lb></lb>agatur <emph type="italics"></emph>KF<emph.end type="italics"></emph.end>parallela Aſymptoto <emph type="italics"></emph>OQ<emph.end type="italics"></emph.end>occurrens Aſymptoto <emph type="italics"></emph>OK<emph.end type="italics"></emph.end>in <lb></lb><emph type="italics"></emph>K,<emph.end type="italics"></emph.end>& perpendiculis <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>QE<emph.end type="italics"></emph.end>in <emph type="italics"></emph>L<emph.end type="italics"></emph.end>& <emph type="italics"></emph>F<emph.end type="italics"></emph.end>) fuerit area Hyperboliea <lb></lb><emph type="italics"></emph>PIEQ<emph.end type="italics"></emph.end>ad aream Hyperbolicam <emph type="italics"></emph>PITC<emph.end type="italics"></emph.end>ut arcus <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>deſcenſu cor<lb></lb>poris deſcriptus ad arcum <emph type="italics"></emph>Ca<emph.end type="italics"></emph.end>aſcenſu deſcriptum, & area <emph type="italics"></emph>IEF<emph.end type="italics"></emph.end>ad <pb xlink:href="039/01/306.jpg" pagenum="278"></pb><arrow.to.target n="note254"></arrow.to.target>aream <emph type="italics"></emph>ILT<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>OQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OC.<emph.end type="italics"></emph.end>Dein perpendiculo <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>abſcindatur <lb></lb>area Hyperbolica <emph type="italics"></emph>PINM<emph.end type="italics"></emph.end>quæ ſit ad aream Hyperbolicam <emph type="italics"></emph>PIEQ<emph.end type="italics"></emph.end><lb></lb>ut arcus <emph type="italics"></emph>CZ<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>deſcenſu deſcriptum. </s> <s>Et ſi perpendicu<lb></lb>lo <emph type="italics"></emph>RG<emph.end type="italics"></emph.end>abſcindatur area Hyperbolica <emph type="italics"></emph>PIGR,<emph.end type="italics"></emph.end>quæ ſit ad aream <lb></lb><emph type="italics"></emph>PIEQ<emph.end type="italics"></emph.end>ut arcus quilibet <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ad arcum <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>deſcenſu toto de<lb></lb>ſcriptum: erit reſiſtentia in loco <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ad vim gravitatis, ut area <lb></lb><emph type="italics"></emph>(OR/OQ)IEF-IGH<emph.end type="italics"></emph.end>ad aream <emph type="italics"></emph>PIENM.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note254"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Nam cum vires a gravitate oriundæ quibus corpus in locis <emph type="italics"></emph>Z, B, D, <lb></lb>a<emph.end type="italics"></emph.end>urgetur, ſint ut arcus <emph type="italics"></emph>CZ, CB, CD, Ca,<emph.end type="italics"></emph.end>& arcus illi ſint ut areæ <lb></lb><emph type="italics"></emph>PINM, PIEQ, PIGR, PITC<emph.end type="italics"></emph.end>; exponantur tum arcus tum vi<lb></lb>res per has areas reſpective. </s> <s>Sit inſuper <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>ſpatium quam minimum <lb></lb>a corpore deſcendente deſcriptum, & exponatur idem per aream <lb></lb>quam minimam <emph type="italics"></emph>RGgr<emph.end type="italics"></emph.end>parallelis <emph type="italics"></emph>RG, rg<emph.end type="italics"></emph.end>comprehenſam; & pro<lb></lb><figure id="id.039.01.306.1.jpg" xlink:href="039/01/306/1.jpg"></figure><lb></lb>ducatur <emph type="italics"></emph>rg<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>h,<emph.end type="italics"></emph.end>ut ſint <emph type="italics"></emph>GHhg,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>RGgr<emph.end type="italics"></emph.end>contemporanea arearum <lb></lb><emph type="italics"></emph>IGH, PIGR<emph.end type="italics"></emph.end>decrementa. </s> <s>Et areæ <emph type="italics"></emph>(OR/OQ)IEF-IGH<emph.end type="italics"></emph.end>incremen<lb></lb>tum <emph type="italics"></emph>GHhg-(Rr/OQ)IEF,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>RrXHG-(Rr/OQ)IEF,<emph.end type="italics"></emph.end>erit ad areæ <lb></lb><emph type="italics"></emph>PIGR<emph.end type="italics"></emph.end>decrementum <emph type="italics"></emph>RGgr<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>RrXRG,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>HG-(IEF/OQ)<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>RG<emph.end type="italics"></emph.end>; adeoque ut <emph type="italics"></emph>ORXHG-(OR/OQ)IEF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ORXGR<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>OPXPI,<emph.end type="italics"></emph.end>hoc eſt (ob æqualia <emph type="italics"></emph>ORXHG, ORXHR-ORXGR, <lb></lb>ORHK-OPIK, PIHR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FIGR+IGH<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>PIGR+ <lb></lb>IGH-(OR/OQ)IEF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OPIK.<emph.end type="italics"></emph.end>Igitur ſi area <emph type="italics"></emph>(OR/OQ)IEF-IGH<emph.end type="italics"></emph.end><pb xlink:href="039/01/307.jpg" pagenum="279"></pb>dicatur Y, atque areæ <emph type="italics"></emph>PIGR<emph.end type="italics"></emph.end>decrementum <emph type="italics"></emph>RGgr<emph.end type="italics"></emph.end>detur, erit <lb></lb><arrow.to.target n="note255"></arrow.to.target>incrementum areæ Y ut <emph type="italics"></emph>PIGR<emph.end type="italics"></emph.end>-Y. </s></p> <p type="margin"> <s><margin.target id="note255"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Quod ſi V deſignet vim a gravitate oriundam, arcui deſcribendo <lb></lb><emph type="italics"></emph>CD<emph.end type="italics"></emph.end>proportionalem, qua corpus urgetur in <emph type="italics"></emph>D:<emph.end type="italics"></emph.end>& R pro reſiſten<lb></lb>tia ponatur: erit V-R vis tota qua corpus urgetur in <emph type="italics"></emph>D.<emph.end type="italics"></emph.end>Eſt <lb></lb>itaQ.E.I.crementum velocitatis ut V-R & particula illa temporis <lb></lb>in qua factum eſt conjunctim: Sed & velocitas ipſa eſt ut incre<lb></lb>mentum contemporaneum ſpatii deſcripti directe & particula ea<lb></lb>dem temporis inverſe. </s> <s>Unde, cum reſiſtentia (per Hypotheſin) <lb></lb>ſit ut quadratum velocitatis, incrementum reſiſtentiæ (per Lem. </s> <s>II) <lb></lb>erit ut velocitas & incrementum velocitatis conjunctim, id eſt, ut <lb></lb>momentum ſpatii & V-R conjunctim; atque adeo, ſi momen<lb></lb>tum ſpatii detur, ut V-R; id eſt, ſi pro vi V ſeribatur ejus ex<lb></lb>ponens <emph type="italics"></emph>PIGR,<emph.end type="italics"></emph.end>& reſiſtentia R exponatur per aliam aliquam are<lb></lb>am Z, ut <emph type="italics"></emph>PIGR<emph.end type="italics"></emph.end>-Z. </s></p> <p type="main"> <s>Igitur area <emph type="italics"></emph>PIGR<emph.end type="italics"></emph.end>per datorum momentorum ſubductionem <lb></lb>uniformiter decreſcente, creſcunt area Y in ratione <emph type="italics"></emph>PIGR<emph.end type="italics"></emph.end>-Y, <lb></lb>& area Z in ratione <emph type="italics"></emph>PIGR<emph.end type="italics"></emph.end>-Z. </s> <s>Et propterea ſi areæ Y & Z ſi<lb></lb>mul incipiant & ſub initio æquales ſint, hæ per additionem æqua<lb></lb>lium momentorum pergent eſſe æquales, & æqualibus itidem mo<lb></lb>mentis ſubinde decreſcentes ſimul evaneſcent. </s> <s>Et viciſſim, ſi ſimul <lb></lb>incipiunt & ſimul evaneſcunt, æqualia habebunt momenta & ſem<lb></lb>per erunt æquales: id adeo quia ſi reſiſtentia Z augeatur, veloci<lb></lb>tas una cum arcu illo <emph type="italics"></emph>Ca,<emph.end type="italics"></emph.end>qui in aſcenſu corporis deſcribitur, dimi<lb></lb>nuetur; & puncto in quo motus omnis una cum reſiſtentia ceſſat <lb></lb>propius accedente ad punctum <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>reſiſtentia citius evaneſcet quam <lb></lb>area Y. </s> <s>Et contrarium eveniet ubi reſiſtentia diminuitur. </s></p> <p type="main"> <s>Jam vero area Z incipit deſinitque ubi reſiſtentia nulla eſt, hoc <lb></lb>eſt, in principio & fine motus, ubi arcus <emph type="italics"></emph>CD, CD<emph.end type="italics"></emph.end>arcubus <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>Ca<emph.end type="italics"></emph.end>æquantur, adeoque ubi recta <emph type="italics"></emph>RG<emph.end type="italics"></emph.end>incidit in rectas <emph type="italics"></emph>QE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CT.<emph.end type="italics"></emph.end><lb></lb>Et area Y ſeu <emph type="italics"></emph>(OR/OQ)IEF-IGH<emph.end type="italics"></emph.end>incipit deſinitque ubi nulla eſt, ad<lb></lb>eoque ubi <emph type="italics"></emph>(OR/OQ)IEF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IGH<emph.end type="italics"></emph.end>æqualia ſunt: hoc eſt (per con<lb></lb>ſtructionem) ubi recta <emph type="italics"></emph>RG<emph.end type="italics"></emph.end>incidit in rectas <emph type="italics"></emph>QE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CT.<emph.end type="italics"></emph.end>Proin<lb></lb>deque areæ illæ ſimul incipiunt & ſimul evaneſcunt, & propterea <lb></lb>ſemper ſunt æquales. </s> <s>Igitur area <emph type="italics"></emph>(OR/OQ)IEF-IGH<emph.end type="italics"></emph.end>æqualis eſt <lb></lb>areæ Z, per quam reſiſtentia exponitur, & propterea eſt ad aream <lb></lb><emph type="italics"></emph>PINM<emph.end type="italics"></emph.end>per quam gravitas exponitur, ut reſiſtentia ad gravita<lb></lb>tem. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/308.jpg" pagenum="280"></pb><arrow.to.target n="note256"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note256"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Eſt igitur reſiſtentia in loco infimo <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ad vim gravitatis, <lb></lb>ut area <emph type="italics"></emph>(OP/OQ) IEF<emph.end type="italics"></emph.end>ad aream <emph type="italics"></emph>PINM.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Fit autem maxima, ubi area <emph type="italics"></emph>PIHR<emph.end type="italics"></emph.end>eſt ad aream <lb></lb><emph type="italics"></emph>IEF<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>OR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="Oq.">Oque</expan><emph.end type="italics"></emph.end>Eo enim in caſu momentum ejus (nimirum <lb></lb><emph type="italics"></emph>PIGR<emph.end type="italics"></emph.end>-Y) evadit nullum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Hinc etiam innoteſcit velocitas in locis ſingulis: quippe <lb></lb>quæ eſt in ſubduplicata ratione reſiſtentiæ, & ipſo motus initio æ<lb></lb>quatur velocitati corporis in eadem Cycloide abſque omni reſiſten<lb></lb>tia oſcillantis. </s></p> <p type="main"> <s>Cæterum ob difficilem calculum quo reſiſtentia & velocitas per <lb></lb>hanc Propoſitionem inveniendæ ſunt, viſum eſt Propoſitionem ſe<lb></lb>quentem ſubjungere, quæ & generalior ſit & ad uſus Philoſophi<lb></lb>cos abunde ſatis accurata. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXX. THEOREMA XXIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si recta<emph.end type="italics"></emph.end>aB <emph type="italics"></emph>æqualis ſit Cycloidis arcui quem corpus oſcillando de<lb></lb>ſcribit, & ad ſingula ejus puncta<emph.end type="italics"></emph.end>D <emph type="italics"></emph>erigantur perpendicula<emph.end type="italics"></emph.end>DK, <lb></lb><emph type="italics"></emph>quæ ſint ad longitudinem Penduli ut reſiſtentia corporis in ar<lb></lb>cus punctis correſpondentibus ad vim gravitatis: dico quod <lb></lb>differentia inter arcum deſcenſu toto deſcriptum, & arcum <lb></lb>aſcenſu toto ſubſequente deſcriptum, ducta in arcuum eorundem <lb></lb>ſemiſummam, æqualis erit areæ<emph.end type="italics"></emph.end>BKaB <emph type="italics"></emph>a perpendiculis omnibus<emph.end type="italics"></emph.end><lb></lb>DK <emph type="italics"></emph>occupatæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Exponatur enim tum Cycloidis arcus, oſcillatione integra de<lb></lb>ſcriptus, per rectam illam ſibi æqualem <emph type="italics"></emph>aB,<emph.end type="italics"></emph.end>tum arcus qui deſcribe<lb></lb>retur in vacuo per longitudinem <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Biſecetur <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>& pun<lb></lb>ctum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>repræſentabit infimum Cycloidis punctum, & erit <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ut <lb></lb>vis a gravitate oriunda, qua corpus in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ſecundum tangentem <lb></lb>Cycloidis urgetur, eamque habebit rationem ad longitudinem Pen<lb></lb>duli quam habet vis in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ad vim gravitatis. </s> <s>Exponatur igitur vis <lb></lb>illa per longitudinem <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end>& vis gravitatis per longitudinem pen<lb></lb>duli, & ſi in <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>in ea ratione ad longitudinem <pb xlink:href="039/01/309.jpg" pagenum="281"></pb>penduli quam habet reſiſtentia ad gravitatem, erit <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>exponens </s></p> <p type="main"> <s><arrow.to.target n="note257"></arrow.to.target>reſiſtentiæ. </s> <s>Centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& intervallo <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>conſtruatur Semi<lb></lb>circulus <emph type="italics"></emph>BEeA.<emph.end type="italics"></emph.end>Deſcribat autem corpus tempore quam minimo <lb></lb>ſpatium <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>& erectis perpendiculis <emph type="italics"></emph>DE, de<emph.end type="italics"></emph.end>circumferentiæ oc<lb></lb>currentibus in <emph type="italics"></emph>E<emph.end type="italics"></emph.end>& <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>erunt hæc ut velocitates quas corpus in va<lb></lb>cuo, deſcendendo a puncto <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>acquireret in locis <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>d.<emph.end type="italics"></emph.end>Patet <lb></lb>hoc per Prop. </s> <s>LII. Lib. </s> <s>1. Exponantur itaque hæ velocitates per <lb></lb>perpendicula illa <emph type="italics"></emph>DE, de<emph.end type="italics"></emph.end>; ſitque <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>velocitas quam acquirit <lb></lb>in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>cadendo de <emph type="italics"></emph>B<emph.end type="italics"></emph.end>in Medio reſiſtente. </s> <s>Et ſi centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& inter<lb></lb>vallo <emph type="italics"></emph>CF<emph.end type="italics"></emph.end>deſcribatur Circulus <emph type="italics"></emph>FfM<emph.end type="italics"></emph.end>occurrens rectis <emph type="italics"></emph>de<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in <lb></lb><emph type="italics"></emph>f<emph.end type="italics"></emph.end>& <emph type="italics"></emph>M,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>M<emph.end type="italics"></emph.end>locus ad quem deinceps abſque ulteriore reſiſten<lb></lb>tia aſcenderet, & <emph type="italics"></emph>df<emph.end type="italics"></emph.end>velocitas quam acquireret in <emph type="italics"></emph>d.<emph.end type="italics"></emph.end>Unde etiam <lb></lb>ſi <emph type="italics"></emph>Fg<emph.end type="italics"></emph.end>deſignet velocitatis momentum quod corpus <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>deſcribendo <lb></lb>ſpatium quam minimum <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>ex reſiſtentia Medii amittit; & ſu<lb></lb>matur <emph type="italics"></emph>CN<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>Cg:<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>N<emph.end type="italics"></emph.end>locus ad quem corpus deinceps <lb></lb>abſque ulteriore reſiſtentia aſcenderet, & <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>erit decrementum <lb></lb>aſcenſus ex velocitatis illius amiſſione oriundum. </s> <s>Ad <emph type="italics"></emph>df<emph.end type="italics"></emph.end>demitta<lb></lb>tur perpendiculum <emph type="italics"></emph>Fm,<emph.end type="italics"></emph.end>& velocitatis <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>decrementum <emph type="italics"></emph>Fg<emph.end type="italics"></emph.end>a <lb></lb>reſiſtentia <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>genitum, erit ad velocitatis ejuſdem incrementum <lb></lb><emph type="italics"></emph>fm<emph.end type="italics"></emph.end>a vi <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>genitum, ut vis generans <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>ad vim generantem <lb></lb><emph type="italics"></emph>CD.<emph.end type="italics"></emph.end>Sed & ob ſimilia <lb></lb><figure id="id.039.01.309.1.jpg" xlink:href="039/01/309/1.jpg"></figure><lb></lb>triangula <emph type="italics"></emph>Fmf, Fhg, <lb></lb>FDC,<emph.end type="italics"></emph.end>eſt <emph type="italics"></emph>fm<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Fm<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>Dd,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>DF<emph.end type="italics"></emph.end>; & ex æquo <emph type="italics"></emph>Fg<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DF.<emph.end type="italics"></emph.end><lb></lb>Item <emph type="italics"></emph>Fh<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Fg<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DF<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>CF<emph.end type="italics"></emph.end>; & ex æquo <lb></lb>perturbate, <emph type="italics"></emph>Fh<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>MN<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CF<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>CM<emph.end type="italics"></emph.end>; ideoque ſumma omnium <emph type="italics"></emph>MNXCM<emph.end type="italics"></emph.end>æqualis erit ſummæ <lb></lb>omnium <emph type="italics"></emph>DdXDK.<emph.end type="italics"></emph.end>Ad punctum mobile <emph type="italics"></emph>M<emph.end type="italics"></emph.end>erigi ſemper intelli<lb></lb>gatur ordinata rectangula æqualis indeterminatæ <emph type="italics"></emph>CM,<emph.end type="italics"></emph.end>quæ motu <lb></lb>continuo ducatur in totam longitudinem <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>; & trapezium ex illo <lb></lb>motu deſcriptum ſive huic æquale rectangulum <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>X1/2<emph type="italics"></emph>aB<emph.end type="italics"></emph.end>æquabitur <lb></lb>ſummæ omnium <emph type="italics"></emph>MNXCM,<emph.end type="italics"></emph.end>adeoque ſummæ omnium <emph type="italics"></emph>DdXDK,<emph.end type="italics"></emph.end><lb></lb>id eſt, areæ <emph type="italics"></emph>BKkVTa. </s> <s>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note257"></margin.target>LIBER <lb></lb>SECUNDUS</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc ex lege reſiſtentiæ & arcuum <emph type="italics"></emph>Ca, CB<emph.end type="italics"></emph.end>differentia <emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end><lb></lb>colligi poteſt proportio reſiſtentiæ ad gravitatem quam proxime. <pb xlink:href="039/01/310.jpg" pagenum="282"></pb><arrow.to.target n="note258"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note258"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Nam ſi uniformis ſit reſiſtentia <emph type="italics"></emph>DK,<emph.end type="italics"></emph.end>Figura <emph type="italics"></emph>aBKkT<emph.end type="italics"></emph.end>rectangu<lb></lb>lum erit ſub <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>; & inde rectangulum ſub 1/2 <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end><lb></lb>erit æquale rectangulo ſub <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DK,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>æqualis erit 1/2 <emph type="italics"></emph>Aa.<emph.end type="italics"></emph.end><lb></lb>Quare cum <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>ſit exponens reſiſtentiæ, & longitudo penduli ex<lb></lb>ponens gravitatis, erit reſiſtentia ad gravitatem ut 1/2 <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ad longi<lb></lb>tudinem Penduli; omnino ut in Prop. </s> <s>XXVIII demonſtratum eſt. </s></p> <p type="main"> <s>Si reſiſtentia ſit ut velocitas, Figura <emph type="italics"></emph>aBKkT<emph.end type="italics"></emph.end>Ellipſis erit quam <lb></lb>proxime. </s> <s>Nam ſi corpus, in Medio non reſiſtente, oſcillatione <lb></lb>integra deſcriberet longitudinem <emph type="italics"></emph>BA,<emph.end type="italics"></emph.end>velocitas in loco quovis <emph type="italics"></emph>D<emph.end type="italics"></emph.end><lb></lb>foret ut Circuli diametro <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>deſcripti ordinatim applicata <emph type="italics"></emph>DE.<emph.end type="italics"></emph.end><lb></lb>Proinde cum <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>in Medio reſiſtente, & <emph type="italics"></emph>BA<emph.end type="italics"></emph.end>in Medio non reſi<lb></lb>ſtente, æqualibus circiter temporibus deſcribantur; adeoque velo<lb></lb>citates in ſingulis ipſius <lb></lb><figure id="id.039.01.310.1.jpg" xlink:href="039/01/310/1.jpg"></figure><lb></lb><emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>punctis, ſint quam <lb></lb>proxime ad velocitates <lb></lb>in punctis correſpon<lb></lb>dentibus longitudinis <lb></lb><emph type="italics"></emph>BA,<emph.end type="italics"></emph.end>ut eſt <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BA<emph.end type="italics"></emph.end>; <lb></lb>erit velocitas <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>in <lb></lb>Medio reſiſtente ut Cir<lb></lb>culi vel Ellipſeos ſuper <lb></lb>diametro <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>deſcripti <lb></lb>ordinatim applicata; adeoque Figura <emph type="italics"></emph>BKVTa<emph.end type="italics"></emph.end>Ellipſis, quam pro<lb></lb>xime. </s> <s>Cum reſiſtentia velocitati proportionalis ſupponatur, ſit <emph type="italics"></emph>OV<emph.end type="italics"></emph.end><lb></lb>exponens reſiſtentiæ in puncto Medio <emph type="italics"></emph>O<emph.end type="italics"></emph.end>; & Ellipſis <emph type="italics"></emph>aBRVS,<emph.end type="italics"></emph.end><lb></lb>centro <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>ſemiaxibus <emph type="italics"></emph>OB, OV<emph.end type="italics"></emph.end>deſcripta, Figuram <emph type="italics"></emph>aBKVT,<emph.end type="italics"></emph.end><lb></lb>eique æquale rectangulum <emph type="italics"></emph>AaXBO,<emph.end type="italics"></emph.end>æquabit quamproxime. </s> <s>Eſt <lb></lb>igitur <emph type="italics"></emph>AaXBO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OVXBO<emph.end type="italics"></emph.end>ut area Ellipſeos hujus ad <emph type="italics"></emph>OVXBO<emph.end type="italics"></emph.end>: <lb></lb>id eſt, <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OV<emph.end type="italics"></emph.end>ut area ſemicirculi ad quadratum radii, ſive ut <lb></lb>11 ad 7 circiter: Et propterea (1/11) <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ad longitudinem penduli ut <lb></lb>corporis oſcillantis reſiſtentia in <emph type="italics"></emph>O<emph.end type="italics"></emph.end>ad ejuſdem gravitatem. </s></p> <p type="main"> <s>Quod ſi reſiſtentia <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>ſit in duplicata ratione velocitatis, Fi<lb></lb>gura <emph type="italics"></emph>BKVTa<emph.end type="italics"></emph.end>Parabola erit verticem habens <emph type="italics"></emph>V<emph.end type="italics"></emph.end>& axem <emph type="italics"></emph>OV,<emph.end type="italics"></emph.end>id<lb></lb>eoque æqualis erit rectangulo ſub 2/3 <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>& <emph type="italics"></emph>OV<emph.end type="italics"></emph.end>quam proxime. </s> <s>Eſt <lb></lb>igitur rectangulum ſub 1/2 <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>æquale rectangulo ſub 2/3 <emph type="italics"></emph>Ba<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>OV,<emph.end type="italics"></emph.end>adeoque <emph type="italics"></emph>OV<emph.end type="italics"></emph.end>æqualis 1/4 <emph type="italics"></emph>Aa:<emph.end type="italics"></emph.end>& propterea corporis oſcillan<lb></lb>tis reſiſtentia in <emph type="italics"></emph>O<emph.end type="italics"></emph.end>ad ipſius gravitatem ut 1/4 <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ad longitudi<lb></lb>nem Penduli. </s></p> <p type="main"> <s>Atque has concluſiones in rebus practicis abunde ſatis accuratas <lb></lb>eſſe cenſeo. </s> <s>Nam cum Ellipſis vel Parabola <emph type="italics"></emph>BRVSa<emph.end type="italics"></emph.end>congruat <pb xlink:href="039/01/311.jpg" pagenum="283"></pb>cum Figura <emph type="italics"></emph>BKVTa<emph.end type="italics"></emph.end>in puncto medio <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>hæc ſi ad partem al</s></p> <p type="main"> <s><arrow.to.target n="note259"></arrow.to.target>terutram <emph type="italics"></emph>BRV<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>VSa<emph.end type="italics"></emph.end>excedit Figuram illam, deficiet ab eadem <lb></lb>ad partem alteram, & ſic eidem æquabitur quam proxime. </s></p> <p type="margin"> <s><margin.target id="note259"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXI. THEOREMA XXV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Corporis oſcillantis reſiſtentia in ſingulis arcuum deſcriptorum <lb></lb>partibus proportionalibus augeatur vel minuatur in data ratio<lb></lb>ne; differentia inter arcum deſcenſu deſcriptum & arcum ſub<lb></lb>ſequente aſcenſu deſcriptum, augebitur vel diminuetur in eadem <lb></lb>ratione.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Oritur enim differentia illa ex retardatione Penduli per reſi<lb></lb>ſtentiam Medii, adeoque eſt ut retardatio tota eique proportio<lb></lb>nalis reſiſtentia retardans. </s> <s>In ſuperiore Propoſitione rectangu<lb></lb>lum ſub recta 1/2 <emph type="italics"></emph>aB<emph.end type="italics"></emph.end>& arcuum illorum <emph type="italics"></emph>CB, Ca<emph.end type="italics"></emph.end>differentia <emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end><lb></lb>æqualis erat areæ <emph type="italics"></emph>BKT.<emph.end type="italics"></emph.end>Et area illa, ſi maneat longitudo <emph type="italics"></emph>aB,<emph.end type="italics"></emph.end><lb></lb>augetur vel diminuitur in ratione ordinatim applicatarum <emph type="italics"></emph>DK<emph.end type="italics"></emph.end>; <lb></lb>hoc eſt, in ratione reſiſtentiæ, adeoque eſt ut longitudo <emph type="italics"></emph>aB<emph.end type="italics"></emph.end>& <lb></lb>reſiſtentia conjunctim. </s> <s>Proindeque rectangulum ſub <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>& 1/2 <emph type="italics"></emph>aB<emph.end type="italics"></emph.end><lb></lb>eſt ut <emph type="italics"></emph>aB<emph.end type="italics"></emph.end>& reſiſtentia conjunctim, & propterea <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>ut reſiſten<lb></lb>tia. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Unde ſi reſiſtentia ſit ut velocitas, differentia arcuum <lb></lb>in eodem Medio erit ut arcus totus deſcriptus: & contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si reſiſtentia ſit in duplicata ratione velocitatis, diffe<lb></lb>rentia illa erit in duplicata ratione arcus totius: & contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et univerſaliter, ſi reſiſtentia ſit in triplicata vel alia <lb></lb>quavis ratione velocitatis, differentia erit in eadem ratione arcus <lb></lb>totius: & contra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Et ſi reſiſtentia ſit partim in ratione ſimplici velocita<lb></lb>tis, partim in ejuſdem ratione duplicata, differentia erit partim in <lb></lb>ratione arcus totius & partim in ejus ratione duplicata: & contra. </s> <s><lb></lb>Eadem erit lex & ratio reſiſtentiæ pro velocitate, quæ eſt differen<lb></lb>tiæ illius pro longitudine arcus. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Ideoque ſi, pendulo inæquales arcus ſucceſſive deſcri<lb></lb>bente, inveniri poteſt ratio incrementi ac decrementi differentiæ hu<lb></lb>jus pro longitudine arcus deſcripti; habebitur etiam ratio incrementi <lb></lb>ac decrementi reſiſtentiæ pro velocitate majore vel minore. <pb xlink:href="039/01/312.jpg" pagenum="284"></pb><arrow.to.target n="note260"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note260"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium Generale.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ex his Propoſitionibus, per oſcillationes Pendulorum in Mediis <lb></lb>quibuſcunque, invenire licet reſiſtentiam Mediorum. </s> <s>Aeris vero <lb></lb>reſiſtentiam inveſtigavi per Experimenta ſequentia. </s> <s>Globum lig<lb></lb>neum pondere unciarum <emph type="italics"></emph>Romanarum<emph.end type="italics"></emph.end>(57 7/22), diametro digitorum <lb></lb><emph type="italics"></emph>Londinenſium<emph.end type="italics"></emph.end>6 7/8 fabricatum, filo tenui ab unco ſatis firmo ſuſpen<lb></lb>di, ita ut inter uncum & centrum oſcillationis Globi diſtantia eſſet <lb></lb>pedum 10 1/2. In filo punctum notavi pedibus decem & uncia una <lb></lb>a centro ſuſpenſionis diſtans; & e regione puncti illius collocavi <lb></lb>Regulam in digitos diſtinctam, quorum ope notarem longitudi<lb></lb>nes arcuum a Pendulo deſcriptas. </s> <s>Deinde numeravi oſcillationes <lb></lb>quibus Globus octavam motus ſui partem amitteret. </s> <s>Si pendu<lb></lb>lum deducebatur a perpendiculo ad diſtantiam duorum digitorum, <lb></lb>& inde demittebatur; ita ut toto ſuo deſcenſu deſcriberet arcum <lb></lb>duorum digitorum, totaque oſcillatione prima, ex deſcenſu & aſcen<lb></lb>ſu ſubſequente compoſita, arcum digitorum fere quatuor: idem <lb></lb>oſcillationibus 164 amiſit octavam motus ſui partem, ſic ut ultimo <lb></lb>ſuo aſcenſu deſcriberet arcum digiti unius cum tribus partibus <lb></lb>quartis digiti. </s> <s>Si primo deſcenſu deſcripſit arcum digitorum qua<lb></lb>tuor; amiſit octavam motus partem oſcillationibus 121, ita ut aſcen<lb></lb>ſu ultimo deſcriberet arcum digitorum 3 1/2. Si primo deſcenſu de<lb></lb>ſcripſit arcum digitorum octo, ſexdecim, triginta duorum vel ſexa<lb></lb>ginta quatuor; amiſit octavam motus partem oſcillationibus 69, 35 1/2, <lb></lb>18 1/2, 9 2/3, reſpective. </s> <s>Igitur differentia inter arcus deſcenſu primo <lb></lb>& aſcenſu ultimo deſcriptos, erat in caſu primo, ſecundo, tertio, <lb></lb>quarto, quinto, ſexto, digitorum 1/4, 1/2, 1, 2, 4, 8 reſpective. </s> <s>Divi<lb></lb>dantur eæ differentiæ per numerum oſcillationum in caſu unoquo<lb></lb>que, & in oſcillatione una mediocri, qua arcus digitorum 3 1/4, 7 1/2, <lb></lb>15, 30, 60, 120 deſcriptus fuit, differentia arcuum deſcenſu & ſub<lb></lb>ſequente aſcenſu deſcriptorum, erit (1/656), (1/242), (1/69), (4/71), (8/37), (24/29) partes di<lb></lb>giti reſpective. </s> <s>Hæ autem in majoribus oſcillationibus ſunt in du<lb></lb>plicata ratione arcuum deſcriptorum quam proxime, in minoribus <lb></lb>vero paulo majores quam in ea ratione; & propterea (per Corol. </s> <s>2. <lb></lb>Prop. </s> <s>XXXI Libri hujus) reſiſtentia Globi, ubi celerius movetur, <lb></lb>eſt in duplicata ratione velocitatis quam proxime; ubi tardius, pau<lb></lb>lo major quam in ea ratione. </s></p><pb xlink:href="039/01/313.jpg" pagenum="285"></pb> <p type="main"> <s>Deſignet jam V velocitatem maximam in oſcillatione quavis, <lb></lb><arrow.to.target n="note261"></arrow.to.target>ſintque A, B, C quantitates datæ, & fingamus quod differentia <lb></lb>arcuum ſit AV+BV 1/2+CV<emph type="sup"></emph>2<emph.end type="sup"></emph.end>. </s> <s>Cum velocitates maximæ ſint in <lb></lb>Cycloide ut ſemiſſes arcuum oſcillando deſcriptorum, in Circu<lb></lb>lo vero ut ſemiſſium arcuum illorum chordæ; adeoque paribus <lb></lb>arcubus majores ſint in Cycloide quam in Circulo, in ratione <lb></lb>ſemiſſium arcuum ad eorundem chordas; tempora autem in Cir<lb></lb>culo ſint majora quam in Cycloide in velocitatis ratione reci<lb></lb>proca; patet arcuum differentias (quæ ſunt ut reſiſtentia & qua<lb></lb>dratum temporis conjunctim) eaſdem fore, quamproxime, in utra<lb></lb>que Curva: deberent enim differentiæ illæ in Cycloide augeri, una <lb></lb>cum reſiſtentia, in duplicata circiter ratione arcus ad chordam, ob <lb></lb>velocitatem in ratione illa ſimplici auctam; & diminui, una cum <lb></lb>quadrato temporis, in eadem duplicata ratione. </s> <s>Itaque ut reductio <lb></lb>fiat ad Cycloidem, eædem ſumendæ ſunt arcuum differentiæ quæ <lb></lb>fuerunt in Circulo obſervatæ, velocitates vero maximæ ponen<lb></lb>dæ ſunt arcubus dimidiatis vel integris, hoc eſt, numeris 1/2, 1, 2, <lb></lb>4, 8, 16 analogæ. </s> <s>Scribamus ergo in caſu ſecundo, quarto & ſex<lb></lb>to numeros 1, 4 & 16 pro V; & prodibit arcuum differentia <lb></lb>(1/2/121)=A+B+C in caſu ſecundo; (2/35 1/2)=4A+8B+16C in caſu <lb></lb>quarto; & (8/9 2/3)=16A+64B+256C in caſu ſexto. </s> <s>Et ex his æ<lb></lb>quationibus, per debitam collationem & reductionem Analyticam, <lb></lb>fit A=0,0000916, B=0,0010847, & C=0,0029558. Eſt igitur <lb></lb>differentia arcuum ut 0,0000916V+0,0010847V1/2+0,0029558V<emph type="sup"></emph>2<emph.end type="sup"></emph.end>: <lb></lb>& propterea cum (per Corollarium Propoſitionis XXX) reſiſtentia <lb></lb>Globi in medio arcus oſcillando deſcripti, ubi velocitas eſt V, <lb></lb>ſit ad ipſius pondus ut (7/11)AV+(16/23)BV1/2+1/4CV<emph type="sup"></emph>2<emph.end type="sup"></emph.end> ad longitudinem <lb></lb>Penduli; ſi pro A, B & C ſcribantur numeri inventi, fiet reſiſtentia <lb></lb>Globi ad ejus pondus, ut 0,0000583V+0,0007546V1/2+0,0022169V<emph type="sup"></emph>2<emph.end type="sup"></emph.end><lb></lb>ad longitudinem Penduli inter centrum ſuſpenſionis & Regulam, <lb></lb>id eſt, ad 121 digitos. </s> <s>Unde cum V in caſu ſecundo deſignet 1, <lb></lb>in quarto 4, in ſexto 16: erit reſiſtentia ad pondus Globi in caſu <lb></lb>ſecundo ut 0,0030298 ad 121, in quarto ut 0,0417402 ad 121, in <lb></lb>ſexto ut 0,61675 ad 121. </s></p> <p type="margin"> <s><margin.target id="note261"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Arcus quem punctum in filo notatum in caſu ſexto deſcripſit, <lb></lb>erat 120-(8/9 2/3) ſeu (119 5/29) digitorum. </s> <s>Et propterea cum radius eſſet <lb></lb>121 digitorum, & longitudo Penduli inter punctum ſuſpenſionis <pb xlink:href="039/01/314.jpg" pagenum="286"></pb><arrow.to.target n="note262"></arrow.to.target>& centrum Globi eſſet 126 digitorum, arcus quem centrum Globi <lb></lb>deſcripſit erat (124 1/31) digitorum. </s> <s>Quoniam corporis oſcillantis ve<lb></lb>locitas maxima, ob reſiſtentiam Aeris, non incidit in punctum infi<lb></lb>mum arcus deſcripti, ſed in medio fere loco arcus totius verſatur: <lb></lb>hæc eadem erit circiter ac ſi Globus deſcenſu ſuo toto in Medio <lb></lb>non reſiſtente deſcriberet arcus illius partem dimidiam digitorum <lb></lb>(62 1/62), idQ.E.I. Cycloide, ad quam motum Penduli ſupra reduxi<lb></lb>mus: & propterea velocitas illa æqualis erit velocitati quam Glo<lb></lb>bus, perpendiculariter cadendo & caſu ſuo deſcribendo altitudinem <lb></lb>arcus illius ſinui verſo æqualem, acquirere poſſet. </s> <s>Eſt autem ſinus <lb></lb>ille verſus in Cycloide ad arcum iſtum (62 1/62) ut arcus idem ad pen<lb></lb>duli longitudinem duplam 252, & propterea æqualis digitis 15,278. <lb></lb>Quare velocitas ea ipſa eſt quam corpus cadendo & caſu ſuo ſpa<lb></lb>tium 15,278 digitorum deſcribendo acquirere poſſet. </s> <s>Tali igitur <lb></lb>cum velocitate Globus reſiſtentiam patitur, quæ ſit ad ejus pondus <lb></lb>ut 0,61675 ad 121, vel (ſi reſiſtentiæ pars illa ſola ſpectetur quæ <lb></lb>eſt in velocitatis ratione duplicata) ut 0,56752 ad 121. </s></p> <p type="margin"> <s><margin.target id="note262"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Experimento autem Hydroſtatico inveni quod pondus Globi hu<lb></lb>jus lignei eſſet ad pondus Globi aquei magnitudinis ejuſdem, ut 55 <lb></lb>ad 97: & propterea cum 121 ſit ad 213,4 in eadem ratione, erit <lb></lb>reſiſtentia Globi aquei præfata cum velocitate progredientis ad ip<lb></lb>ſius pondus, ut 0,56752 ad 213,4 id eſt, ut 1 ad (376 1/50). Unde cum <lb></lb>pondus Globi aquei, quo tempore Globus cum velocitate unifor<lb></lb>miter continuata deſcribat longitudinem digitorum 30,556, veloci<lb></lb>tatem illam omnem in Globo cadente generare poſſet; manifeſtum <lb></lb>eſt quod vis reſiſtentiæ eodem tempore uniformiter continuata tol<lb></lb>lere poſſet velocitatem minorem in ratione 1 ad (376 1/50), hoc eſt, ve<lb></lb>locitatis totius partem (1/(376 1/50)). Et propterea quo tempore Globus, <lb></lb>ea cum velocitate uniformiter continuata, longitudinem ſemidiame<lb></lb>tri ſuæ, ſeu digitorum (3 7/16), deſcribere poſſet, eodem amitteret mo<lb></lb>tus ſui partem (1/3342). </s></p> <p type="main"> <s>Numerabam etiam oſcillationes quibus Pendulum quartam mo<lb></lb>tus ſui partem amiſit. </s> <s>In ſequente Tabula numeri ſupremi deno<lb></lb>tant longitudinem arcus deſcenſu primo deſcripti, in digitis & par<lb></lb>tibus digiti expreſſam: numeri medii ſignificant longitudinem ar<lb></lb>cus aſcenſu ultimo deſcripti; & loco infimo ſtant numeri oſcilla<lb></lb>tionum. </s> <s>Experimentum deſcripſi tanquam magis accuratum quam <lb></lb>cum motus pars tantum octava amitteretur. </s> <s>Calculum tentet qui <lb></lb>volet. <pb xlink:href="039/01/315.jpg" pagenum="287"></pb><arrow.to.target n="note263"></arrow.to.target><arrow.to.target n="table1"></arrow.to.target><arrow.to.target n="table2"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note263"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p><table><table.target id="table1"></table.target><row><cell><emph type="italics"></emph>Deſcenſus primus<emph.end type="italics"></emph.end></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"></emph>Aſcenſus ultimus<emph.end type="italics"></emph.end></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"></emph>Numerus Oſcillat.<emph.end type="italics"></emph.end></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>Poſtea Globum plumbeum, diametro digitorum 2, & pondere <lb></lb> unciarum <emph type="italics"></emph>Romanarum<emph.end type="italics"></emph.end>26 1/4, ſuſpendi filo eodem, ſic ut inter cen<lb></lb>trum Globi & punctum ſuſpenſionis intervallum eſſet pedum 10 1/2, <lb></lb> & numerabam oſcillationes quibus data motus pars amitteretur. <lb></lb> Tabularum ſubſequentium prior exhibet numerum oſcillationum <lb></lb> quibus pars octava motus totius ceſſavit; ſecunda numerum oſcil<lb></lb>lationum quibus ejuſdem pars quarta amiſſa fuit. <lb></lb></s></p> <table><row><cell><emph type="italics"></emph>Deſcenſus primus<emph.end type="italics"></emph.end></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"></emph>Aſcenſus ultimus<emph.end type="italics"></emph.end></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"></emph>Numerus Oſcillat.<emph.end type="italics"></emph.end></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><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell><emph type="italics"></emph>Deſcenſus primus<emph.end type="italics"></emph.end></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"></emph>Aſcenſus ultimus<emph.end type="italics"></emph.end></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"></emph>Numerus Oſcillat.<emph.end type="italics"></emph.end></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 ſeligendo ex obſervationibus tertiam, quintam <lb></lb> & ſeptimam, & exponendo velocitates maximas in his obſerva<lb></lb>tionibus particulatim per numeros 1, 4, 16 reſpective, & genera<lb></lb>liter per quantitatem V ut ſupra: emerget in obſervatione tertia <lb></lb> (1/2/193)=A+B+C, in quinta (2/90 1/2)=4A+8B+16C, in ſeptima <lb></lb> (8/30)=16A+64B+256C. Hæ vero æquationes reductæ dant <lb></lb> A=0,001414, B=0,000297, C=0,000879. Et inde prodit reſi<lb></lb>ſtentia Globi cum velocitate V moti, in ea ratione ad pondus ſuum <lb></lb> unciarum 26 1/4, quam habet 0,0009V+0,000207V1/2+0,000659V<emph type="sup"></emph>2<emph.end type="sup"></emph.end><lb></lb>ad penduli longitudinem 121 digitorum. </s> <s>Et ſi ſpectemus eam ſo<lb></lb>lummodo reſiſtentiæ partem quæ eſt in duplicata ratione velocitatis, <lb></lb> hæc erit ad pondus Globi ut 0,000659V<emph type="sup"></emph>2<emph.end type="sup"></emph.end> ad 121 digitos. </s> <s>Erat au<lb></lb>tem hæc pars reſiſtentiæ in experimento primo ad pondus Globi <lb></lb> lignei unciarum (57 7/22), ut 0,002217V<emph type="sup"></emph>2<emph.end type="sup"></emph.end> ad 121: & inde fit reſiſtentia <lb></lb> Globi lignei ad reſiſtentiam Globi plumbei (paribus eorum velocita<lb></lb>tibus) ut (57 7/22) in 0,002217 ad 26 1/4 in 0,000659, id eſt, ut 7 1/3 ad 1. <lb></lb> Diametri Globorum duorum erant 6 7/8 & 2 digitorum, & harum <lb></lb> quadrata ſunt ad invicem ut 47 1/4 & 4, ſeu (11 11/16) & 1 quamproxime. <lb></lb> Ergo reſiſtentiæ Globorum æquivelocium erant in minore ratione <lb></lb> quam duplicata diametrorum. </s> <s>At nondum conſideravimus reſi-<pb xlink:href="039/01/316.jpg" pagenum="288"></pb><lb></lb><arrow.to.target n="note264"></arrow.to.target>ſtentiam fili, quæ certe permagna erat, ac de pendulorum inventa <lb></lb> reſiſtentia ſubduci debet. </s> <s>Hanc accurate definire non potui, ſed <lb></lb> majorem tamen inveni quam partem tertiam reſiſtentiæ totius mi<lb></lb>noris penduli; & inde didici quod reſiſtentiæ Globorum, dempta <lb></lb> fili reſiſtentia, ſunt quam proxime in duplicata ratione diametro<lb></lb>rum. </s> <s>Nam ratio 7 1/3-1/3 ad 1-1/3, ſeu 10 1/2 ad 1, non longe abeſt a <lb></lb> diametrorum ratione duplicata (11 11/16) ad 1. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note264"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s>Cum reſiſtentia fili in Globis majoribus minoris ſit momenti, <lb></lb> tentavi etiam experimentum in Globo cujus diameter erat 18 1/4 di<lb></lb>gitorum. </s> <s>Longitudo penduli inter punctum ſuſpenſionis & cen<lb></lb>trum oſcillationis erat digitorum 122 1/2, inter punctum ſuſpenſionis <lb></lb> & nodum in filo 109 1/2 dig. Arcus primo penduli deſcenſu a no<lb></lb>do deſcriptus, 32 dig. Arcus aſcenſu ultimo poſt oſcillationes <lb></lb> quinque ab eodem nodo deſcriptus, 28 dig. Summa arcuum ſeu <lb></lb> arcus totus oſcillatione mediocri deſcriptus, 60 dig. Differentia <lb></lb> arcuum 4 dig. Ejus pars decima ſeu differentia inter deſcenſum & <lb></lb> aſcenſum in oſcillatione mediocri 2/5 dig. Ut radius 109 1/2 ad radi<lb></lb>um 122 1/2, ita arcus totus 60 dig. oſcillatione mediocri a nodo de<lb></lb>ſcriptus, ad arcum totum 67 1/8 dig. oſcillatione mediocri a centro <lb></lb> Globi deſcriptum: & ita differentia 2/5 ad differentiam novam 0,4475. <lb></lb> Si longitudo penduli, manente longitudine arcus deſcripti, augere<lb></lb>tur in ratione 126 ad 122 1/2; tempus oſcillationis augeretur & velo<lb></lb>citas penduli diminueretur in ratione illa ſubduplicata, maneret <lb></lb> vero arcuum deſcenſu & ſubſequente aſcenſu deſcriptorum diffe<lb></lb>rentia 0,4475. Deinde ſi arcus deſcriptus augeretur in ratione <lb></lb> (124 1/31) ad 67 1/8, differentia iſta 0,4475 augeretur in duplicata illa ra<lb></lb>tione, adeoque evaderet 1,5295. Hæc ita ſe haberent, ex hy<lb></lb>potheſi quod reſiſtentia Penduli eſſet in duplicata ratione velo<lb></lb>citatis. </s> <s>Ergo ſi pendulum deſcriberet arcum totum (124 1/31) di<lb></lb>gitorum, & longitudo ejus inter punctum ſuſpenſionis & cen<lb></lb>trum oſcillationis eſſet 126 digitorum, differentia arcuum de<lb></lb>ſcenſu & ſubſequente aſcenſu deſcriptorum foret 1,5295 digito<lb></lb>rum. </s> <s>Et hæc differentia ducta in pondus Globi penduli, quod erat <lb></lb> unciarum 208, producit 318,136. Rurſus ubi pendulum ſuperius <lb></lb> ex Globo ligneo conſtructum, centro oſcillationis, quod a puncto <lb></lb> ſuſpenſionis digitos 126 diſtabat, deſcribebat arcum totum (124 1/31) <lb></lb> digitorum, differentia arcuum deſcenſu & aſcenſu deſcriptum fuit <lb></lb> (126/121) in (8/9 2/3), quæ ducta in pondus Globi, quod erat unciarum (57 1/22), <lb></lb> producit 49,396. Duxi autem differentias haſce in pondera Glo<lb></lb>borum ut invenirem eorum reſiſtentias. </s> <s>Nam differentiæ ori-<pb xlink:href="039/01/317.jpg" pagenum="289"></pb><lb></lb>untur ex reſiſtentiis, ſuntque ut reſiſtentiæ directe & pondera in<lb></lb><arrow.to.target n="note265"></arrow.to.target>verſe. </s> <s>Sunt igitur reſiſtentiæ ut numeri 318,136 & 49,396. Pars <lb></lb> autem reſiſtentiæ Globi minoris, quæ eſt in duplicata ratione velo<lb></lb>citatis, erat ad reſiſtentiam totam, ut 0,56752 ad 0,61675, id eſt, ut <lb></lb> 45,453 ad 49,396; & pars reſiſtentiæ Globi majoris propemodum <lb></lb> æquatur ipſius reſiſtentiæ toti; adeoque partes illæ ſunt ut 318,136 <lb></lb> & 45,453 quamproxime, id eſt, ut 7 & 1. Sunt autem Globorum <lb></lb> diametri 18 1/4 & 6 7/8; & harum quadrata (351 9/16) & (47 17/64) ſunt ut 7,438 <lb></lb> & 1, id eſt, ut Globorum reſiſtentiæ 7 & 1 quamproxime. </s> <s>Diffe<lb></lb>rentia rationum haud major eſt quam quæ ex fili reſiſtentia oriri po<lb></lb>tuit. </s> <s>Igitur reſiſtentiarum partes illæ quæ ſunt, paribus Globis, ut <lb></lb> quadrata velocitatum; ſunt etiam, paribus velocitatibus, ut qua<lb></lb>drata diametrorum Globorum. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note265"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s>Cæterum Globorum, quibus uſus ſum in his experimentis, max<lb></lb>imus non erat perfecte Sphæricus, & propterea in calculo hic allato <lb></lb> minutias quaſdam brevitatis gratia neglexi; de calculo accurato in <lb></lb> experimento non ſatis accurato minime ſollicitus. </s> <s>Optarim itaque <lb></lb> (cum demonſtratio Vacui ex his dependeat) ut experimenta cum <lb></lb> Globis & pluribus & majoribus & magis accuratis tentarentur. </s> <s>Si <lb></lb> Globi ſumantur in proportione Geometrica, puta quorum diametri <lb></lb> ſint digitorum 4, 8, 16, 32; ex progreſſione experimentorum col<lb></lb>ligetur quid in Globis adhuc majoribus evenire debeat. <lb></lb> </s></p> <p type="main"> <s>Jam vero conferendo reſiſtentias diverſorum Fluidorum inter ſe <lb></lb> tentavi ſequentia. </s> <s>Arcam ligneam paravi longitudine pedum qua<lb></lb>tuor, latitudine & altitudine pedis unius. </s> <s>Hanc operculo nuda<lb></lb>tam implevi aqua fontana, fecique ut immerſa pendula in medio <lb></lb> aquæ oſcillando moverentur. </s> <s>Globus autem plumbeus pondere <lb></lb> 166 1/6 unciarum, diametro 3 5/8 digitorum, movebatur ut in Tabula <lb></lb> ſequente deſcripſimus, exiſtente videlicet longitudine penduli a <lb></lb> puncto ſuſpenſionis ad punctum quoddam in filo notatum 126 di<lb></lb>gitorum, ad oſcillationis autem centrum 134 1/8 digitorum.</s></p><table><row><cell><emph type="italics"></emph>Arcus deſcenſu primo a puncto in <lb></lb> filo notato deſcriptus, digitorum<emph.end type="italics"></emph.end></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"></emph>Arcus aſcenſu ultimo deſcriptus, <lb></lb> digitorum<emph.end type="italics"></emph.end></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"></emph>Arcuum differentia motui amiſſo <lb></lb> proportionalis, digitorum<emph.end type="italics"></emph.end></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"></emph>Numerus Oſcillationum in aqua<emph.end type="italics"></emph.end></cell><cell></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"></emph>Numerus Oſcillationum in aere<emph.end type="italics"></emph.end></cell><cell>85 1/2</cell><cell></cell><cell>287</cell><cell>535</cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell></row></table><pb xlink:href="039/01/318.jpg" pagenum="290"></pb> <p type="margin"> <s><margin.target id="note266"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s>In experimento columnæ quartæ, motus æquales oſcillationibus <lb></lb> 535 in aere, & 1 1/5 in aqua amiſſi ſunt. </s> <s>Erant quidem oſcillationes <lb></lb> in aere paulo celeriores quam in aqua. </s> <s>At ſi oſcillationes in aqua <lb></lb> in ea ratione accelerarentur ut motus pendulorum in Medio utro<lb></lb>que fierent æquiveloces, maneret numerus idem oſcillationum 1 1/5 <lb></lb> in aqua, quibus motus idem ac prius amitteretur; ob reſiſtentiam <lb></lb> auctam & ſimul quadratum temporis diminutum in eadem ratione <lb></lb> illa duplicata. </s> <s>Paribus igitur pendulorum velocitatibus motus æ<lb></lb>quales in aere oſcillationibus 535 & in aqua oſcillationibus 1 1/5 amiſſi <lb></lb> ſunt; ideoque reſiſtentia penduli in aqua eſt ad ejus reſiſtentiam in <lb></lb> aere ut 535 ad 1 1/5. Hæc eſt proportio reſiſtentiarum totarum in <lb></lb> caſu columnæ quartæ. <lb></lb> </s></p> <p type="main"> <s>Deſignet jam AV+CV differentiam arcuum in deſcenſu & ſub<lb></lb>ſequente aſcenſu deſcriptorum a Globo, in Aere cum velocitate maxi<lb></lb>ma V moto; & cum velocitas maxima, in caſu columnæ quartæ, ſit <lb></lb> ad velocitatem maximam in caſu columnæ primæ, ut 1 ad 8; & diffe<lb></lb>rentia illa arcuum, in caſu columnæ quartæ, ad differentiam in caſu <lb></lb> columnæ primæ ut (2/535) ad (16/85 1/2), ſeu ut 85 1/2 ad 4280: ſeribamus in <lb></lb> his caſibus 1 & 8 pro velocitatibus, atque 85 1/2 & 4280 pro dif<lb></lb>ferentiis arcuum, & fiet A+C=85 1/2 & 8A+64C=4280 ſeu <lb></lb> A+8C=535; indeque per reductionem æquationum proveniet <lb></lb> 7C=449 1/2 & C=(64 1/14) & A=21 1/7: atque adeo reſiſtentia, cum <lb></lb> ſit ut (7/11) AV+1/4 CV<emph type="sup"></emph>2<emph.end type="sup"></emph.end>, erit ut (13 6/11)V+(48 1/56)V<emph type="sup"></emph>2<emph.end type="sup"></emph.end>. Quare in caſu co<lb></lb>lumnæ quartæ, ubi velocitas erat 1, reſiſtentia tota eſt ad partem <lb></lb> ſuam quadrato velocitatis proportionalem, ut (13 6/11)+(48 2/56) ſeu <lb></lb> (61 12/17) ad (48 9/56); & idcirco reſiſtentia penduli in aqua eſt ad reſiſten<lb></lb>tiæ partem illam in aere quæ quadrato velocitatis proportionalis <lb></lb> eſt, quæque ſola in motibus velocioribus conſideranda venit, ut (61 12/17) <lb></lb> ad (48 9/56) & 535 ad 1 1/5 conjunctim, id eſt, ut 571 ad 1. Si penduli <lb></lb> in aqua oſcillantis filum totum fuiſſet immerſum, reſiſtentia ejus <lb></lb> fuiſſet adhuc major; adeo ut penduli in aere oſcillantis reſiſtentia <lb></lb> illa quæ velocitatis quadrato proportionalis eſt, quæque ſola in <lb></lb> corporibus velocioribus conſideranda venit, ſit ad reſiſtentiam e<lb></lb>juſdem penduli totius, eadem cum velocitate, in aqua oſcillantis, <lb></lb> ut 800 vel 900 ad 1 circiter, hoc eſt, ut denſitas aquæ ad denſita<lb></lb>tatem aeris quamproxime. <lb></lb> </s></p> <p type="main"> <s>In hoc calculo ſumi quoQ.E.D.beret pars illa reſiſtentiæ penduli <lb></lb> in aqua, quæ eſſet ut quadratum velocitatis, ſed (quod mirum for<lb></lb>te videatur) reſiſtentia in aqua augebatur in ratione velocitatis <pb xlink:href="039/01/319.jpg" pagenum="291"></pb><lb></lb>pluſquam duplicata. </s> <s>Ejus rei cauſam inveſtigando, in hanc in<lb></lb><arrow.to.target n="note267"></arrow.to.target>cidi, quod Arca nimis anguſta eſſet pro magnitudine Globi pen<lb></lb>duli, & motum aquæ cedentis præ anguſtia ſua nimis impedie<lb></lb>bat. </s> <s>Nam ſi Globus pendulus, cujus diameter erat digiti u<lb></lb>nius, immergeretur; reſiſtentia augebatur in duplicata ratione ve<lb></lb>locitatis quam proxime. </s> <s>Id tentabam conſtruendo pendulum ex <lb></lb> Globis duobus, quorum inferior & minor oſcillaretur in aqua, ſu<lb></lb>perior & major proxime ſupra aquam filo affixus eſſet, & in Aere <lb></lb> oſcillando, adjuvaret motum penduli eumQ.E.D.uturniorem redde<lb></lb>ret. </s> <s>Experimenta autem hoc modo inſtituta ſe habebant ut in Ta<lb></lb>bula ſequente deſcribitur. <lb></lb></s></p><table><row><cell><emph type="italics"></emph>Arcus deſcenſu primo deſcriptus<emph.end type="italics"></emph.end></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"></emph>Arcus aſcenſu ultimo deſcriptus<emph.end type="italics"></emph.end></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"></emph>Arcuum diff. motui amiſſo proport.<emph.end type="italics"></emph.end></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"></emph>Numerus Oſcillationum<emph.end type="italics"></emph.end></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"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s>Conferendo reſiſtentias Mediorum inter ſe, effeci etiam ut pen<lb></lb>dula ferrea oſcillarentur in argento vivo. </s> <s>Longitudo fili ferrei erat <lb></lb> pedum quaſi trium, & diameter Globi penduli quaſi tertia pars di<lb></lb>giti. </s> <s>Ad filum autem proxime ſupra Mercurium affixus erat Glo<lb></lb>bus alius plumbeus ſatis magnus ad motum penduli diutius conti<lb></lb>nuandum. </s> <s>Tum vaſculum, quod capiebat quaſi libras tres argenti <lb></lb> vivi, implebam vicibus alternis argento vivo & aqua communi, ut <lb></lb> pendulo in Fluido utroque ſucceſſive oſcillante, invenirem propor<lb></lb>tionem reſiſtentiarum: & prodiit reſiſtentia argenti vivi ad reſi<lb></lb>ſtentiam aquæ, ut 13 vel 14 ad 1 circiter: id eſt, ut denſitas argen<lb></lb>ti vivi ad denſitatem aquæ. Ubi Globum pendulum paulo majo<lb></lb>rem adhibebam, puta cujus diameter eſſet quaſi 1/3 vel 2/3 partes di<lb></lb>giti, prodibat reſiſtentia argenti vivi in ea ratione ad reſiſtentiam <lb></lb> aquæ quam habet numerus 12 vel 10 ad 1 circiter. </s> <s>Sed experi<lb></lb>mento priori magis fidendum eſt, propterea quod in his ultimis <lb></lb> Vas nimis anguſtum fuit pro magnitudine Globi immerſi. </s> <s>Am<lb></lb>pliato Globo, deberet etiam Vas ampliari. </s> <s>Conſtitueram quidem <lb></lb> hujuſmodi experimenta in vaſis majoribus & in liquoribus tum <lb></lb> Metallorum fuſorum, tum aliis quibuſdam tam calidis quam fri<lb></lb>gidis repetere: ſed omnia experiri non vacat, & ex jam deſcriptis <lb></lb> ſatis liquet reſiſtentiam corporum celeriter motorum denſitati Flu<lb></lb>idorum in quibus moventur proportionalem eſſe quam proxime. <lb></lb> Non dico accurate. </s> <s>Nam Fluida tenaciora, pari denſitate, procul-<pb xlink:href="039/01/320.jpg" pagenum="292"></pb><lb></lb><arrow.to.target n="note268"></arrow.to.target>dubio magis reſiſtunt quam liquidiora, ut Oleum frigidum quam <lb></lb> calidum, calidum quam aqua pluvialis, aqua quam Spiritus Vini. <lb></lb> Verum in liquoribus qui ad ſenſum ſatis fluidi ſunt, ut in Aere, in <lb></lb> Aqua ſeu dulci ſeu ſalſa, in Spiritibus Vini, Terebinthi & Salium, <lb></lb> in Oleo a fæcibus per deſtillationem liberato & calefacto, Oleoque <lb></lb> Vitrioli & Mercurio, ac Metallis liquefactis, & ſiqui ſint alii, qui <lb></lb> tam fluidi ſunt ut in vaſis agitati motum impreſſum diutius con<lb></lb>ſervent, effuſique liberrime in guttas decurrendo reſolvantur, nul<lb></lb>lus dubito quin regula allata ſatis accurate obtineat: præſertim ſi <lb></lb> experimenta in corporibus pendulis & majoribus & velocius motis <lb></lb> inſtituantur. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note268"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s>Denique cum receptiſſima Philoſophorum ætatis hujus opinio <lb></lb> ſit, Medium quoddam æthereum & longe ſubtiliſſimum extare, <lb></lb> quod omnes omnium corporum poros & meatus liberrime per<lb></lb>meet; a tali autem Medio per corporum poros fluente reſiſtentia <lb></lb> oriri debeat: ut tentarem an reſiſtentia, quam in motis corporibus <lb></lb> experimur, tota ſit in eorum externa ſuperficie, an vero partes eti<lb></lb>am internæ in ſuperficiebus propriis reſiſtentiam notabilem ſenti<lb></lb>ant, excogitavi experimentum tale. </s> <s>Filo pedum undecim longitu<lb></lb>dinis, ab unco chalyoeo ſatis firmo, mediante annulo chalybeo, ſu<lb></lb>ſpendebam pyxidem abiegnam rotundam, ad conſtituendum pen<lb></lb>dulum longitudinis prædictæ. Uncus ſurſum præacutus erat acie <lb></lb> concava, ut annulus arcu ſuo ſuperiore aciei innixus liberrime mo<lb></lb>veretur. </s> <s>Arcui autem inferiori annectebatur filum. </s> <s>Pendulum ita <lb></lb> conſtitutum deducebam a perpendiculo ad diſtantiam quaſi pedum <lb></lb> ſex, idque ſecundum planum aciei unci perpendiculare, ne annu<lb></lb>lus, oſcillante pendulo, ſupra aciem unci ultro citroque laberetur. <lb></lb> Nam punctum ſuſpenſionis, in quo annulus uncum tangit, immo<lb></lb>tum manere debet. </s> <s>Locum igitur accurate notabam, ad quem de<lb></lb>duxeram pendulum, dein pendulo demiſſo notabam alia tria loca ad <lb></lb> quæ redibat in fine oſcillationis primæ, ſecundæ ac tertiæ. Hoc re<lb></lb>petebam ſæpius, ut loca illa quam potui accuratiſſime invenirem. <lb></lb> Tum pyxidem plumbo & gravioribus, quæ ad manus erant, me<lb></lb>tallis implebam. </s> <s>Sed prius ponderabam pyxidem vacuam, una <lb></lb> cum parte fili quæ circum pyxidem volvebatur ac dimidio par<lb></lb>tis reliquæ inter uncum & pyxidem pendulam tendebatur. <lb></lb> (Nam filum tenſum dimidio ponderis ſui pendulum a perpendiculo <lb></lb> digreſſum ſemper urget.) Huic ponderi addebam pondus Aeris <lb></lb> quem pyxis capiebat. </s> <s>Et pondus totum erat quaſi pars ſeptuage<lb></lb>ſima octava pyxidis metallorum plenæ. Tum quoniam pyxis me-<pb xlink:href="039/01/321.jpg" pagenum="293"></pb><lb></lb>tallorum plena, pondere ſuo tendendo filum, augebat longitudi<lb></lb><arrow.to.target n="note269"></arrow.to.target>nem penduli, contrahebam filum ut penduli jam oſcillantis eadem <lb></lb> eſſet longitudo ac prius. </s> <s>Dein pendulo ad locum primo notatum <lb></lb> retracto ac dimiſſo, numerabam oſcillationes quaſi ſeptuaginta & <lb></lb> ſeptem, donec pyxis ad locum ſecundo notatum rediret, totidem<lb></lb>que ſubinde donec pyxis ad locum tertio notatum rediret, atque <lb></lb> rurſus totidem donec pyxis reditu ſuo attingeret locum quartum. <lb></lb> Unde concludo quod reſiſtentia tota pyxidis plenæ non majorem <lb></lb> habebat proportionem ad reſiſtentiam pyxidis vacuæ quam 78 ad <lb></lb> 77. Nam ſi æquales eſſent ambarum reſiſtentiæ, pyxis plena ob <lb></lb> vim ſuam inſitam ſeptuagies & octies majorem vi inſita pyxidis <lb></lb> vacuæ, motum ſuum oſcillatorium tanto diutius conſervare debe<lb></lb>ret, atque adeo completis ſemper oſcillationibus 78 ad loca illa <lb></lb> notata redire. </s> <s>Rediit autem ad eadem completis oſcillationibus 77. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note269"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s>Deſignet igitur A reſiſtentiam pyxidis in ipſius ſuperficie exter<lb></lb>na, & B reſiſtentiam pyxidis vacuæ in partibus internis; & ſi reſi<lb></lb>ſtentiæ corporum æquivelocium in partibus internis ſint ut mate<lb></lb>ria, ſeu numerus particularum quibus reſiſtitur: erit 78 B reſiſten<lb></lb>tia pyxidis plenæ in ipſius partibus internis: adeoque pyxidis va<lb></lb>cuæ reſiſtentia tota A+B erit ad pyxidis plenæ reſiſtentiam to<lb></lb>tam A+78 B ut 77 ad 78, & diviſim A+B ad 77 B, ut 77 ad 1, <lb></lb> indeque A+B ad B ut 77X77 ad 1, & diviſim A ad B ut 5928 <lb></lb> ad 1. Eſt igitur reſiſtentia pyxidis vacuæ in partibus internis <lb></lb> quinquies millies minor quam ejuſdem reſiſtentia in externa ſuper<lb></lb>ficie, & amplius. </s> <s>Sic vero diſputamus ex Hypotheſi quod ma<lb></lb>jor illa reſiſtentia pyxidis plenæ, non ab alia aliqua cauſa latente <lb></lb> oriatur, ſed ab actione ſola Fluidi alicujus ſubtilis in metallum <lb></lb> incluſum. <lb></lb> </s></p> <p type="main"> <s>Hoc experimentum recitavi memoriter. </s> <s>Nam charta, in qua il<lb></lb>lud aliquando deſcripſeram, intercidit. </s> <s>Unde fractas quaſdam nu<lb></lb>merorum partes, quæ memoria exciderunt, omittere compulſus <lb></lb> ſum. </s> <s>Nam omnia denuo tentare non vacat. </s> <s>Prima vice, cum un<lb></lb>co infirmo uſus eſſem, pyxis plena citius retardabatur. </s> <s>Cauſam <lb></lb> quærendo, reperi quod uncus infirmus cedebat ponderi pyxidis, & <lb></lb> ejus oſcillationibus obſeQ.E.D. in partes omnes flectebatur. </s> <s>Para<lb></lb>bam igitur uncum firmum, ut punctum ſuſpenſionis immotum ma<lb></lb>neret, & tunc omnia ita evenerunt uti ſupra deſcripſimus. <pb xlink:href="039/01/322.jpg" pagenum="294"></pb><lb></lb></s></p></subchap2><subchap2><p> <s><arrow.to.target n="note270"></arrow.to.target><emph type="center"></emph>SECTIO VII.<emph.end type="center"></emph.end><lb></lb><emph type="center"></emph><emph type="italics"></emph>De Motu Fluidorum & Reſiſtentia Projectilium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end><lb></lb><emph type="center"></emph>PROPOSITIO XXXII. THEOREMA XXVI.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note270"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Si Corporum Syſtemata duo ſimilia ex æquali particularum numero <lb></lb> conſtent, & particulæ correſpondentes ſimiles ſint & propor<lb></lb>tionales, ſingulæ in uno Syſtemate ſingulis in altero, & ſimiliter <lb></lb> ſitæ inter ſe, ac datam habeant rationem denſitatis ad invicem, <lb></lb> & inter ſe temporibus proportionalibus ſimiliter moveri inci<lb></lb>piant, (eæ inter ſe quæ in uno ſunt Syſtemate & eæ inter ſe quæ <lb></lb> ſunt in altero) & ſi non tangant ſe mutuo quæ in eodem ſunt <lb></lb> Syſtemate, niſi in momentis reflexionum, neque attrahant vel fu<lb></lb>gent ſe mutuo, niſi viribus acceleratricibus quæ ſint ut particu<lb></lb>larum correſpondentium diametri inverſe & quadrata velocita. <lb></lb> tum directe: dico quod Syſtematum particulæ illæ pergent inter <lb></lb> ſe temporibus proportionalibus ſimiliter moveri.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Corpora ſimilia & ſimiliter ſita temporibus proportionalibus in<lb></lb>ter ſe ſimiliter moveri dico, quorum ſitus ad invicem in fine tem<lb></lb>porum illorum ſemper ſunt ſimiles: puta ſi particulæ unius Syſte<lb></lb>matis cum alterius particulis correſpondentibus conferantur. </s> <s>Un<lb></lb>de tempora erunt proportionalia, in quibus ſimiles & proportiona<lb></lb>les Figurarum ſimilium partes a particulis correſpondentibus de<lb></lb>ſcribuntur. </s> <s>Igitur ſi duo ſint ejuſmodi Syſtemata, particulæ cor<lb></lb>reſpondentes, ob ſimilitudinem incæptorum motuum, pergent ſi<lb></lb>militer moveri uſQ.E.D.nec ſibi mutuo occurrant. </s> <s>Nam ſi nullis <lb></lb> agitantur viribus, progredientur uniformiter in lineis rectis per mo<lb></lb>tus Leg. 1. Si viribus aliquibus ſe mutuo agitant, & vires illæ ſint <lb></lb> ut particularum correſpondentium diametri inverſe & quadrata ve<lb></lb>locitatum directe; quoniam particularum ſitus ſunt ſimiles & vires <lb></lb> proportionales, vires totæ quibus particulæ correſpondentes agi<lb></lb>tantur, ex viribus ſingulis agitantibus (per Legum Corollarium <pb xlink:href="039/01/323.jpg" pagenum="295"></pb><lb></lb>fecundum) compoſitæ, ſimiles habebunt determinationes, perin<lb></lb><arrow.to.target n="note271"></arrow.to.target>de ac ſi centra inter particulas ſimiliter ſita reſpicerent; & erunt <lb></lb> vires illæ totæ ad invicem ut vires ſingulæ componentes, hoc eſt, <lb></lb> ut correſpondentium particularum diametri inverſe, & quadrata <lb></lb> velocitatum directe: & propterea efficient ut correſpondentes par<lb></lb>ticulæ figuras ſimiles deſcribere pergant. </s> <s>Hæc ita ſe habebunt per <lb></lb> Corol. 1, & 8 Prop. IV, Lib. 1. ſi modo centra illa quieſcant. <lb></lb> Sin moveantur, quoniam ob tranſlationum ſimilitudinem, ſimiles <lb></lb> manent eorum ſitus inter Syſtematum particulas; ſimiles indu<lb></lb>centur mutationes in figuris quas particulæ deſcribunt. </s> <s>Similes igi<lb></lb>tur erunt correſpondentium & ſimilium particularum motus uſ<lb></lb>que ad occurſus ſuos primos, & propterea ſimiles occurſus, & ſi<lb></lb>miles reflexiones, & ſubinde (per jam oſtenſa) ſimiles motus in<lb></lb>ter ſe donec iterum in ſe mutuo inciderint, & ſic deinceps in in<lb></lb>finitum. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note271"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi corpora duo quævis, quæ ſimilia ſint & ad <lb></lb> Syſtematum particulas correſpondentes ſimiliter ſita, inter ipſas <lb></lb> temporibus proportionalibus ſimiliter moveri incipiant, ſintque <lb></lb> eorum magnitudines ac denſitates ad invicem ut magnitudines ac <lb></lb> denſitates correſpondentium particularum: hæc pergent tempori<lb></lb>bus proportionalibus ſimiliter moveri. </s> <s>Eſt enim eadem ratio par<lb></lb>tium majorum Syſtematis utriuſque atque particularum. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et ſi ſimiles & ſimiliter poſitæ Syſtematum partes om<lb></lb>nes quieſcant inter ſe: & earum duæ, quæ cæteris majores ſint, & <lb></lb> ſibi mutuo in utroque Syſtemate correſpondeant, ſecundum lineas <lb></lb> ſimiliter ſitas ſimili cum motu utcunque moveri incipiant: hæ ſi<lb></lb>miles in reliquis Syſtematum partibus excitabunt motus, & pergent <lb></lb> inter ipſas temporibus proportionalibus ſimiliter moveri; atque <lb></lb> adeo ſpatia diametris ſuis proportionalia deſcribere. <lb></lb> <emph type="center"></emph>PROPOSITIO XXXIII. THEOREMA XXVII.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, dico quod Syſtematum partes majores reſiſtituntur <lb></lb> in ratione compoſita ex duplicata ratione velocitatum ſuarum & <lb></lb> duplicata ratione diametrorum & ratione denſitatis partium <lb></lb> Syſtematum.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam reſiſtentia oritur partim ex viribus centripetis vel centri<lb></lb>fugis quibus particulæ Syſtematum ſe mutuo agitant, partim ex <lb></lb> occurſibus & reflexionibus particularum & partium majorum. <pb xlink:href="039/01/324.jpg" pagenum="296"></pb><lb></lb><arrow.to.target n="note272"></arrow.to.target>Prioris autem generis reſiſtentiæ ſunt ad invicem ut vires totæ mo<lb></lb>trices a quibus oriuntur, id eſt, ut vires totæ acceleratrices & quan<lb></lb>titates materiæ in partibus correſpondentibus; hoc eſt (per Hy<lb></lb>potheſin) ut quadrata velocitatum directe & diſtantiæ particula<lb></lb>rum correſpondentium inverſe & quantitates materiæ in partibus <lb></lb> correſpondentibus directe: ideoque (cum diſtantiæ particularum Sy<lb></lb>ſtematis unius ſint ad diſtantias correſpondentes particularum alte<lb></lb>rius, ut diameter particulæ vel partis in Syſtemate priore ad dia<lb></lb>metrum particulæ vel partis correſpondentis in altero, & quantita<lb></lb>tes materiæ ſint ut denſitates partium & cubi diametrorum) reſi<lb></lb>ſtentiæ ſunt ad invicem ut quadrata velocitatum & quadrata dia<lb></lb>metrorum & denſitates partium Syſtematum. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end>Poſte<lb></lb>rioris generis reſiſtentiæ ſunt ut reflexionum correſpondentium nu<lb></lb>meri & vires conjunctim. </s> <s>Numeri autem reflexionum ſunt ad in<lb></lb>vicem ut velocitates partium correſpondentium directe, & ſpatia <lb></lb> inter earum reflexiones inverſe. </s> <s>Et vires reflexionum ſunt ut ve<lb></lb>locitates & magnitudines & denſitates partium correſpondentium <lb></lb> conjunctim; id eſt, ut velocitates & diametrorum cubi & denſita<lb></lb>tes partium. </s> <s>Et conjunctis his omnibus rationibus, reſiſtentiæ <lb></lb> partium correſpondentium ſunt ad invicem ut quadrata veloci<lb></lb>tum & quadrata diametrorum & denſitates partium conjunctim. <lb></lb> <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note272"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur ſi Syſtemata illa ſint Fluida duo Elaſtica ad <lb></lb> modum Aeris, & partes eorum quieſcant inter ſe: corpora autem <lb></lb> duo ſimilia & partibus fluidorum quoad magnitudinem & denſita<lb></lb>tem proportionalia, & inter partes illas ſimiliter poſita, ſecundum <lb></lb> lineas ſimiliter poſitas utcunque projiciantur; vires autem acce<lb></lb>leratrices, quibus particulæ Fluidorum ſe mutuo agitant, ſint ut <lb></lb> corporum projectorum diametri inverſe, & quadrata velocitatum <lb></lb> directe: corpora illa temporibus proportionalibus ſimiles excita<lb></lb>bunt motus in Fluidis, & ſpatia ſimilia ac diametris ſuis propor<lb></lb>tionalia deſcribent. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Proinde in eodem Fluido projectile velox reſiſtentiam pa<lb></lb>titur quæ eſt in duplicata ratione velocitatis quam proxime. </s> <s>Nam <lb></lb> ſi vires, quibus particulæ diſtantes ſe mutuo agitant, augerentur in <lb></lb> duplicata ratione velocitatis, reſiſtentia foret in eadem ratione du<lb></lb>plicata accurate; ideoQ.E.I. Medio, cujus partes ab invicem diſtan<lb></lb>tes ſeſe viribus nullis agitant, reſiſtentia eſt in duplicata ratione ve<lb></lb>locitatis accurate. </s> <s>Sunto igitur Media tria <emph type="italics"></emph>A, B, C<emph.end type="italics"></emph.end>ex partibus <lb></lb> ſimilibus & æqualibus & ſecundum diſtantias æquales regulariter <pb xlink:href="039/01/325.jpg" pagenum="297"></pb><lb></lb>diſpoſitis conſtantia. </s> <s>Partes Mediorum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B<emph.end type="italics"></emph.end>fugiant ſe mutuo <lb></lb> <arrow.to.target n="note273"></arrow.to.target>viribus quæ ſint ad invicem ut <emph type="italics"></emph>T<emph.end type="italics"></emph.end>& <emph type="italics"></emph>V,<emph.end type="italics"></emph.end>illæ Medii <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ejuſmo<lb></lb>di viribus omnino deſtituantur. </s> <s>Et ſi corpora quatuor æqualia <lb></lb> <emph type="italics"></emph>D, E, F, G<emph.end type="italics"></emph.end>in his Mediis moveantur, priora duo <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>E<emph.end type="italics"></emph.end>in pri<lb></lb>oribus duobus <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& altera duo <emph type="italics"></emph>F<emph.end type="italics"></emph.end>& <emph type="italics"></emph>G<emph.end type="italics"></emph.end>in tertio <emph type="italics"></emph>G<emph.end type="italics"></emph.end>; ſitque ve<lb></lb>locitas corporis <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ad velocitatem corporis <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>& velocitas corpo<lb></lb>ris <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ad velocitatem corporis <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>in ſubduplicata ratione virium <emph type="italics"></emph>T<emph.end type="italics"></emph.end><lb></lb>ad vires <emph type="italics"></emph>V<emph.end type="italics"></emph.end>: reſiſtentia corporis <emph type="italics"></emph>D<emph.end type="italics"></emph.end>erit ad reſiſtentiam corporis <emph type="italics"></emph>E,<emph.end type="italics"></emph.end><lb></lb>& reſiſtentia corporis <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ad reſiſtentiam corporis <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>in velocitatum <lb></lb> ratione duplicata; & propterea reſiſtentia corporis <emph type="italics"></emph>D<emph.end type="italics"></emph.end>erit ad reſi<lb></lb>ſtentiam corporis <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ut reſiſtentia corporis <emph type="italics"></emph>E<emph.end type="italics"></emph.end>ad reſiſtentiam corpo<lb></lb>ris <emph type="italics"></emph>G.<emph.end type="italics"></emph.end>Sunto corpora <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>F<emph.end type="italics"></emph.end>æquivelocia ut & corpora <emph type="italics"></emph>E<emph.end type="italics"></emph.end>& <emph type="italics"></emph>G<emph.end type="italics"></emph.end>; <lb></lb> & augendo velocitates corporum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>F<emph.end type="italics"></emph.end>in ratione quacunque, ac <lb></lb> diminuendo vires particularum Medii <emph type="italics"></emph>B<emph.end type="italics"></emph.end>in eadem ratione duplicata, <lb></lb> accedet Medium <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad formam & conditionem Medii <emph type="italics"></emph>C<emph.end type="italics"></emph.end>pro lubi<lb></lb>tu, & idcirco reſiſtentiæ corporum æqualium & æquivelocium <emph type="italics"></emph>E<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>G<emph.end type="italics"></emph.end>in his Mediis, perpetuo accedent ad æqualitatem, ita ut ea<lb></lb>rum differentia evadat tandem minor quam data quævis. </s> <s>Proinde <lb></lb> cum reſiſtentiæ corporum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ſint ad invicem ut reſiſtentiæ cor<lb></lb>porum <emph type="italics"></emph>E<emph.end type="italics"></emph.end>& <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>accedent etiam hæ ſimiliter ad rationem æqualita<lb></lb>tis. </s> <s>Corporum igitur <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>ubi velociſſime moventur, reſiſten<lb></lb>tiæ ſunt æquales quam proxime: & propterea cum reſiſtentia cor<lb></lb>poris <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ſit in duplicata ratione velocitatis, erit reſiſtentia corporis <lb></lb> <emph type="italics"></emph>D<emph.end type="italics"></emph.end>in eadem ratione quam proxime. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note273"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Igitur corporis in Fluido quovis Elaſtico velociſſime <lb></lb> moti eadem fere eſt reſiſtentia ac ſi partes Fluidi viribus ſuis <lb></lb> centrifugis deſtituerentur, ſeque mutuo non fugerent: ſi modo <lb></lb> Fluidi vis Elaſtica ex particularum viribus centrifugis oriatur, & <lb></lb> velocitas adeo magna ſit ut vires non habeant ſatis temporis ad <lb></lb> agendum. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Proinde cum reſiſtentiæ ſimilium & æquivelocium cor<lb></lb>porum, in Medio cujus partes diſtantes ſe mutuo non fugiunt, ſint <lb></lb> ut quadrata diametrorum; ſunt etiam æquivelocium & celerrime <lb></lb> motorum corporum reſiſtentiæ in Fluido Elaſtico ut quadrata <lb></lb> diametrorum quam proxime. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et cum corpora ſimilia, æqualia & æquivelocia, in <lb></lb> Mediis ejuſdem denſitatis, quorum particulæ ſe mutuo non fu<lb></lb>giunt, ſive particulæ illæ ſint plures & minores, ſive pauciores & <lb></lb> majores, in æqualem materiæ quantitatem temporibus æqualibus <lb></lb> inpingant, eique æqualem motus quantitatem imprimant, & vi-<pb xlink:href="039/01/326.jpg" pagenum="298"></pb><lb></lb><arrow.to.target n="note274"></arrow.to.target>ciſſim (per motus Legem tertiam) æqualem ab eadem reactionem <lb></lb> patiantur, hoc eſt, æqualiter reſiſtantur: manifeſtum eſt etiam <lb></lb> quod in ejuſdem denſitatis Fluidis Elaſticis, ubi velociſſime mo<lb></lb>ventur, æquales ſint eorum reſiſtentiæ quam proxime; ſive Fluida <lb></lb> illa ex particulis craſſioribus conſtent, ſive ex omnium ſubtiliſſi<lb></lb>mis conſtituantur. </s> <s>Ex Medii ſubtilitate reſiſtentia projectilium ce<lb></lb>lerrime motorum non multum diminuitur. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note274"></margin.target>DE MOTU <lb></lb> CORPORUM.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Hæc omnia ita ſe habent in Fluidis, quorum vis Ela<lb></lb>ſtica ex particularum viribus centrifugis originem ducit. </s> <s>Quod ſi <lb></lb> vis illa aliunde oriatur, veluti ex particularum expanſione ad inſtar <lb></lb> Lanæ vel ramorum Arborum, aut ex alia quavis cauſa, qua motus <lb></lb> particularum inter ſe redduntur minus liberi: reſiſtentia, ob mi<lb></lb>norem Medii fluiditatem, erit major quam in ſuperioribus Co<lb></lb>rollariis. <lb></lb> <emph type="center"></emph>PROPOSITIO XXXIV. THEOREMA XXVIII.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Si Globus & Cylindrus æqualibus diametris deſcripti, in Medio <lb></lb> raro ex particulis æqualibus & ad æquales ab invicem diſtan<lb></lb>tias libere diſpoſitis conſtante, ſecundum plagam axis Cylindri, <lb></lb> æquali cum velocitate moveantur: erit reſiſtentia Globi duplo <lb></lb> minor quam reſiſtentia Cylindri.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam quoniam actio Medii in corpus eadem eſt (per Legum <lb></lb> Corol, 5.) ſive corpus in Medio quieſcente moveatur, ſive Medii <lb></lb> particulæ eadem cum velocitate impingant in corpus quieſcens: <lb></lb> conſideremus corpus tanquam quieſcens, & videamus quo impetu <lb></lb> urgebitur a Medio movente. <lb></lb> <figure id="id.039.01.326.1.jpg" xlink:href="039/01/326/1.jpg"></figure><lb></lb>Deſignet igitur <emph type="italics"></emph>ABKI<emph.end type="italics"></emph.end>cor<lb></lb>pus Sphæricum centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ſe<lb></lb>midiametro <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>deſcriptum, <lb></lb> & incidant particulæ Medii <lb></lb> data cum velocitate in cor<lb></lb>pus illud Sphæricum, ſecun<lb></lb>dum rectas ipſi <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>paralle<lb></lb>las: Sitque <emph type="italics"></emph>FB<emph.end type="italics"></emph.end>ejuſmodi <lb></lb> recta. </s> <s>In ea capiatur <emph type="italics"></emph>LB<emph.end type="italics"></emph.end><lb></lb>ſemidiametro <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>æqualis, <lb></lb> & ducatur <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>quæ Sphæram tangat in <emph type="italics"></emph>B.<emph.end type="italics"></emph.end>In <emph type="italics"></emph>KC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>de-<pb xlink:href="039/01/327.jpg" pagenum="299"></pb><lb></lb>mittantur perpendiculares <emph type="italics"></emph>BE, DL,<emph.end type="italics"></emph.end>& vis qua particula Medii, <lb></lb> <arrow.to.target n="note275"></arrow.to.target>ſecundum rectam <emph type="italics"></emph>FB<emph.end type="italics"></emph.end>obliQ.E.I.cidendo, Globum ferit in <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>erit <lb></lb> ad vim qua particula eadem Cylindrum <emph type="italics"></emph>ONGQ<emph.end type="italics"></emph.end>axe <emph type="italics"></emph>ACI<emph.end type="italics"></emph.end>circa <lb></lb> Globum deſcriptum perpendiculariter feriret in <emph type="italics"></emph>b,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>LD<emph.end type="italics"></emph.end>ad <lb></lb> <emph type="italics"></emph>LB<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BC.<emph.end type="italics"></emph.end>Rurſus efficacia hujus vis ad movendum <lb></lb> Globum ſecundum incidentiæ ſuæ plagam <emph type="italics"></emph>FB<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>eſt ad ejuſ<lb></lb>dem efficaciam ad movendum Globum ſecundum plagam determi<lb></lb>nationis ſuæ, id eſt, ſecundum plagam rectæ <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>qua Globum di<lb></lb>recte urget, ut <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BC.<emph.end type="italics"></emph.end>Et conjunctis rationibus, efficacia <lb></lb> particulæ, in Globum ſecundum rectam <emph type="italics"></emph>FB<emph.end type="italics"></emph.end>obliQ.E.I.cidentis, ad <lb></lb> movendum eundem ſecundum plagam incidentiæ ſuæ, eſt ad effi<lb></lb>caciam particulæ ejuſdem ſecundum eandem rectam in Cylindrum <lb></lb> perpendiculariter incidentis, ad ipſum movendum in plagam ean<lb></lb>dem, ut <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>quadratum ad <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>quadratum. </s> <s>Quare ſi ad Cylin<lb></lb>dri baſem circularem <emph type="italics"></emph>NAO<emph.end type="italics"></emph.end>erigatur perpendiculum <emph type="italics"></emph>bHE,<emph.end type="italics"></emph.end>& ſit <lb></lb> <emph type="italics"></emph>bE<emph.end type="italics"></emph.end>æqualis radio <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>bH<emph.end type="italics"></emph.end>æqualis (<emph type="italics"></emph>BE quad/CB<emph.end type="italics"></emph.end>): erit <emph type="italics"></emph>bH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>bE<emph.end type="italics"></emph.end><lb></lb>ut effectus particulæ in Globum ad effectum particulæ in Cylin<lb></lb>drum. </s> <s>Et propterea ſolidum quod à rectis omnibus <emph type="italics"></emph>bH<emph.end type="italics"></emph.end>occu<lb></lb>patur erit ad ſolidum quod à rectis omnibus <emph type="italics"></emph>bE<emph.end type="italics"></emph.end>occupatur, ut <lb></lb> effectus particularum omnium in Globum ad effectum particu<lb></lb>larum omnium in Cylindrum. </s> <s>Sed ſolidum prius eſt Parabolois <lb></lb> vertice <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>axe <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>& latere recto <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>deſcriptum, & ſolidum <lb></lb> poſterius eſt Cylindrus Paraboloidi circumſcriptus: & notum eſt <lb></lb> quod Parabolois ſit ſemiſſis Cylindri circumſcripti. </s> <s>Ergo vis <lb></lb> tota Medii in Globum eſt duplo minor quam ejuſdem vis tota <lb></lb> in Cylindrum. </s> <s>Et propterea ſi particulæ Medii quieſcerent, & <lb></lb> Cylindrus ac Globus æquali cum velocitate moverentur, foret re<lb></lb>ſiſtentia Globi duplo minor quam reſiſtentia Cylindri. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end><lb></lb><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note275"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s>Eadem methodo Figuræ aliæ inter ſe quo<lb></lb><figure id="id.039.01.327.1.jpg" xlink:href="039/01/327/1.jpg"></figure><lb></lb>ad reſiſtentiam comparari poſſunt, eæQ.E.I.<lb></lb>veniri quæ ad motus ſuos in Mediis reſiſten<lb></lb>tibus continuandos aptiores ſunt. </s> <s>Ut ſi baſe <lb></lb> circulari <emph type="italics"></emph>CEBH,<emph.end type="italics"></emph.end>quæ centro <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>OC<emph.end type="italics"></emph.end><lb></lb>deſcribitur, & altitudine <emph type="italics"></emph>OD,<emph.end type="italics"></emph.end>conſtruen<lb></lb>dum ſit fruſtum Coni <emph type="italics"></emph>CBGF,<emph.end type="italics"></emph.end>quod omni<lb></lb>um eadem baſi & altitudine conſtructorum & ſecundum plagam <pb xlink:href="039/01/328.jpg" pagenum="300"></pb><lb></lb><arrow.to.target n="note276"></arrow.to.target>axis ſui verſus <emph type="italics"></emph>D<emph.end type="italics"></emph.end>progredientium fruſtorum minime reſiſtatur: bi<lb></lb>ſeca altitudinem <emph type="italics"></emph>OD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>& produc <emph type="italics"></emph>OQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>QS<emph.end type="italics"></emph.end>æqua<lb></lb>lis <emph type="italics"></emph>QC,<emph.end type="italics"></emph.end>& erit <emph type="italics"></emph>S<emph.end type="italics"></emph.end>vertex Coni cujus fruſtum quæritur. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note276"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s>Unde obiter, cum angulus <emph type="italics"></emph>CSB<emph.end type="italics"></emph.end>ſemper ſit acutus, conſequens <lb></lb> eſt, quod ſi ſolidum <emph type="italics"></emph>ADBE<emph.end type="italics"></emph.end>convolutione figuræ Ellipticæ vel <lb></lb> Ovalis <emph type="italics"></emph>ADBE<emph.end type="italics"></emph.end>circa axem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>facta generetur, & tangatur figura <lb></lb> generans à rectis tribus <emph type="italics"></emph>FG, GH, HI<emph.end type="italics"></emph.end>in punctis <emph type="italics"></emph>F, B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>ea <lb></lb> lege ut <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>ſit perpendicularis ad axem in puncto contactus <emph type="italics"></emph>B,<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>FG, HI<emph.end type="italics"></emph.end>cum eadem <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>contineant angulos <emph type="italics"></emph>FGB, BHI<emph.end type="italics"></emph.end><lb></lb>graduum 135: ſolidum, quod convolutione figuræ <emph type="italics"></emph>ADFGHIE<emph.end type="italics"></emph.end><lb></lb>circa axem eundem <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>generatur, minus reſiſtitur quam ſolidum <lb></lb> prius; ſi modo utrumque ſecundum plagam axis ſui <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>progre<lb></lb>diatur, & utriuſque terminus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>præcedat. </s> <s>Quam quidem propoſi<lb></lb>tionem in conſtruendis Navibus non inutilem futuram eſſe cenſeo. <lb></lb> </s></p> <p type="main"> <s>Quod ſi Figura <emph type="italics"></emph>DNFG<emph.end type="italics"></emph.end><lb></lb>ejuſmodi ſit curva ut, ſi ab <lb></lb> <figure id="id.039.01.328.1.jpg" xlink:href="039/01/328/1.jpg"></figure><lb></lb>ejus puncto quovis <emph type="italics"></emph>N<emph.end type="italics"></emph.end>ad <lb></lb> axem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>demittatur per<lb></lb>pendiculum <emph type="italics"></emph>NM,<emph.end type="italics"></emph.end>& à pun<lb></lb>cto dato <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ducatur recta <lb></lb> <emph type="italics"></emph>GR<emph.end type="italics"></emph.end>quæ parallela ſit rectæ <lb></lb> figuram tangenti in <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>& <lb></lb> axem productum ſecet in <lb></lb> <emph type="italics"></emph>R,<emph.end type="italics"></emph.end>fuerit <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GR<emph.end type="italics"></emph.end>ut <lb></lb> <emph type="italics"></emph>GR cub<emph.end type="italics"></emph.end>ad 4 <emph type="italics"></emph>BRXGBq<emph.end type="italics"></emph.end>: <lb></lb> Solidum quod figuræ hujus revolutione circa axem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>facta de<lb></lb>ſcribitur, in Medio raro prædicto ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>movendo, minus <lb></lb> reſiſtetur quam aliud quodvis eadem longitudine & latitudine de<lb></lb>ſcriptum Solidum circulare. <lb></lb> <emph type="center"></emph>PROPOSITIO XXXV. PROBLEMA VII.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Si Medium rarum ex particulis quam minimis quieſcentibus æqua<lb></lb>libus & ad æquales ab invicem diſtantias libere diſpoſitis con<lb></lb>ſtet: invenire reſiſtentiam Globi in hoc Medio uniformitor pro<lb></lb>gredientis.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Cylindrus eadem diametro & altitudine deſcriptus pro<lb></lb>gredi intelligatur eadem velocitate ſecundum longitudinem axis <lb></lb> ſui in eodem Medio. </s> <s>Et ponamus quod particulæ Medii in quas <pb xlink:href="039/01/329.jpg" pagenum="301"></pb><lb></lb>Globus vel Cylindrus incidit, vi reflexionis quam maxima reſiliant. <lb></lb> <arrow.to.target n="note277"></arrow.to.target>Et cum reſiſtentia Globi (per Propoſitionem noviſſimam) ſit duplo <lb></lb> minor quam reſiſtentia Cylindri, & Globus ſit ad Cylindrum ut <lb></lb> duo ad tria, & Cylindrus incidendo perpendiculariter in particulas <lb></lb> ipſaſque quam maxime reflectendo, duplam ſui ipſius velocitatem <lb></lb> ipſis communicet: Cylindrus quo tempore dimidiam longitudinem <lb></lb> axis ſui deſcribit communicabit motum particulis qui ſit ad totum <lb></lb> Cylindri motum ut denſitas Medii ad denſitatem Cylindri; & Glo<lb></lb>bus quo tempore totam longitudinem diametri ſuæ deſcribit, com<lb></lb>municabit motum eundem particulis; & quo tempore duas tertias <lb></lb> partes diametri ſuæ deſcribit communicabit motum particulis qui <lb></lb> ſit ad totum Globi motum ut denſitas Medii ad denſitatem Globi. <lb></lb> Et propterea Globus reſiſtentiam patitur quæ ſit ad vim qua totus <lb></lb> ejus motus vel auferri poſſit vel generari quo tempore duas tertias <lb></lb> partes diametri ſuæ deſcribit, ut denſitas Medii ad denſitatem <lb></lb> Globi. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note277"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Ponamus quod particulæ Medii in Globum vel Cylin<lb></lb>drum incidentes non reflectantur; & Cylindrus incidendo perpen<lb></lb>diculariter in particulas ſimplicem ſuam velocitatem ipſis commu<lb></lb>nicabit, ideoque reſiſtentiam patitur duplo minorem quam in pri<lb></lb>ore caſu, & reſiſtentia Globi erit etiam duplo minor quam prius. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Ponamus quod particulæ Medii vi reflexionis neque ma<lb></lb>xima neque nulla, ſed mediocri aliqua reſiliant a Globo; & reſi<lb></lb>ſtentia Globi erit in eadem ratione mediocri inter reſiſtentiam in <lb></lb> primo caſu & reſiſtentiam in ſecundo. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi Globus & particulæ ſint infinite dura, & vi om<lb></lb>ni elaſtica & propterea etiam vi omni reflexionis deſtituta: re<lb></lb>ſiſtentia Globi erit ad vim qua totus ejus motus vel auferri poſſit <lb></lb> vel generari, quo tempore Globus quatuor tertias partes diametri <lb></lb> ſuæ deſcribit, ut denſitas Medii ad denſitatem Globi. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Reſiſtentia Globi, cæteris paribus, eſt in duplicata ra<lb></lb>tione velocitatis. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Reſiſtentia Globi, cæteris paribus, eſt in duplicata ra<lb></lb>tione diametri. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Reſiſtentia Globi, cæteris paribus, eſt ut denſitas Medii. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Reſiſtentia Globi eſt in ratione quæ componitur ex du<lb></lb>plicata ratione velocitatis & duplicata ratione diametri & ratione <lb></lb> denſitatis Medii. <pb xlink:href="039/01/330.jpg" pagenum="302"></pb><lb></lb><arrow.to.target n="note278"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note278"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Et motus Globi cum ejus reſiſtentia ſic exponi poteſt. <lb></lb> Sit <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>tempus quo Globus per reſiſtentiam ſuam uniformiter con<lb></lb>tinuatam totum ſuum motum amit<lb></lb><figure id="id.039.01.330.1.jpg" xlink:href="039/01/330/1.jpg"></figure><lb></lb>tere poteſt. </s> <s>Ad <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>erigantur per<lb></lb>pendicula <emph type="italics"></emph>AD, BC.<emph.end type="italics"></emph.end>Sitque <emph type="italics"></emph>BC<emph.end type="italics"></emph.end><lb></lb>motus ille totus, & per punctum <emph type="italics"></emph>C<emph.end type="italics"></emph.end><lb></lb>Aſymptotis <emph type="italics"></emph>AD, AB<emph.end type="italics"></emph.end>deſcribatur <lb></lb> Hyperbola <emph type="italics"></emph>CF.<emph.end type="italics"></emph.end>Producatur <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <lb></lb> punctum quodvis <emph type="italics"></emph>E.<emph.end type="italics"></emph.end>Erigatur per<lb></lb>pendiculum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>Hyperbolæ occur<lb></lb>rens in <emph type="italics"></emph>F.<emph.end type="italics"></emph.end>Compleatur parallelo<lb></lb>grammum <emph type="italics"></emph>CBEG,<emph.end type="italics"></emph.end>& agatur <emph type="italics"></emph>AF<emph.end type="italics"></emph.end><lb></lb>ipſi <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>occurrens in <emph type="italics"></emph>H.<emph.end type="italics"></emph.end>Et ſi Globus tempore quovis <emph type="italics"></emph>BE,<emph.end type="italics"></emph.end>motu <lb></lb> ſuo primo <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>uniformiter continuato, in Medio non reſiſtente de<lb></lb>ſcribat ſpatium <emph type="italics"></emph>CBEG<emph.end type="italics"></emph.end>per aream parallelogrammi expoſitum, idem <lb></lb> in Medio reſiſtente deſcribet ſpatium <emph type="italics"></emph>CBEF<emph.end type="italics"></emph.end>per aream Hyper<lb></lb>bolæ expoſitum, & motus ejus in fine temporis illius exponetur <lb></lb> per Hyperbolæ ordinatam <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>amiſſa motus ejus parte <emph type="italics"></emph>FG.<emph.end type="italics"></emph.end>Et <lb></lb> reſiſtentia ejus in fine temporis ejuſdem exponetur per longitudi<lb></lb>nem <emph type="italics"></emph>BH,<emph.end type="italics"></emph.end>amiſſa reſiſtentiæ parte <emph type="italics"></emph>CH.<emph.end type="italics"></emph.end>Patent hæc omnia per <lb></lb> Corol. 1. Prop. v. Lib. II. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Hinc ſi Globus tempore T per reſiſtentiam R unifor<lb></lb>miter continuatam amittat motum ſuum totum M: idem Globus tem<lb></lb>pore <emph type="italics"></emph>t<emph.end type="italics"></emph.end>in Medio reſiſtente, per reſiſtentiam R in duplicata velocitatis <lb></lb> ratione decreſcentem, amittet motus ſui M partem (<emph type="italics"></emph>t<emph.end type="italics"></emph.end>M/T+<emph type="italics"></emph>t<emph.end type="italics"></emph.end>), manente <lb></lb> parte (TM/T+<emph type="italics"></emph>t<emph.end type="italics"></emph.end>), & deſcribet ſpatium quod ſit ad ſpatium motu uni<lb></lb>formi M eodem tempore <emph type="italics"></emph>t<emph.end type="italics"></emph.end>deſcriptum, ut Logarithmus numeri <lb></lb> (T+<emph type="italics"></emph>t<emph.end type="italics"></emph.end>/T) multiplicatus per numerum 2,302585092994 eſt ad nume<lb></lb>rum <emph type="italics"></emph>t<emph.end type="italics"></emph.end>/T. Nam area Hyperbolica <emph type="italics"></emph>BCFE<emph.end type="italics"></emph.end>eſt ad rectangulum <lb></lb> <emph type="italics"></emph>BCGE<emph.end type="italics"></emph.end>in hac proportione. <lb></lb> <emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s>In hac Propoſitione expoſui reſiſtentiam & retardationem Pro<lb></lb>jectilium Sphærieorum in Mediis non continuis, & oſtendi quod <lb></lb> hæc reſiſtentia ſit ad vim qua totus Globi motus vel tolli poſſit vel <pb xlink:href="039/01/331.jpg" pagenum="303"></pb><lb></lb>generari quo tempore Globus duas tertias diametri ſuæ partes, ve<lb></lb><arrow.to.target n="note279"></arrow.to.target>locitate uniformiter continuata deſcribat, ut denſitas Medii ad <lb></lb> denſitatem Globi, ſi modo Globus & particulæ Medii ſint ſumme <lb></lb> elaſtica & vi maxima reflectendi polleant: quodque hæc vis ſit <lb></lb> duplo minor ubi Globus & particulæ Medii ſunt infinite dura & <lb></lb> vi reflectendi prorſus deſtituta. </s> <s>In Medus autem continuis qualia <lb></lb> ſunt Aqua, Oleum calidum, & Argentum vivum, in quibus Globus <lb></lb> non incidit immediate in omnes fluidi particulas reſiſtentiam gene<lb></lb>rantes, ſed premit tantum proximas particulas & hæ premunt alias <lb></lb> & hæ alias, reſiſtentia eſt adhuc duplo minor. </s> <s>Globus utiQ.E.I. <lb></lb> hujuſmodi Mediis fluidiſſimis reſiſtentiam patitur quæ eſt ad vim <lb></lb> qua totus ejus motus vel tolli poſſit vel generari quo tempore, <lb></lb> motu illo uniformiter continuato, partes octo tertias diametri ſuæ <lb></lb> deſcribat, ut denſitas Medii ad denſitatem Globi. </s> <s>Id quod in ſe<lb></lb>quentibus conabimur oſtendere. <lb></lb> <emph type="center"></emph>PROPOSITIO XXXVI. PROBLEMA VIII.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note279"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Aquæ de vaſe Cylindrico per foramen in fundo factum effluentis <lb></lb> definire motum.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>ACDB<emph.end type="italics"></emph.end>vas cylindricum, <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ejus orificium ſuperius, <emph type="italics"></emph>CD<emph.end type="italics"></emph.end><lb></lb>fundum horizonti parallelum, <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>foramen circulare in medio <lb></lb> fundi, <emph type="italics"></emph>G<emph.end type="italics"></emph.end>centrum foraminis, & <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>axis cylindri horizonti per<lb></lb>pendicularis. </s> <s>Et concipe cylindrum gla<lb></lb><figure id="id.039.01.331.1.jpg" xlink:href="039/01/331/1.jpg"></figure><lb></lb>ciei <emph type="italics"></emph>APQB<emph.end type="italics"></emph.end>ejuſdem eſſe latitudinis <lb></lb> cum cavitate vaſis, & axem eundem ha<lb></lb>bere, & uniformi cum motu perpetuo <lb></lb> deſcendere, & partes ejus quam primum <lb></lb> attingunt ſuperficiem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>liqueſcere, & <lb></lb> in aquam converſas gravitate ſua defluere <lb></lb> in vas, & cataractam vel columnam aquæ <lb></lb> <emph type="italics"></emph>ABNFEM<emph.end type="italics"></emph.end>cadendo formare, & per <lb></lb> foramen <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>tranſire, idemque adæquate <lb></lb> implere. </s> <s>Ea vero ſit uniformis veloci<lb></lb>tas glaciei deſcendentis ut & aquæ con<lb></lb>tiguæ in circulo <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>quam aqua caden<lb></lb>do & caſu ſuo deſcribendo altitudinem <lb></lb> <emph type="italics"></emph>IH<emph.end type="italics"></emph.end>acquirere poteſt; & jaceant <emph type="italics"></emph>IH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>HG<emph.end type="italics"></emph.end>in directum, & per <lb></lb> punctum <emph type="italics"></emph>I<emph.end type="italics"></emph.end>ducatur recta <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>horizonti parallela & lateribus gla-<pb xlink:href="039/01/332.jpg" pagenum="304"></pb><lb></lb><arrow.to.target n="note280"></arrow.to.target>ciei occurrens in <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>L.<emph.end type="italics"></emph.end>Et velocitas aquæ effluentis per fora<lb></lb>men <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ea erit quam aqua cadendo ab <emph type="italics"></emph>I<emph.end type="italics"></emph.end>& caſu ſuo deſcribendo <lb></lb> altitudinem <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>acquirere poteſt. </s> <s>Ideoque per Theoremata <emph type="italics"></emph>Galilæi<emph.end type="italics"></emph.end><lb></lb>erit <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>IH<emph.end type="italics"></emph.end>in duplicata ratione velocitatis aquæ per foramen <lb></lb> effluentis ad velocitatem aquæ in circulo <emph type="italics"></emph>AB,<emph.end type="italics"></emph.end>hoc eſt, in dupli<lb></lb>cata ratione circuli <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad circulum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>; nam hi circuli ſunt re<lb></lb>ciproce ut velocitates aquarum quæ per ipſos, eodem tempore & <lb></lb> æquali quantitate, adæquate tranſeunt. </s> <s>De velocitate aquæ hori<lb></lb>zontem verſus hic agitur. </s> <s>Et motus horizonti parallelus quo par<lb></lb>tes aquæ cadentis ad invicem accedunt, cum non oriatur a gravi<lb></lb>tate, nec motum horizonti perpendicularem à gravitate oriundum <lb></lb> mutet, hic non conſideratur. </s> <s>Supponimus quidem quod partes <lb></lb> aquæ aliquantulum cohærent, & per cohæſionem ſuam inter ca<lb></lb>dendum accedant ad invicem per motus horizonti parallelos, ut <lb></lb> unicam tantum efforment cataractam & non in plures cataractas <lb></lb> dividantur: ſed motum horizonti parallelum, a cohæſione illa ori<lb></lb>undum, hic non conſideramus. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note280"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Concipe jam cavitatem totam in vaſe, in circuitu aquæ <lb></lb> cadentis <emph type="italics"></emph>ABNFEM,<emph.end type="italics"></emph.end>glacie plenam eſſe, ut aqua per glaciem <lb></lb> tanquam per infundibulum tranſeat. </s> <s>Et ſi aqua glaciem tantum <lb></lb> non tangat vel, quod perinde eſt, ſi tangat & per glaciem propter <lb></lb> ſummam ejus polituram quam liberrime & ſine omni reſiſtentia la<lb></lb>batur; hæc defluet per foramen <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>eadem velocitate ac prius, & <lb></lb> pondus totum columnæ aquæ <emph type="italics"></emph>ABNFEM<emph.end type="italics"></emph.end>impendetur in deflu<lb></lb>xum ejus generandum uti prius, & fundum vaſis ſuſtinebit pon<lb></lb>dus glaciei columnam ambientis. <lb></lb> </s></p> <p type="main"> <s>Liqueſcat jam glacies in vaſe; & effluxus aquæ quoad velocita<lb></lb>tem, idem manebit ac prius. </s> <s>Non minor erit, quia glacies in aquam <lb></lb> reſoluta conabitur deſcendere: non major, quia glacies in aquam <lb></lb> reſoluta non poteſt deſcendere niſi impediendo deſcenſum aquæ <lb></lb> alterius deſcenſui ſuo æqualem. </s> <s>Eadem vis eandem aquæ effluen<lb></lb>tis velocitatem generare debet. <lb></lb> </s></p> <p type="main"> <s>Sed foramen in fundo vaſis, propter obliquos motus particula<lb></lb>rum aquæ effluentis, paulo majus eſſe debet quam prius. </s> <s>Nam par<lb></lb>ticulæ aquæ jam non tranſeunt omnes per foramen perpendicula<lb></lb>riter; ſed a lateribus vaſis undique confluentes & in foramen con<lb></lb>vergentes, obliquis tranſeunt motibus; & curſum ſuum deorſum <lb></lb> flectentes in venam aquæ exilientis conſpirant, quæ exilior eſt pau<lb></lb>lo infra foramen quam in ipſo foramine, exiſtente ejus diametro <lb></lb> ad diametrum foraminis ut 5 ad 6, vel 5 1/2 ad 6 1/2 quam proxime, ſi <pb xlink:href="039/01/333.jpg" pagenum="305"></pb><lb></lb>modo diametros recte dimenſus ſum. </s> <s>Parabam utique laminam <lb></lb> <arrow.to.target n="note281"></arrow.to.target>planam pertenuem in medio perforatam, exiſtente circularis fora<lb></lb>minis diametro partium quinque octavarum digiti. </s> <s>Et ne vena <lb></lb> aquæ exilientis cadendo acceleraretur & acceleratione redderetur <lb></lb> anguſtior, hanc laminam non fundo ſed lateri vaſis affixi ſic, ut <lb></lb> vena illa egrederetur ſecundum lineam horizonti parallelam. </s> <s>Dein <lb></lb> ubi vas aquæ plenum eſſet, aperui foramen ut aqua efflueret; & <lb></lb> venæ diameter, ad diſtantiam quaſi dimidii digiti â ſoramine quam <lb></lb> accuratiſſime menſurata, prodiit partium viginti & unius quadrageſi<lb></lb>marum digiti. </s> <s>Erat igitur diameter foraminis hujus circularis ad <lb></lb> diametrum venæ ut 25 ad 21 quamproxime. </s> <s>Per experimenta vero <lb></lb> conſtat quod quantitas aquæ quæ per foramen circulare in fundo <lb></lb> vaſis factum effluit, ea eſt quæ, pro diametro venæ, cum velocitate <lb></lb> prædicta effluere debet. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note281"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s>In ſequentibus igitur, plano foraminis parallelum duci intelliga<lb></lb>tur planum aliud ſuperius ad diſtantiam diametro foraminis æqua<lb></lb>lem vel paulo majorem & foramine majore pertuſum, per quod <lb></lb> utique vena cadat quæ adæquate impleat <lb></lb> <figure id="id.039.01.333.1.jpg" xlink:href="039/01/333/1.jpg"></figure><lb></lb>foramen inferius <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>atque adeo cujus <lb></lb> diameter ſit ad diametrum foraminis in<lb></lb>ferioris ut 25 ad 21 circiter. </s> <s>Sic enim <lb></lb> vena per foramen inferius perpendicu<lb></lb>lariter tranſibit; & quantitas aquæ ef<lb></lb>fluentis, pro magnitudine foraminis hu<lb></lb>jus, ea erit quam ſolutio Problematis po<lb></lb>ſtulat quamproxime. </s> <s>Spatium vero quod <lb></lb> planis duobus & vena cadente clauditur, <lb></lb> pro fundo vaſis haberi poteſt. </s> <s>Sed ut <lb></lb> ſolutio Problematis ſimplicior ſit & ma<lb></lb>gis Mathematica, præſtat adhibere pla<lb></lb>num ſolum inferius pro fundo vaſis, & <lb></lb> fingere quod aqua quæ per glaciem ceu per infundibulum deflue<lb></lb>bat, & è vaſe per foramen <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>egrediebatur, motum ſuum per<lb></lb>petuo ſervet & glacies quietem ſuam etiamſ in aquam fluidam <lb></lb> reſolvatur. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Si foramen <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>non ſit in medio fundi vaſis, ſed fun<lb></lb>dum alibi perforetur: aqua effluet eadem cum velocitate ac prius, <lb></lb> ſi modo eadem ſit foraminis magnitudo. </s> <s>Nam grave majori qui<lb></lb>dem tempore deſcendit ad eandem profunditatem per lineam ob<lb></lb>liquam quam per lineam perpendicularem, ſed deſcendendo ean-<pb xlink:href="039/01/334.jpg" pagenum="306"></pb><lb></lb><arrow.to.target n="note282"></arrow.to.target>dem velocitatem acquirit in utroque caſu, ut <emph type="italics"></emph>Galilæus<emph.end type="italics"></emph.end>demon<lb></lb>ſtravit. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note282"></margin.target>DE MOTU <lb></lb> CORPORUM.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Eadem eſt aquæ velocitas effluentis per foramen in la<lb></lb>tere vaſis. </s> <s>Nam ſi foramen parvum ſit, ut intervallum inter ſuper<lb></lb>ficies <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>quoad ſenſum evaneſcat, & vena aquæ hori<lb></lb>zontaliter exilientis figuram Parabolicam efformet: ex latere recto <lb></lb> hujus Parabolæ colligetur, quod velocitas aquæ effluentis ea ſit <lb></lb> quam corpus ab aquæ in vaſe ſtagnantis altitudine <emph type="italics"></emph>HG<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>ca<lb></lb>dendo acquirere potuiſſet. </s> <s>Facto utique experimento inveni quod, <lb></lb> ſi altitudo aquæ ſtagnantis ſupra foramen eſſet viginti digitorum <lb></lb> & altitudo foraminis ſupra planum horizonti parallelum eſſet quo<lb></lb>que viginti digitorum, vena aquæ proſilientis incideret in planum <lb></lb> illud ad diſtantiam digitorum 37 circiter à perpendiculo quod in <lb></lb> planum illud à foramine demittebatur captam. </s> <s>Nam ſine reſiſten<lb></lb>tia vena incidere debuiſſet in planum illud ad diſtantiam digitorum <lb></lb> 40, exiſtente venæ Parabolicæ latere recto digitorum 80. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>4. Quinetiam aqua effluens, ſi ſurſum feratur, eadem egre<lb></lb>ditur cum velocitate. </s> <s>Aſcendit enim aquæ exilientis vena parva <lb></lb> motu perpendiculari ad aquæ in vaſe ſtagnantis altitudinem <emph type="italics"></emph>GH<emph.end type="italics"></emph.end><lb></lb>vel <emph type="italics"></emph>GI,<emph.end type="italics"></emph.end>niſi quatenus aſcenſus ejus ab aeris reſiſtentia aliquantu<lb></lb>lum impediatur; ac proinde ea effluit cum velocitate quam ab al<lb></lb>titudine illa cadendo acquirere potuiſſet. <lb></lb> <figure id="id.039.01.334.1.jpg" xlink:href="039/01/334/1.jpg"></figure><lb></lb>Aquæ ſtagnantis particula unaquæque <lb></lb> undique premitur æqualiter, per Prop. <lb></lb> XIX. Lib. II, & preſſioni cedendo æquali <lb></lb> impetu in omnes partes fertur, ſive de<lb></lb>ſcendat per foramen in fundo vaſis, ſive <lb></lb> horizontaliter effluat per foramen in ejus <lb></lb> latere, ſive egrediatur in canalem & inde <lb></lb> aſcendat per foramen parvum in ſuperiore <lb></lb> canalis parte factum. </s> <s>Et velocitatem qua <lb></lb> aqua effluit, eam eſſe quam in hac Pro<lb></lb>poſitione aſſignavimus, non ſolum rati<lb></lb>one colligitus, ſed etiam per experimenta <lb></lb> notiſſima jam deſcripta manifeſtum eſt. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>5. Eadem eſt aquæ effluentis velocitas ſive figura foraminis <lb></lb> ſit circularis ſive quadrata vel triangularis aut alia quæcunque cir<lb></lb>culari æqualis. </s> <s>Nam velocitas aquæ effluentis non pendet à figura <lb></lb> foraminis ſed ab ejus altitudine infra planum <emph type="italics"></emph>KL.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>6. Si vaſis <emph type="italics"></emph>ABDC<emph.end type="italics"></emph.end>pars inferior in aquam ſtagnantem im-<pb xlink:href="039/01/335.jpg" pagenum="307"></pb><lb></lb>mergatur, & altitudo aquæ ſtagnantis ſupra fundum vaſis ſit <emph type="italics"></emph>GR<emph.end type="italics"></emph.end>: <lb></lb> <arrow.to.target n="note283"></arrow.to.target>velocitas quacum aqua quæ in vaſe eſt, effluet per foramen <emph type="italics"></emph>EF<emph.end type="italics"></emph.end><lb></lb>in aquam ſtagnantem, ea erit quam aqua cadendo & caſu ſuo de<lb></lb>ſcribendo altitudinem <emph type="italics"></emph>IR<emph.end type="italics"></emph.end>acquirere poteſt. </s> <s>Nam pondus aquæ <lb></lb> omnis in vaſe quæ inferior eſt ſuperficie aquæ ſtagnantis, ſuſtine<lb></lb>bitur in æquilibrio per pondus aquæ ſtagnantis, ideoque motum <lb></lb> aquæ deſcendentis in vaſe minime accelerabit. </s> <s>Patebit etiam & <lb></lb> hic Caſus per Experimenta, menſurando ſcilicet tempora qui<lb></lb>bus aqua effluit. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note283"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi aquæ altitudo <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>producatur ad <emph type="italics"></emph>K,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>AK<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>in duplicata ratione areæ foraminis in quavis fundi parte <lb></lb> facti, ad aream circuli <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>: velocitas aquæ effluentis æqualis erit <lb></lb> velocitati quam aqua cadendo & caſu ſuo deſcribendo altitudinera <lb></lb> <emph type="italics"></emph>KC<emph.end type="italics"></emph.end>acquirere poteſt. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et vis qua totus aquæ exilientis motus generari poteſt, <lb></lb> æqualis eſt ponderi Cylindricæ columnæ aquæ cujus baſis eſt fora<lb></lb>men <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>& altitudo 2<emph type="italics"></emph>GI<emph.end type="italics"></emph.end>vel 2<emph type="italics"></emph>CK.<emph.end type="italics"></emph.end>Nam aqua exiliens quo <lb></lb> tempore hanc columnam æquat, pondere ſuo ab altitudine <emph type="italics"></emph>GI<emph.end type="italics"></emph.end>ca<lb></lb>dendo, velocitatem ſuam qua exilit, acquirere poteſt. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Pondus aquæ totius in vaſe <emph type="italics"></emph>ABDC,<emph.end type="italics"></emph.end>eſt ad ponderis <lb></lb> partem quæ in defluxum aquæ impenditur, ut ſumma circulorum <lb></lb> <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ad duplum circulum <emph type="italics"></emph>EF.<emph.end type="italics"></emph.end>Sit enim <emph type="italics"></emph>IO<emph.end type="italics"></emph.end>media pro<lb></lb>portionalis inter <emph type="italics"></emph>IH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>; & aqua per foramen <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>egrediens, <lb></lb> quo tempore gutta cadendo ab <emph type="italics"></emph>I<emph.end type="italics"></emph.end>deſcribere poſſet altitudinem <emph type="italics"></emph>IG,<emph.end type="italics"></emph.end><lb></lb>æqualis erit Cylindro cujus baſis eſt circulus <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>& altitudo eſt 2<emph type="italics"></emph>IG,<emph.end type="italics"></emph.end><lb></lb>id eſt, Cylindro cujus baſis eſt circulus <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& altitudo eſt 2<emph type="italics"></emph>IO,<emph.end type="italics"></emph.end><lb></lb>nam circulus <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>eſt ad circulum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>in ſubduplicata ratione <lb></lb> altitudinis <emph type="italics"></emph>IH<emph.end type="italics"></emph.end>ad altitudinem <emph type="italics"></emph>IG,<emph.end type="italics"></emph.end>hoc eſt, in ſimplici ratione me<lb></lb>diæ proportionalis <emph type="italics"></emph>IO<emph.end type="italics"></emph.end>ad altitudinem <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>: & quo tempore gutta <lb></lb> cadendo ab <emph type="italics"></emph>I<emph.end type="italics"></emph.end>deſcribere poteſt altitudinem <emph type="italics"></emph>IH,<emph.end type="italics"></emph.end>aqua egrediens <lb></lb> æqualis erit Cylindro cujus baſis eſt circulus <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& altitudo eſt <lb></lb> 2<emph type="italics"></emph>IH<emph.end type="italics"></emph.end>: & quo tempore gutta cadendo ab <emph type="italics"></emph>I<emph.end type="italics"></emph.end>per <emph type="italics"></emph>H<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>G<emph.end type="italics"></emph.end>deſcribit <lb></lb> altitudinum differentiam <emph type="italics"></emph>HG,<emph.end type="italics"></emph.end>aqua egrediens, id eſt, aqua tota in <lb></lb> ſolido <emph type="italics"></emph>ABNFEM<emph.end type="italics"></emph.end>æqualis erit differentiæ Cylindrorum, id eſt, <lb></lb> Cylindro cujus baſis eſt <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& altitudo 2<emph type="italics"></emph>HO.<emph.end type="italics"></emph.end>Et propterea <lb></lb> aqua tota in vaſe <emph type="italics"></emph>ABDC<emph.end type="italics"></emph.end>eſt ad aquam totam cadentem in <lb></lb> ſolido <emph type="italics"></emph>ABNFEM<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>HG<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>HO,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>HO+OG<emph.end type="italics"></emph.end><lb></lb>ad 2<emph type="italics"></emph>HO,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>IH+IO<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>IH.<emph.end type="italics"></emph.end>Sed pondus aquæ totius in <lb></lb> ſolido <emph type="italics"></emph>ABNFEM<emph.end type="italics"></emph.end>in aquæ defluxum impenditur: ac pro-<pb xlink:href="039/01/336.jpg" pagenum="308"></pb><lb></lb><arrow.to.target n="note284"></arrow.to.target>inde pondus aquæ totius in vaſe eſt ad ponderis partem quæ in <lb></lb> defluxum aquæ impenditur, ut <emph type="italics"></emph>IH+IO<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>IH,<emph.end type="italics"></emph.end>atque adeo ut <lb></lb> ſumma circulorum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad duplum circulum <emph type="italics"></emph>EF.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note284"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Et hinc pondus aquæ totius in vaſe <emph type="italics"></emph>ABDC,<emph.end type="italics"></emph.end>eſt ad <lb></lb> ponderis partem alteram quam fundum vaſis ſuſtinet, ut ſumma <lb></lb> circulorum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ad differentiam eorundem circulorum. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et ponderis pars quam fundum vaſis ſuſtinet, eſt ad <lb></lb> ponderis partem alteram quæ in defluxum aquæ impenditur, ut <lb></lb> differentia circulorum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ad duplum circulum minorem <lb></lb> <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ſive ut area fundi ad duplum foramen. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Ponderis autem pars qua ſola fundum urgetur, eſt ad <lb></lb> pondus aquæ totius quæ fundo perpendiculariter incumbit, ut cir<lb></lb>culus <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad ſummam circulorum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ſive ut circulus <lb></lb> <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad exceſſum dupli circuli <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ſupra fundum. </s> <s>Nam ponderis <lb></lb> pars qua ſola fundum urgetur, eſt ad pondus aquæ totius in vaſe, <lb></lb> ut differentia circulorum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ad ſummam eorundem cir<lb></lb>culorum, per Cor.4; & pondus aquæ totius in vaſe eſt ad pondus <lb></lb> aquæ totius quæ fundo perpendiculariter incumbit, ut circulus <lb></lb> <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad differentiam circulorum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF.<emph.end type="italics"></emph.end>Itaque ex æquo <lb></lb> perturbate, ponderis pars qua ſola fundum urgetur, eſt ad pondus <lb></lb> aquæ totius quæ fundo perpendiculariter incumbit, ut circulus <lb></lb> <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad ſummam circulorum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>vel exceſſum dupli cir<lb></lb>culi <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ſupra fundum. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Si in medio foraminis <emph type="italics"></emph>EF<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.336.1.jpg" xlink:href="039/01/336/1.jpg"></figure><lb></lb>locetur Circellus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>centro <emph type="italics"></emph>G<emph.end type="italics"></emph.end>deſcri<lb></lb>ptus & horizonti parallelus: pondus <lb></lb> aquæ quam circellus ille ſuſtinet, majus <lb></lb> eſt pondere tertiæ partis Cylindri a<lb></lb>quæ cujus baſis eſt circellus ille & al<lb></lb>titudo eſt <emph type="italics"></emph>GH.<emph.end type="italics"></emph.end>Sit enim <emph type="italics"></emph>ABNFEM<emph.end type="italics"></emph.end><lb></lb>cataracta vel columna aquæ cadentis <lb></lb> axem habens <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>ut ſupra, & conge<lb></lb>lari intelligatur aqua omnis in vaſe, tam <lb></lb> in circuitu cataractæ quam ſupra cir<lb></lb>cellum, cujus fluiditas ad promptiſſimum <lb></lb> & celerrimum aquæ deſcenſum non requiritur. </s> <s>Et ſit <emph type="italics"></emph>PHQ<emph.end type="italics"></emph.end>co<lb></lb>lumna aquæ ſupra circellum congelata, verticem habens <emph type="italics"></emph>H<emph.end type="italics"></emph.end>& alti<lb></lb>tudinem <emph type="italics"></emph>GH.<emph.end type="italics"></emph.end>Et quemadmodum aqua in circuitu cataractæ con<lb></lb>gelata <emph type="italics"></emph>AMEC, BNFD<emph.end type="italics"></emph.end>convexa eſt in ſuperficie interna <emph type="italics"></emph>AME, <lb></lb> BNF<emph.end type="italics"></emph.end>verſus cataractam cadentem, ſic etiam hæc columna <emph type="italics"></emph>PHQ<emph.end type="italics"></emph.end><pb xlink:href="039/01/337.jpg" pagenum="309"></pb><lb></lb>convexa erit verſus cataractam, & propterea major Cono cujus ba<lb></lb><arrow.to.target n="note285"></arrow.to.target>ſis eſt circellus ille <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>& altitudo <emph type="italics"></emph>GH,<emph.end type="italics"></emph.end>id eſt, major tertia parte <lb></lb> Cylindri eadem baſe & altitudine deſcripti. </s> <s>Suſtinet autem cir<lb></lb>cellus ille pondus hujus columnæ, id eſt, pondus quod pondere <lb></lb> Coni ſeu tertiæ partis Cylindri illius majus eſt. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note285"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Pondus aquæ quam circellus valde parvus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ſuſtinet, <lb></lb> minor eſt pondere duarum tertiarum partium Cylindri aquæ cujus <lb></lb> baſis eſt circellus ille & altitudo eſt <emph type="italics"></emph>HG.<emph.end type="italics"></emph.end>Nam ſtantibus jam po<lb></lb>ſitis, deſcribi intelligatur dimidium Sphæroidis cujus baſis eſt cir<lb></lb>cellus ille & ſemiaxis ſive altitudo eſt <emph type="italics"></emph>HG.<emph.end type="italics"></emph.end>Et hæc figura æqualis <lb></lb> erit duabus tertiis partibus Cylindri illius & comprehendet colum<lb></lb>nam aquæ congelatæ <emph type="italics"></emph>PHQ<emph.end type="italics"></emph.end>cujus pondus circellus ille ſuſtinet. <lb></lb> Nam ut motus aquæ ſit maxime directus, columnæ illius ſuper<lb></lb>ficies externa concurret cum baſi <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>in angulo nonnihil acuto, <lb></lb> propterea quod aqua cadendo perpetuo acceleratur & propter ac<lb></lb>celerationem fit tenuior; & cum angulus ille ſit recto minor, hæc <lb></lb> columna ad inferiores ejus partes jacebit intra dimidium Sphæroi<lb></lb>dis. </s> <s>Eadem vero ſurſum acuta erit ſeu cuſpidata, ne horizontalis <lb></lb> motus aquæ ad verticem Sphæroidis ſit infinite velocior quam ejus <lb></lb> motus horizontem verſus. </s> <s>Et quo minor eſt circellus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>eo <lb></lb> acutior erit vertex columnæ; & circello in infinitum diminuto, an<lb></lb>gulus <emph type="italics"></emph>PHQ<emph.end type="italics"></emph.end>in infinitum diminuetur, & propterea columna ja<lb></lb>cebit intra dimidium Sphæroidis. </s> <s>Eſt igitur columna illa minor <lb></lb> dimidio Sphæroidis, ſeu duabus tertiis partibus Cylindri cujus baſis <lb></lb> eſt circellus ille & altitudo <emph type="italics"></emph>GH.<emph.end type="italics"></emph.end>Suſtinet autem circellus vim aquæ <lb></lb> ponderi hujus columnæ æqualem, cum pondus aquæ ambientis in <lb></lb> defluxum ejus impendatur. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>9. Pondus aquæ quam circellus valde parvus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ſuſti<lb></lb>net, æquale ſet ponderi Cylindri aquæ cujus baſis eſt circellus ille <lb></lb> & altitudo eſt 1/2<emph type="italics"></emph>GH<emph.end type="italics"></emph.end>quamproxime. </s> <s>Nam pondus hocce eſt me<lb></lb>dium Arithmeticum inter pondera Coni & Hemiſphæroidis præ<lb></lb>dictæ. At ſi circellus ille non ſit valde parvus, ſed augeatur donec <lb></lb> æquet foramen <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>; hic ſuſtinebit pondus aquæ totius ſibi per<lb></lb>pendiculariter imminentis, id eſt, pondus Cylindri aquæ cujus ba<lb></lb>ſis eſt circellus ille & altitudo eſt <emph type="italics"></emph>GH.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>10. Et (quantum ſentio) pondus quod circellus ſuſtinet, <lb></lb> eſt ſemper ad pondus Cylindri aquæ cujus baſis eſt circellus ille & <lb></lb> altitudo eſt 1/2<emph type="italics"></emph>GH,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>-1/2<emph type="italics"></emph>PQq,<emph.end type="italics"></emph.end>ſive ut circulus <lb></lb> <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ad exceſſum circuli hujus ſupra ſemiſſem circelli <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>quam<lb></lb>proxime. <pb xlink:href="039/01/338.jpg" pagenum="310"></pb><lb></lb><arrow.to.target n="note286"></arrow.to.target><emph type="center"></emph>LEMMA IV.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note286"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cylindri, qui ſecundum longitudinem ſuam uniformiter progreditur, <lb></lb> reſiſtentia ex aucta vel diminuta ejus longitudine non mutatur; <lb></lb> ideoque eadem eſt cum reſiſtentia Circuli eadem diametro de<lb></lb>ſcripti & eadem velocitate ſecundum lineam rectam plano ip<lb></lb>ſius perpendicularem progredientis.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam latera Cylindri motui ejus minime opponuntur: & Cy<lb></lb>lindrus, longitudine ejus in infinitum diminuta, in Circulum <lb></lb> vertitur. <lb></lb> <emph type="center"></emph>PROPOSITIO XXXVII. THEOREMA XXIX.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Cylindri, qui in fluide compreſſo infinito & non elaſtico ſecundum <lb></lb> longitudinem ſuam uniformiter progreditur, reſiſtentia quæ ori<lb></lb>tur a magnitudine ſectionis tranſverſæ, eſt ad vim qua totus <lb></lb> ejus motus interea dum quadruplum longitudinis ſuæ deſcribit, <lb></lb> vel tolli poſſit vel generari, ut denſitas Medii ad denſitatem <lb></lb> Cylindri quamproxime.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam ſi vas <emph type="italics"></emph>ABDC<emph.end type="italics"></emph.end>fundo ſuo <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>ſuperficiem aquæ ſtagnan<lb></lb>tis tangat, & aqua ex hoc vaſe per ca<lb></lb><figure id="id.039.01.338.1.jpg" xlink:href="039/01/338/1.jpg"></figure><lb></lb>nalem Cylindricum <emph type="italics"></emph>EFTS<emph.end type="italics"></emph.end>horizonti <lb></lb> perpendicularem in aquam ſtagnantem <lb></lb> effluat, locetur autem Circellus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ho<lb></lb>rizonti parallelus ubivis in medio ca<lb></lb>nalis, & producatur <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>K,<emph.end type="italics"></emph.end>ut ſit <lb></lb> <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>in duplicata ratione quam <lb></lb> habet exceſſus orificii canalis <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ſupra <lb></lb> circellum <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad circulum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>: mani<lb></lb>feſtum eſt (per Caſ.5, Caſ.6, & Cor. 1. <lb></lb> Prop.XXXVI.) quod velocitas aquæ tran<lb></lb>ſeuntis per ſpatium annulare inter cir<lb></lb>cellum & latera vaſis, ea erit quam aqua <lb></lb> cadendo & caſu ſuo deſcribendo altitudinem <emph type="italics"></emph>KC<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>acquirere <lb></lb> poteſt. <pb xlink:href="039/01/339.jpg" pagenum="311"></pb><lb></lb></s></p> <p type="main"> <s>Et (per Cor. 10, Prop.XXXVI) ſi vaſis latitudo ſit infinita, ut li<lb></lb><arrow.to.target n="note287"></arrow.to.target>neola <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>evaneſcat & altitudines <emph type="italics"></emph>IG, HG<emph.end type="italics"></emph.end>æquentur: vis aquæ <lb></lb> defluentis in circellum erit ad pondus Cylindri cujus baſis eſt cir<lb></lb>cellus ille & altitudo eſt 1/2 <emph type="italics"></emph>IG,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>-1/2 <emph type="italics"></emph>PQq<emph.end type="italics"></emph.end>quam <lb></lb> proxime. </s> <s>Nam vis aquæ, uniformi motu defluentis per totum ca<lb></lb>nalem, eadem erit in circellum <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>in quacunque canalis parte <lb></lb> locatum. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note287"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s>Claudantur jam canalis orificia <emph type="italics"></emph>EF, ST,<emph.end type="italics"></emph.end>& aſcendat circellus in <lb></lb> fluido undique compreſſo & aſcenſu ſuo cogat aquam ſuperiorem <lb></lb> deſcendere per ſpatium annulare inter circellum & latera cana<lb></lb>lis: & velocitas circelli aſcendentis erit ad velocitatem aquæ <lb></lb> deſcendentis ut differentia circulorum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad circulum <lb></lb> <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>& velocitas circelli aſcendentis ad ſummam velocitatum, <lb></lb> hoc eſt, ad velocitatem relativam aquæ deſcendentis qua præ<lb></lb>terfluit circellum aſcendentem, ut differentia circulorum <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>& <lb></lb> <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad circulum <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ſive ut <emph type="italics"></emph>EFq-PQq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq.<emph.end type="italics"></emph.end>Sit illa <lb></lb> velocitas relativa æqualis velocitati qua ſupra oſtenſum eſt <lb></lb> aquam tranſire per idem ſpatium annulare dum circellus interea <lb></lb> immotus manet, id eſt, velocitati quam aqua cadendo & caſu ſuo <lb></lb> deſcribendo altitudinem <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>acquirere poteſt: & vis aquæ in cir<lb></lb>cellum aſcendentem eadem erit ac prius, per Legum Cor. 5, id eſt, <lb></lb> Reſiſtentia circelli aſcendentis erit ad pondus Cylindri aquæ cujus <lb></lb> baſis eſt circellus ille & altitudo eſt 1/2 <emph type="italics"></emph>IG,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>-1/2 <emph type="italics"></emph>PQq<emph.end type="italics"></emph.end><lb></lb>quamproxime. </s> <s>Velocitas autem circelli erit ad velocitatem quam <lb></lb> aqua cadendo & caſu ſuo deſcribendo altitudinem <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>acquirit, <lb></lb> ut <emph type="italics"></emph>EFq-PQq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Augeatur amplitudo canalis in infinitum: & rationes illæ inter <lb></lb> <emph type="italics"></emph>EFq-PQq<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EFq,<emph.end type="italics"></emph.end>interque <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>-1/2 <emph type="italics"></emph>PQq<emph.end type="italics"></emph.end>acce<lb></lb>dent ultimo ad rationes æqualitatis. </s> <s>Et propterea Velocitas cir<lb></lb>celli ea nunc erit quam aqua cadendo & caſu ſuo deſcribendo al<lb></lb>titudinem <emph type="italics"></emph>IG<emph.end type="italics"></emph.end>acquirere poteſt, Reſiſtentia vero ejus æqualis eva<lb></lb>det ponderi Cylindri cujus baſis eſt circellus ille & altitudo di<lb></lb>midium eſt altitudinis <emph type="italics"></emph>IG,<emph.end type="italics"></emph.end>a qua Cylindrus cadere debet ut velo<lb></lb>citatem circelli aſcendentis acquirat; & hac velocitate Cylindrus, <lb></lb> tempore cadendi, quadruplum longitudinis ſuæ deſcribet. </s> <s>Reſi<lb></lb>ſtentia autem Cylindri, hac velocitate ſecundum longitudinem ſuam <lb></lb> progredientis, eadem eſt cum Reſiſtentia circelli per Lemma IV; <lb></lb> ideoque æqualis eſt Vi qua motus ejus, interea dum quadruplum <lb></lb> longitudinis ſuæ deſcribit, generari poteſt quamproxime. <pb xlink:href="039/01/340.jpg" pagenum="312"></pb><lb></lb><arrow.to.target n="note288"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note288"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s>Si longitudo Cylindri augeatur vel minuatur: motus ejus ut & <lb></lb> tempus quo quadruplum longitudinis ſuæ deſcribit, augebitur vel <lb></lb> minuetur in eadem ratione; adeoque Vis illa qua motus auctus vel <lb></lb> diminutus, tempore pariter aucto vel diminuto, generari vel tolli <lb></lb> poſſit, non mutabitur; ac proinde etiamnum æqualis eſt reſi<lb></lb>ſtentiæ Cylindri, nam & hæc quoQ.E.I.mutata manet per Lem<lb></lb>ma IV. <lb></lb> </s></p> <p type="main"> <s>Si denſitas Cylindri augeatur vel minuatur: motus ejus ut & <lb></lb> Vis qua motus eodem tempore generari vel tolli poteſt, in eadem <lb></lb> ratione augebitur vel minuetur. </s> <s>Reſiſtentia itaque Cylindri cu<lb></lb>juſcunque erit ad Vim qua totus ejus motus, interea dum quadru<lb></lb>plum longitudinis ſuæ deſcribit, vel generari poſſit vel tolli, ut <lb></lb> denſitas Medii ad denſitatem Cylindri quamproxime. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Fluidum autem comprimi debet ut ſit continuum, continuum <lb></lb> vero eſſe & non elaſticum ut preſſio omnis quæ ab ejus compreſſi<lb></lb>one oritur propagetur in inſtanti &, in omnes moti corporis partes <lb></lb> æqualiter agendo, reſiſtentiam non mutet. </s> <s>Preſſio utique quæ a <lb></lb> motu corporis oritur, impenditur in motum partium fluidi gene<lb></lb>randum & Reſiſtentiam creat. </s> <s>Preſſio autem quæ oritur a com<lb></lb>preſſione fluidi, utcunque fortis ſit, ſi propagetur in inſtanti, nul<lb></lb>lum generat motum in partibus fluidi continui, nullam omnino in<lb></lb>ducit motus mutationem; ideoque reſiſtentiam nec auget nec mi<lb></lb>nuit. </s> <s>Certe Actio fluidi, quæ ab ejus compreſſione oritur, fortior <lb></lb> eſſe non poteſt in partes poſticas corporis moti quam in ejus par<lb></lb>tes anticas, ideoque reſiſtentiam in hac Propoſitione deſcriptam <lb></lb> minuere non poteſt: & fortior non erit in partes anticas quam in <lb></lb> poſticas, ſi modo propagatio ejus infinite velocior ſit quam motus <lb></lb> corporis preſſi. </s> <s>Infinite autem velocior erit & propagabitur in in<lb></lb>ſtanti, ſi modo fluidum ſit continuum & non elaſticum. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Cylindrorum, qui ſecundum longitudines ſuas in Mediis <lb></lb> continuis infinitis uniformiter progrediuntur, reſiſtentiæ ſunt in ra<lb></lb>tione quæ componitur ex duplicata ratione velocitatum & dupli<lb></lb>cata ratione diametrorum & ratione denſitatis Mediorum. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si amplitudo canalis non augeatur in infinitum, ſed Cy<lb></lb>lindrus in Medio quieſcente incluſo ſecundum longitudinem ſuam <lb></lb> progrediatur, & interea axis ejus cum axe canalis coincidat: Reſi<lb></lb>ſtentia ejus erit ad vim qua totus ejus motus, quo tempore qua<lb></lb>druplum longitudinis ſuæ deſcribit, vel generari poſſit vel tolli, <lb></lb> in ratione quæ componitur ex ratione <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>-1/2 <emph type="italics"></emph>PQq<emph.end type="italics"></emph.end><pb xlink:href="039/01/341.jpg" pagenum="313"></pb><lb></lb>ſemel, & ratione <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq-PQq<emph.end type="italics"></emph.end>bis, & ratione denſitatis <lb></lb> <arrow.to.target n="note289"></arrow.to.target>Medii ad denſitatem Cylindri. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note289"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Iiſdem poſitis, & quod longitudo L ſit ad quadru<lb></lb>plum longitudinis Cylindri in ratione quæ componitur ex ratione <lb></lb> <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>-1/2 <emph type="italics"></emph>PQq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end>ſemel, & ratione <emph type="italics"></emph>EFq-PQq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EFq<emph.end type="italics"></emph.end><lb></lb>bis: reſiſtentia Cylindri erit ad vim qua totus ejus motus, interea <lb></lb> dum longitudinem L deſcribit, vel tolli poſſit vel generari, ut <lb></lb> denſitas Medii ad denſitatem Cylindri. <lb></lb> <emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s>In hac Propoſitione reſiſtentiam inveſtigavimus quæ oritur a <lb></lb> ſola magnitudine tranſverſæ ſectionis Cylindri, neglecta reſiſtentiæ <lb></lb> parte quæ ab obliquitate motuum oriri poſſit. </s> <s>Nam quemadmo<lb></lb>dum in caſu primo Propoſitionis XXXVI, obliquitas motuum qui<lb></lb>bus partes aquæ in vaſe, undique convergebant in foramen <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end><lb></lb>impedivit effluxum aquæ illius per foramen: ſic in hac Propoſiti<lb></lb>one, obliquitas motuum quibus partes aquæ ab anteriore Cylindri <lb></lb> termino preſſæ, cedunt preſſioni & undiQ.E.D.vergunt, retardat eo<lb></lb>rum tranſitum per loca in circuitu termini illius antecedentis ver<lb></lb>ſus poſteriores partes Cylindri, efficitque ut fluidum ad majorem <lb></lb> diſtantiam commoveatur & reſiſtentiam auget, idQ.E.I. ea fere <lb></lb> ratione qua effluxum aquæ e vaſe diminuit, id eſt, in ratione du<lb></lb>plicata 25 ad 21 circiter. </s> <s>Et quemadmodum, in Propoſitionis illius <lb></lb> caſu primo, effecimus ut partes aquæ perpendiculariter & maxima <lb></lb> copia tranſirent per foramen <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ponendo quod aqua omnis in <lb></lb> vaſe quæ in circuitu cataractæ congelata fuerat, & cujus motus <lb></lb> obliquus erat & inutilis, maneret ſine motu: ſic in hac Propoſi<lb></lb>tione, ut obliquitas motuum tollatur, & partes aquæ motu maxime <lb></lb> directo & breviſſimo cedentes facillimum præbeant tranſitum Cy<lb></lb>lindro, & ſola maneat reſiſtentia quæ oritur a magnitudine ſecti<lb></lb>onis tranſverſæ, quæQ.E.D.minui non poteſt niſi diminuendo dia<lb></lb>metrum Cylindri, concipiendum eſt quod partes fluidi quarum <lb></lb> motus ſunt obliqui & inutiles & reſiſtentiam creant, quieſcant in<lb></lb>ter ſe ad utrumque Cylindri ter<lb></lb><figure id="id.039.01.341.1.jpg" xlink:href="039/01/341/1.jpg"></figure><lb></lb>minum, & cohæreant & Cylindro <lb></lb> jungantur. </s> <s>Sit <emph type="italics"></emph>ABCD<emph.end type="italics"></emph.end>rectan<lb></lb>gulum, & ſint <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>arcus <lb></lb> duo Parabolici axe <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>deſcripti, <lb></lb> latere autem recto quod ſit ad ſpa-<pb xlink:href="039/01/342.jpg" pagenum="314"></pb><lb></lb><arrow.to.target n="note290"></arrow.to.target>tium <emph type="italics"></emph>HG,<emph.end type="italics"></emph.end>deſcribendum a Cylindro <lb></lb> <figure id="id.039.01.342.1.jpg" xlink:href="039/01/342/1.jpg"></figure><lb></lb>cadente dum velocitatem ſuam ac<lb></lb>quirit, ut <emph type="italics"></emph>HG<emph.end type="italics"></emph.end>ad 1/2 <emph type="italics"></emph>AB.<emph.end type="italics"></emph.end>Sint etiam <lb></lb> <emph type="italics"></emph>CF<emph.end type="italics"></emph.end>& <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>arcus alii duo Para<lb></lb>bolici, axe <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>& latere recto <lb></lb> quod ſit prioris lateris recti qua<lb></lb>druplum deſcripti; & convolutione figuræ circum axem <emph type="italics"></emph>EF<emph.end type="italics"></emph.end>ge<lb></lb>neretur ſolidum cujus media pars <emph type="italics"></emph>ABDC<emph.end type="italics"></emph.end>ſit Cylindrus de quo <lb></lb> agimus, & partes extremæ <emph type="italics"></emph>ABE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CDF<emph.end type="italics"></emph.end>contineant partes fluidi <lb></lb> inter ſe quieſcentes & in corpora duo rigida concretas, quæ Cy<lb></lb>lindro utrinque tanquam caput & cauda adhæreant. </s> <s>Et ſolidi <lb></lb> <emph type="italics"></emph>EACFDB,<emph.end type="italics"></emph.end>ſecundum longitudinem axis ſui <emph type="italics"></emph>FE<emph.end type="italics"></emph.end>in partes ver<lb></lb>ſus <emph type="italics"></emph>E<emph.end type="italics"></emph.end>progredientis, reſiſtentia ea erit quamproxime quam in hac <lb></lb> Propoſitione deſcripſimus, id eſt, quæ rationem illam habet ad <lb></lb> vim qua totus Cylindri motus, interea dum longitudo 4 <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>motu <lb></lb> illo uniformiter continuato deſcribatur, vel tolli poſſit vel generari, <lb></lb> quam denſitas Fluidi habet ad denſitatem Cylindri quamproxime. <lb></lb> Et hac vi Reſiſtentia minor eſſe non poteſt quam in ratione 2 ad 3, <lb></lb> per Corol. 7. Prop. XXXVI. <lb></lb> <emph type="center"></emph>LEMMA V.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note290"></margin.target>DE MOTU <lb></lb> CORPORUM.</s></p> <p type="main"> <s><emph type="italics"></emph>Si Cylindrus, Sphæra & Sphærois, quorum latitudines ſunt æqua<lb></lb>les, in medio canalis Cylindrici ita locentur ſucceſſive ut eo<lb></lb>rum axes cum axe canalis coincidant: hæc corpora fluxum <lb></lb> aquæ per canalem æqualiter impedient.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam ſpatia inter Canalem & Cylindrum, Sphæram, & Sphæroi<lb></lb>dem per quæ aqua tranſit, ſunt æqualia: & aqua per æqualia ſpa<lb></lb>tia æqualiter tranſit. <lb></lb> <emph type="center"></emph>LEMMA VI.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis, corpora prædicta æqualiter urgentur ab aqua per <lb></lb> canalem fluente.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Patet per Lemma v & Motus Legem tertiam. </s> <s>Aqua utique & <lb></lb> corpora in ſe mutuo æqualiter agunt. <pb xlink:href="039/01/343.jpg" pagenum="315"></pb><lb></lb><arrow.to.target n="note291"></arrow.to.target><emph type="center"></emph>LEMMA VII.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note291"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Si aqua quieſcat in canali, & hæc corpora in partes contrarias <lb></lb> æquali velocitate per canalem ferantur: æquales erunt eorum <lb></lb> reſiſtentiæ inter ſe.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Conſtat ex Lemmate ſuperiore, nam motus relativi iidem inter <lb></lb> ſe manent. <lb></lb> <emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s>Eadem eſt ratio corporum omnium convexorum & rotundo<lb></lb>rum, quorum axes cum axe canalis coincidunt. </s> <s>Differentia aliqua <lb></lb> ex majore vel minore frictione oriri poteſt; ſed in his Lemmatis <lb></lb> corpora eſſe politiſſima ſupponimus, & Medii tenacitatem & frictio<lb></lb>nem eſſe nullam, & quod partes fluidi, quæ motibus ſuis obliquis <lb></lb> & ſuperfluis fluxum aquæ per canalem perturbare, impedire, & re<lb></lb>tardare poſſunt, quieſcant inter ſe tanquam gelu conſtrictæ, & cor<lb></lb>poribus ad ipſorum partes anticas & poſticas adhæreant, perinde <lb></lb> ut in Scholio Propoſitionis præcedentis expoſui. </s> <s>Agitur enim in <lb></lb> ſequentibus de reſiſtentia omnium minima quam corpora rotunda, <lb></lb> datis maximis ſectionibus tranſverſis deſcripta, habere poſſunt. <lb></lb> </s></p> <p type="main"> <s>Corpora fluidis innatantia, ubi moventur in directum, efficiunt <lb></lb> ut fluidum ad partem anticam aſcendat, ad poſticam ſubſidat, præ<lb></lb>ſertim ſi figura ſint obtuſa; & inde reſiſtentiam paulo majorem <lb></lb> ſentiunt quam ſi capite & cauda ſint acutis. </s> <s>Et corpora in fluidis <lb></lb> elaſticis mota, ſi ante & poſt obtuſa ſint, fluidum paulo magis <lb></lb> condenſant ad anticam partem & paulo magis relaxant ad poſticam; <lb></lb> & inde reſiſtentiam paulo majorem ſentiunt quam ſi capite & cau<lb></lb>da ſint acutis. </s> <s>Sed nos in his Lemmatis & Propoſitionibus non <lb></lb> agimus de fluidis elaſticis, ſed de non elaſticis; non de inſidentibus <lb></lb> fluido, ſed de alte immerſis. </s> <s>Et ubi reſiſtentia corporum in fluidis <lb></lb> non elaſticis innoteſcit, augenda erit hæc reſiſtentia aliquantulum <lb></lb> tam in fluidis elaſticis, qualis eſt Aer, quam in ſuperficiebus fluido<lb></lb>rum ſtagnantium, qualia ſunt maria & paludes. <pb xlink:href="039/01/344.jpg" pagenum="316"></pb><lb></lb><arrow.to.target n="note292"></arrow.to.target><emph type="center"></emph>PROPOSITIO XXXVIII. THEOREMA XXX.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note292"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Globi, in Fluido compreſſo infinito & non elaſtico uniformiter progre<lb></lb>dientis, reſiſtentia eſt ad vim qua totus ejus motus, quo tempore <lb></lb> octo tertias partes diametri ſuæ deſcribit, vel tolli poſſit vel <lb></lb> generari, ut denſitas Fluidi ad denſitatem Globi quamproxime.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam Globus eſt ad Cylindrum circumſcriptum ut duo ad tria; <lb></lb> & propterea Vis illa, quæ tollere poſſit motum omnem Cylindri <lb></lb> interea dum Cylindrus deſcribat longitudinem quatuor diametro<lb></lb>rum, Globi motum omnem tollet interea dum Globus deſcribat <lb></lb> duas tertias partes hujus longitudinis, id eſt, octo tertias partes <lb></lb> diametri propriæ. Reſiſtentia autem Cylindri eſt ad hanc Vim <lb></lb> quamproxime ut denſitas Fluidi ad denſitatem Cylindri vel Globi, <lb></lb> per Prop.XXXVII; & Reſiſtentia Globi æqualis eſt Reſiſtentiæ Cy<lb></lb>lindri, per Lem. V, VI, VII. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Globorum, in Mediis compreſſis infinitis, reſiſtentiæ ſunt <lb></lb> in ratione quæ componitur ex duplicata ratione velocitatis, & du<lb></lb>plicata ratione diametri, & duplicata ratione denſitatis Mediorum. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2 Velocitas maxima quacum Globus, vi ponderis ſui com<lb></lb>parativi, in fluido reſiſtente poteſt deſcendere, ea eſt quam acqui<lb></lb>rere poteſt Globus idem, eodem pondere, abſque reſiſtentia caden<lb></lb>do & caſu ſuo deſcribendo ſpatium quod ſit ad quatuor tertias <lb></lb> partes diametri ſuæ ut denſitas Globi ad denſitatem Fluidi. </s> <s>Nam <lb></lb> Globus tempore caſus ſui, cum velocitate cadendo acquiſita, de<lb></lb>ſcribet ſpatium quod erit ad octo tertias diametri ſuæ, ut denſitas <lb></lb> Globi ad denſitatem Fluidi; & vis ponderis motum hunc generans, <lb></lb> erit ad vim quæ motum eundem generare poſſit quo tempore Glo<lb></lb>bus octo tertias diametri ſuæ eadem velocitate deſcribit, ut denſitas <lb></lb> Fluidi ad denſitatem Globi: ideoque per hanc Propoſitionem, vis <lb></lb> ponderis æqualis erit vi Reſiſtentiæ, & propterea Globum accele<lb></lb>rare non poteſt. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Data & denſitate Globi & velocitate ejus ſub initio <lb></lb> motus, ut & denſitate fluidi compreſſi quieſcentis in qua Globus <lb></lb> movetur; datur ad omne tempus & velocitas Globi & ejus reſi<lb></lb>ftentia & ſpatium ab eo deſcriptum, per Corol. 7. Prop. XXXV. <pb xlink:href="039/01/345.jpg" pagenum="317"></pb><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Globus in fluido compreſſo quieſcente ejuſdem ſecum <lb></lb> <arrow.to.target n="note293"></arrow.to.target>denſitatis movendo, dimidiam motus ſui partem prius amittet <lb></lb> quam longitudinem duarum ipſius diametrorum deſcripſerit, per <lb></lb> idem Corol. 7. <lb></lb> <emph type="center"></emph>PROPOSITIO XXXIX. THEOREMA XXXI.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note293"></margin.target>LIBER <lb></lb> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Globi, per Fluidum in canali Cylindrico clauſum & compreſſum uni<lb></lb>formiter progredientis, reſiſtentia eſt ad vim qua totus ejus motus, <lb></lb> interea dum octo tertias partes diametri ſuæ deſcribit, vel ge<lb></lb>nerari poſſit vel tolli, in ratione quæ componitur ex ratione ori<lb></lb>ficii canalis ad exceſſum hujus orificii ſupra dimidium circuli <lb></lb> maximi Globi, & ratione duplicata orificii canalis ad exceſſum <lb></lb> hujus orificii ſupra circulum maximum Globi, & ratione den<lb></lb>ſitatis Fluidi ad denſitatem Globi quamproxime.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Patet per Corol. 2. Prop. XXXVII; procedit vero demonſtratio <lb></lb> quemadmodum in Propoſitione præcedente. <lb></lb> <emph type="center"></emph>PROPOSITIO XL. PROBLEMA IX.<emph.end type="center"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Globi, in Medio fluidiſſimo compreſſo progredientis, invenire reſi<lb></lb>ſtentiam per Phænomena.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Sit A pondus Globi in vacuo, B pondus ejus in Medio reſi<lb></lb>ſtente, D diameter Globi, F ſpatium quod ſit ad 4/3 D ut denſitas <lb></lb> Globi ad denſitatem Medii, id eſt, ut A ad A-B, G tempus quo <lb></lb> Globus pondere B abſque reſiſtentia cadendo deſcribit ſpatium F, <lb></lb> & H velocitas quam Globus hocce caſu ſuo acquirit. </s> <s>Et erit H <lb></lb> velocitas maxima quacum Globus, pondere ſuo B, in Medio reſi<lb></lb>ſtente poteſt deſcendere, per Corol. 2, Prop. XXXVIII; & reſi<lb></lb>ſtentia quam Globus ea cum velocitate deſcendens patitur, æqua<lb></lb>lis erit ejus ponderi B: reſiſtentia vero quam patitur in alia qua<lb></lb>cunque velocitate, erit ad pondus B in duplicata ratione velo<lb></lb>citatis hujus ad velocitatem illam maximam <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>&c. G, per Corol. 1, <lb></lb> Prop. XXXVIII. <pb xlink:href="039/01/346.jpg" pagenum="318"></pb><lb></lb><arrow.to.target n="note294"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note294"></margin.target>DE MOTU <lb></lb> CORPORUM</s></p> <p type="main"> <s>Hæc eſt reſiſtentia quæ oritur ab inertia materiæ Fluidi. </s> <s>Ea <lb></lb> vero quæ oritur ab elaſticitate, tenacitate, & frictione partium <lb></lb> ejus, ſic inveſtigabitur. <lb></lb> </s></p> <p type="main"> <s>Demittatur Globus ut pondere ſuo B in Fluido deſcendat; <lb></lb> & ſit P tempus cadendi, idQ.E.I. minutis ſecundis ſi tempus <lb></lb> G in minutis ſecundis habeatur. </s> <s>Inveniatur numerus abſo<lb></lb>lutus N qui congruit Logarithmo 0,4342944819(2P/G), ſitque L <lb></lb> Logarithmus numer; (N+1/N): & velocitas cadendo acquiſita erit <lb></lb> (N-1/N+1)H, altitudo autem deſcripta erit (2PF/G)-1,3862943611 F+ <lb></lb> 4,605170186LF. Si Fluidum ſatis profundum ſit, negligi poteſt <lb></lb> terminus 4,605170186LF; & erit (2PF/G)-1,3862943611 F altitude <lb></lb> deſcripta quamproxime. </s> <s>Patent hæc per Libri ſecundi Propo<lb></lb>ſitionem nonam & ejus Corollaria, ex Hypotheſi quod Glo<lb></lb>bus nullam aliam patiatur reſiſtentiam niſi quæ oritur ab inertia <lb></lb> materiæ. Si vero aliam inſuper reſiſtentiam patiatur, deſcen<lb></lb>ſus erit tardior, & ex retardatione innoteſcet quantitas hujus re<lb></lb>ſiſtentiæ. <lb></lb> </s></p> <p type="main"> <s>Ut corporis in Fluido cadentis velocitas & deſcenſus facilius in<lb></lb>noteſcant, compoſui Tabulam ſequentem, cujus columna prima <lb></lb> denotat tempora deſcenſus, ſecunda exhibet velocitates cadendo <lb></lb> acquiſitas exiſtente velocitate maxima 100000000, tertia exhibet <lb></lb> ſpatia temporibus illis cadendo deſcripta, exiſtente 2 F ſpatio quod <lb></lb> corpus tempore G cum velocitate maxima deſcribit, & quarta ex<lb></lb>hibet ſpatia iiſdem temporibus cum velocitate maxima deſcripta. <lb></lb> Numeri in quarta columna ſunt (2P/G), & ſubducendo numerum <lb></lb> 1,3862944-4,6051702 L, inveniuntur numeri in tertia columna, & <lb></lb> multiplicandi ſunt hi numeri per ſpatium F ut habeantur ſpatia <lb></lb> cadendo deſcripta. </s> <s>Quinta his inſuper adjecta eſt columna, quæ <lb></lb> continet ſpatia deſcripta iiſdem temporibus a corpore, vi ponderis <lb></lb> ſui comparativi B, in vacuo cadente. <pb xlink:href="039/01/347.jpg" pagenum="319"></pb><lb></lb><arrow.to.target n="note295"></arrow.to.target></s></p><table><row><cell><emph type="italics"></emph>Tempora<emph.end type="italics"></emph.end><lb></lb>P</cell><cell><emph type="italics"></emph>Velocitates <lb></lb> cadentis in <lb></lb> fluido<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Spatia caden<lb></lb>do deſcripta <lb></lb> in fluido<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Spatia motu <lb></lb> maximo de<lb></lb>ſcripta.<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Spatia caden<lb></lb>do deſcripta <lb></lb> in vacuo.<emph.end type="italics"></emph.end></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><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ut reſiſtentias Fluidorum inveſtigarem per Experimenta, paravi <lb></lb>vas ligneum quadratum, longitudine & latitudine interna digito<lb></lb>rum novem pedis <emph type="italics"></emph>Londinenſis,<emph.end type="italics"></emph.end>profunditate pedum novem cum <lb></lb>ſemiſſe, idemQ.E.I.plevi aqua pluviali; & globis ex cera & plum<lb></lb>bo incluſo formatis, notavi tempora deſcenſus globorum, exiſtente <lb></lb>deſcenſus altitudine 112 digitorum pedis. </s> <s>Pes ſolidus cubicus <lb></lb><emph type="italics"></emph>Londinenſis<emph.end type="italics"></emph.end>continet 76 libras <emph type="italics"></emph>Romanas<emph.end type="italics"></emph.end>aquæ pluvialis, & pedis hu<lb></lb>jus digitus ſolidus continet (19/36) uncias libræ hujus ſeu grana 253 1/3; <lb></lb>& globus aqueus diametro digiti unius deſcriptus continet grana <pb xlink:href="039/01/348.jpg" pagenum="320"></pb><arrow.to.target n="note328"></arrow.to.target>132,645 in Medio aeris, vel grana 132,8 in vacuo; & globus qui<lb></lb>libet alius eſt ut exceſſus ponderis ejus in vacuo ſupra pondus ejus <lb></lb>in aqua. </s></p> <p type="margin"> <s><margin.target id="note328"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>1. Globus, cujus pondus erat 156 1/4 granorum in aere & <lb></lb>77 granorum in aqua, altitudinem totam digitorum 112 tempore <lb></lb>minutorum quatuor ſecundorum deſcripſit. </s> <s>Et experimento repe<lb></lb>tito, globus iterum cecidit eodem tempore minutorum quatuor ſe<lb></lb>cundorum. </s></p> <p type="main"> <s>Pondus globi in vacuo eſt (156 11/38) <emph type="italics"></emph>gran,<emph.end type="italics"></emph.end>& exceſſus hujus ponde<lb></lb>ris ſupra pondus globi in aqua eſt (79 11/38) <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>Unde prodit globi <lb></lb>diameter 0,84224 partium digiti. </s> <s>Eſt autem ut exceſſus ille ad <lb></lb>pondus globi in vacuo, ita denſitas aquæ ad denſitatem globi, <lb></lb>& ita partes octo tertiæ diametri globi (<emph type="italics"></emph>viz.<emph.end type="italics"></emph.end>2,24597 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>) ad ſpa<lb></lb>tium 2 F, quod proinde erit 4,4256 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>Globus tempore minuti <lb></lb>unius ſecundi, toto ſuo pondere granorum (156 11/38), cadendo in va<lb></lb>cuo deſcribet digitos 193 1/3; & pondere granorum 77, eodem tem<lb></lb>pore, abſque reſiſtentia cadendo in aqua deſcribet digitos 95,219; <lb></lb>& tempore G, quod ſit ad minutum unum ſecundum in ſubduplicata <lb></lb>ratione ſpatii F ſeu 2,2128 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>ad 95,219 <emph type="italics"></emph>dig,<emph.end type="italics"></emph.end>deſcribet 2,2128 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end><lb></lb>& velocitatem maximam H acquiret quacum poteſt in aqua de<lb></lb>ſcendere. </s> <s>Eſt igitur tempus G 0″,15244. Et hoc tempore G, <lb></lb>cum velocitate illa maxima H, globus deſcribet ſpatium 2 F digi<lb></lb>torum 4,4256; ideoque tempore minutorum quatuor ſecundo<lb></lb>rum deſcribet ſpatium digitorum 116,1245. Subducatur ſpatium <lb></lb>1,3862944 F ſeu 3,0676 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>& manebit ſpatium 113,0569 digito<lb></lb>rum quod globus cadendo in aqua, in vaſe ampliſſimo, tempore <lb></lb>minutorum quatuor ſecundorum deſcribet. </s> <s>Hoc ſpatium, ob an<lb></lb>guſtiam vaſis lignei prædicti, minui debet in ratione quæ compo<lb></lb>nitur ex ſubduplicata ratione orificii vaſis ad exceſſum orificii hu<lb></lb>jus ſupra ſemicirculum maximum globi & ex ſimplici ratione ori<lb></lb>ficii ejuſdem ad exceſſum ejus ſupra circulum maximum globi, id <lb></lb>eſt, in ratione 1 ad 0,9914. Quo facto, habebitur ſpatium 112,08 <lb></lb>digitorum, quod Globus cadendo in aqua in hoc vaſe ligneo tem<lb></lb>pore minutorum quatuor ſecundorum per Theoriam deſcribere <lb></lb>debuit quamproxime. </s> <s>Deſcripſit vero digitos 112 per Experi<lb></lb>mentum. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>2. Tres Globi æquales, quorum pondera ſeorſim erant <lb></lb>76 1/3 granorum in aere & (5 1/16) granorum in aqua, ſucceſſive demitte<lb></lb>bantur; & unuſquiſque cecidit in aqua tempore minutorum ſecun<lb></lb>dorum quindecim, caſu ſuo deſcribens altitudinem digitorum 112. </s></p><pb xlink:href="039/01/349.jpg" pagenum="321"></pb> <p type="main"> <s>Computum ineundo prodcunt pondus globi in vacuo (76 1/12) <emph type="italics"></emph>gran,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note329"></arrow.to.target>exceſſus hujus ponderis ſupra pondus in aqua (71 17/48) <emph type="italics"></emph>gran,<emph.end type="italics"></emph.end>diameter <lb></lb>globi 0,81296 <emph type="italics"></emph>dig,<emph.end type="italics"></emph.end>octo tertiæ partes hujus diametri 2,16789 <emph type="italics"></emph>dig,<emph.end type="italics"></emph.end><lb></lb>ſpatium 2 F 2,3217 <emph type="italics"></emph>dig,<emph.end type="italics"></emph.end>ſpatium quod globus pondere (5 1/16) <emph type="italics"></emph>gran,<emph.end type="italics"></emph.end><lb></lb>tempore 1″, abſque reſiſtentia cadendo deſcribat 12,808 <emph type="italics"></emph>dig,<emph.end type="italics"></emph.end>& <lb></lb>tempus G 0′,301056. Globus igitur, velocitate maxima quacum <lb></lb>poteſt in aqua vi ponderis (5 1/16) <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>deſcendere, tempore 0′,301056 <lb></lb>deſcribet ſpatium 2,3217 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>& tempore 15″ ſpatium 115,678 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end><lb></lb>Subducatur ſpatium 1,3862944 F ſeu 1,609 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>& manebit ſpatium <lb></lb>114,069 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>quod proinde globus eodem tempore in vaſe latiſli<lb></lb>mo cadendo deſcribere debet. </s> <s>Propter anguſtiam vaſis noſtri de<lb></lb>trahi debet ſpatium 0,895 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>circiter. </s> <s>Et ſic manebit ſpatium <lb></lb>113,174 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>quod globus cadendo in hoc vaſe, tempore 15″ de<lb></lb>ſcribere debuit per Theoriam quamproxime. </s> <s>Deſcripſit vero digi<lb></lb>tos 112 per Experimentum. </s> <s>Differentia eſt inſenſibilis. </s></p> <p type="margin"> <s><margin.target id="note329"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>3. Globi tres æquales, quorum pondera ſeorſim erant <lb></lb>121 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>in aere & 1 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>in aqua, ſucceſſive demittebantur; & <lb></lb>cadebant in aqua temporibus 46″, 47″, & 50″, deſcribentes alti<lb></lb>tudinem digitorum 112. </s></p> <p type="main"> <s>Per Theoriam hi globi cadere debuerunt tempore 40″ circiter. </s> <s><lb></lb>Quod tardius ceciderunt, vel bullulis nonnullis globo adhærenti<lb></lb>bus, vel rarefactioni ceræ ad calorem vel tempeſtatis vel manus <lb></lb>globum demittentis, vel erroribus inſenſibilibus in ponderandis <lb></lb>globis in aqua, vel denique minori proportioni reſiſtentiæ quæ a <lb></lb>vi inertiæ in tardis motibus oritur ad reſiſtentiam quæ oritur ab <lb></lb>aliis cauſis, tribuendum eſſe puto. </s> <s>Ideoque pondus globi in aqua <lb></lb>debet eſſe plurium granorum ut experimentum certum & fide dig<lb></lb>num reddatur. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>4. Experimenta hactenus deſcripta cæpi ut inveſtigarem <lb></lb>reſiſtentias fluidorum antequam Theoria, in Propoſitionibus pro<lb></lb>xime præcedentibus expoſita, mihi innoteſceret. </s> <s>Poſtea, ut Theo<lb></lb>riam inventam examinarem, paravi vas ligneum latitudine interna <lb></lb>digitorum 8 2/3, profunditate pedum quindecim cum triente. </s> <s>De<lb></lb>inde ex cera & plumbo incluſo globos quatuor formavi, ſingulos <lb></lb>pondere 139 1/4 granorum in aere & 7 1/8 granorum in aqua. </s> <s>Et hos <lb></lb>demiſi ut tempora cadendi in aqua per pendulum, ad ſemi-minuta <lb></lb>ſecunda oſcillans, menſurarem. </s> <s>Globi, ubi ponderabantur & po<lb></lb>ſtea cadebant, frigidi erant & aliquamdiu frigidi manſerant; quia <lb></lb>calor ceram rarefacit, & per rarefactionem diminuit pondus globi <lb></lb>in aqua, & cera rarefacta non ſtatim ad denſitatem priſtinam per <pb xlink:href="039/01/350.jpg" pagenum="322"></pb><arrow.to.target n="note330"></arrow.to.target>frigus reducitur. </s> <s>Antequam caderent, immergebantur penitus in <lb></lb>aquam; ne pondere partis alicujus ex aqua extantis deſcenſus eo<lb></lb>rum ſub initio acceleraretur. </s> <s>Et ubi penitus immerſi quieſcebant, <lb></lb>demittebantur quam cautiſſime, ne impulſum aliquem a manu de<lb></lb>mittente acciperent. </s> <s>Ceciderunt autem ſucceſſive temporibus <lb></lb>oſcillationum 47 1/2, 48 1/2, 50 & 51, deſcribentes altitudinem pedum <lb></lb>quindecim & digitorum duorum. </s> <s>Sed tempeſtas jam paulo frigi<lb></lb>dior erat quam cum globi ponderabantur, ideoQ.E.I.eravi experi<lb></lb>mentum alio die, & globi ceciderunt temporibus oſcillationum <lb></lb>49, 49 1/2, 50 & 53, ac tertio temporibus oſcillationum 49 1/2, 50, 51 <lb></lb>& 53. Et experimento ſæpius capto, Globi ceciderunt maxima <lb></lb>ex parte temporibus oſcillationum 49 1/2 & 50. Ubi tardius ce<lb></lb>cidere, ſuſpicor eoſdem retardatos fuiſſe impingendo in latera <lb></lb>vaſis. </s></p> <p type="margin"> <s><margin.target id="note330"></margin.target>DE MOTU <lb></lb>CORPORUM.</s></p> <p type="main"> <s>Jam computum per Theoriam ineundo, prodeunt pondus globi <lb></lb>in vacuo 139 2/5 granorum. </s> <s>Exceſſus hujus ponderis ſupra pondus <lb></lb>globi in aqua (132 11/40) <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>Diameter globi 0,99868 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>Octo ter<lb></lb>tiæ partes diametri 2,66315 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>Spatium 2 F 2,8066 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>Spatium <lb></lb>quod globus pondere 7 1/8 granorum, tempore minuti unius ſe<lb></lb>cundi abſque reſiſtentia cadendo deſcribit 9,88164 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>Et tempus <lb></lb>G 0″,376843. Globus igitur, velocitate maxima quacum poteſt in <lb></lb>aqua vi ponderis 7 1/8 granorum deſcendere, tempore 0″,376843 de<lb></lb>ſcribit ſpatium 2,8066 digitorum, & tempore 1″ ſpatium 7,44766 di<lb></lb>gitorum, & tempore 25″ ſeu oſcillationum 50 ſpatium 186,1915 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end><lb></lb>Subducatur ſpatium 1,386294 F, ſeu 1,9454 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>& manebit ſpa<lb></lb>tium 184,2461 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>quod globus eodem tempore in vaſe latiſſimo <lb></lb>deſcribet. </s> <s>Ob anguſtiam vaſis noſtri, minuatur hoc ſpatium in ra<lb></lb>tione quæ componitur ex ſubduplicata ratione orificii vaſis ad <lb></lb>exceſſum hujus orificii ſupra ſemicirculum maximum globi, & ſim<lb></lb>plici ratione ejuſdem orificii ad exceſſum ejus ſupra circulum ma<lb></lb>ximum globi; & habebitur ſpatium 181,86 digitorum, quod glo<lb></lb>bus in hoc vaſe tempore oſcillationum 50 deſcribere debuit per <lb></lb>Theoriam quamproxime. </s> <s>Deſcripſit vero ſpatium 182 digitorum <lb></lb>tempore oſcillationum 49 1/2 vel 50 per Experimentum. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>5. Globi quatuor pondere 154 1/8 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>in aere & 21 1/2 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end><lb></lb>in aqua, ſæpe demiſſi, cadebant tempore oſcillationum 28 1/2, 29, <lb></lb>29<gap></gap> & 30, & nonnunquam 31, 32 & 33, deſcribentes altitudinem <lb></lb>pedum quindecim & digitorum duorum. </s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore oſcillationum 29 <lb></lb>quamproxime. </s></p><pb xlink:href="039/01/351.jpg" pagenum="323"></pb> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>6. Globi quinque pondere 212 1/8 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>in aere & 79 1/2 in <lb></lb><arrow.to.target n="note331"></arrow.to.target>aqua, ſæpe demiſſi, cadebant tempore oſcillationum 15, 15 1/2, 16, <lb></lb>17 & 18, deſcribentes altitudinem pedum quindecim & digitorum <lb></lb>duorum. </s></p> <p type="margin"> <s><margin.target id="note331"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore oſcillationum 15 <lb></lb>quamproxime. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>7. Globi quatuor pondere 293 1/8 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>in aere & 35 1/8 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end><lb></lb>in aqua, ſæpe demiſſi, cadebant tempore oſcillationum 29 1/2, 30, <lb></lb>30 1/2, 31, 32 & 33, deſcribentes altitudinem pedum quindecim & <lb></lb>digiti unius cum ſemiſſe. </s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore oſcillationum 28 <lb></lb>quamproxime. </s></p> <p type="main"> <s>Cauſam inveſtigando cur globorum, ejuſdem ponderis & magNI<lb></lb>tudinis, aliqui citius alii tardius caderent, in hanc incidi; quod glo<lb></lb>bi, ubi primum demittebantur & cadere incipiebant, oſcillarent cir<lb></lb>cum centra, latere illo quod forte gravius eſſet, primum deſcen<lb></lb>dente, & motum oſcillatorium generante. </s> <s>Nam per oſcillationes <lb></lb>ſuas, globus majorem motum communicat aquæ, quam ſi ſine oſcil<lb></lb>lationibus deſcenderet; & communicando, amittit partem motus <lb></lb>proprii quo deſcendere deberet: & pro majore vel minore oſcil<lb></lb>latione, magis vel minus retardatur. </s> <s>Quinetiam globus recedit <lb></lb>ſemper a latere ſuo quod per oſcillationem deſcendit, & receden<lb></lb>do appropinquat lateribus vaſis & in latera nonnunquam impin<lb></lb>gitur. </s> <s>Et hæc oſcillatio in globis gravioribus fortior eſt, & in <lb></lb>majoribus aquam magis agitat. </s> <s>Quapropter, ut oſcillatio globo<lb></lb>rum minor redderetur, globos novos ex cera & plumbo conſtruxi, <lb></lb>infigendo plumbum in latus aliquod globi prope ſuperficiem ejus; <lb></lb>& globum ita demiſi, ut latus gravius, quoad fieri potuit, eſſet in<lb></lb>fimum ab initio deſcenſus. </s> <s>Sic oſcillationes factæ ſunt multo mi<lb></lb>nores quam prius, & globi temporibus minus inæqualibus cecide<lb></lb>runt, ut in experimentis ſequentibus. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>8. Globi quatuor pondere granorum 139 in aere & 6 1/2 in <lb></lb>aqua, ſæpe demiſſi, ceciderunt temporibus oſcillationum non plu<lb></lb>rium quam 52, non pauciorum quam 50, & maxima ex parte <lb></lb>tempore oſcillationum 51 circiter, deſcribentes altitudinem digi<lb></lb>torum 182. </s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore oſcillationum 52 <lb></lb>circiter. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>9. Globi quatuor pondere granorum 273 1/4 in aere & <lb></lb>140 1/4 in aqua, ſæpius demiſſi, ceciderunt temporibus oſcillationum <pb xlink:href="039/01/352.jpg" pagenum="324"></pb><arrow.to.target n="note332"></arrow.to.target>non pauciorum quam 12, non plurium quam 13, deſcribentes al<lb></lb>titudinem digitorum 182. </s></p> <p type="margin"> <s><margin.target id="note332"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Per Theoriam vero hi globi cadere debuerunt tempore oſcilla<lb></lb>tionum 11 1/3 quamproxime. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>10. Globi quatuor pondere granorum 384 in aere & <lb></lb>119 1/2 in aqua, ſæpe demiſſi, cadebant temporibus oſcillationum <lb></lb>17 1/4, 18, 18 1/2 & 19, deſcribentes altitudinem digitorum 181 1/2. Et <lb></lb>ubi ceciderunt tempore oſcillationum 19, nonnunquam audivi im<lb></lb>pulſum eorum in latera vaſis antequam ad fundum pervenerunt. </s></p> <p type="main"> <s>Per Theoriam vero cadere debuerunt tempore oſcillationum <lb></lb>15 3/9 quamproxime. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>11. Globi tres æquales, pondere granorum 48 in aere <lb></lb>& (3 29/32) in aqua, ſæpe demiſſi, ceciderunt temporibus oſcillationum <lb></lb>43 1/2, 44, 44 1/2, 45 & 46, & maxima ex parte 44 & 45, deſcribentes <lb></lb>altitudinem digitorum 182 1/2 quamproxime. </s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore oſcillationum 46 5/9 <lb></lb>circiter. </s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>12. Globi tres æquales, pondere granorum 141 in aere <lb></lb>& 4 3/8 in aqua, aliquoties demiſſi, ceciderunt temporibus oſcillatio<lb></lb>num 61, 62, 63, 64 & 65, deſcribentes altitudinem digitorum 182. </s></p> <p type="main"> <s>Et per Theoriam cadere debuerunt tempore oſcillationum <lb></lb>64 1/2 quamproxime. </s></p> <p type="main"> <s>Per hæc Experimenta manifeſtum eſt quod, ubi globi tarde ceci<lb></lb>derunt, ut in experimentis ſecundis, quartis, quintis, octavis, un<lb></lb>decimis ac duodecimis, tempora cadendi recte exhibentur per <lb></lb>Theoriam: at ubi globi velocius ceciderunt, ut in experimentis <lb></lb>ſextis, nonis ac decimis, reſiſtentia paulo major extitit quam in <lb></lb>duplicata ratione velocitatis. </s> <s>Nam globi inter cadendum oſcillant <lb></lb>aliquantulum; & hæc oſcillatio in globis levioribus & tardius ca<lb></lb>dentibus, ob motus languorem cito ceſſat; in gravioribus autem & <lb></lb>majoribus, ob motus fortitudinem diutius durat, & non niſi poſt <lb></lb>plures oſcillationes ab aqua ambiente cohiberi poteſt. </s> <s>Quinetiam <lb></lb>globi, quo velociores ſunt, eo minus premuntur a fluido ad po<lb></lb>ſticas ſuas partes; & ſi velocitas perpetuo augeatur, ſpatium va<lb></lb>cuum tandem a tergo relinquent, niſi compreſſio fluidi ſimul au<lb></lb>geatur. </s> <s>Debet autem compreſſio fluidi (per Prop. </s> <s>XXXII & XXXIII) <lb></lb>augeri in duplicata ratione velocitatis, ut reſiſtentia ſit in eadem <lb></lb>duplicata ratione. </s> <s>Quoniam hoc non fit, globi velociores paulo <lb></lb>minus premuntur a tergo, & defectu preſſionis hujus, reſiſtentia <lb></lb>eorum fit paulo major quam in duplicata ratione velocitatis. </s></p><pb xlink:href="039/01/353.jpg" pagenum="325"></pb> <p type="main"> <s>Congruit igitur Theoria cum phænomenis corporum caden<lb></lb><arrow.to.target n="note333"></arrow.to.target>tium in Aqua, reliquum eſt ut examinemus phænomena caden<lb></lb>tium in Aere. </s></p> <p type="margin"> <s><margin.target id="note333"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Exper.<emph.end type="italics"></emph.end>13. A culmine Eccleſiæ <emph type="italics"></emph>S<emph type="sup"></emph>ti<emph.end type="sup"></emph.end> Pauli,<emph.end type="italics"></emph.end>in urbe <emph type="italics"></emph>Londini,<emph.end type="italics"></emph.end>globi <lb></lb>duo vitrei ſimul demittebantur, unus argenti vivi plenus, alter <lb></lb>aeris; & cadendo deſcribebant altitudinem pedum <emph type="italics"></emph>Londinenſium<emph.end type="italics"></emph.end><lb></lb>220. Tabula lignea ad unum ejus terminum polis ferreis ſuſpen<lb></lb>debatur, ad alterum peſſulo ligneo incumbebat; & globi duo huic <lb></lb>Tabulæ impoſiti ſimul demittebantur, ſubtrahendo peſſulum, ut Ta<lb></lb>bula polis ferreis ſolummodo innixa ſuper iiſdem devolveretur, & <lb></lb>codem temporis momento pendulum ad minuta ſecunda oſcillans, <lb></lb>per filum ferreum a peſſulo ad imam Eccleſiæ partem tendens, <lb></lb>dimitteretur & oſcillare inciperet. </s> <s>Diametri & pondera globorum <lb></lb>ac tempora cadendi exhibentur in Tabula ſequente. <lb></lb><arrow.to.target n="table3"></arrow.to.target> </s></p><table><table.target id="table3"></table.target><row><cell><emph type="italics"></emph>Globorum mercurio plenorum.<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Globorum aere plenorum.<emph.end type="italics"></emph.end></cell></row><row><cell><emph type="italics"></emph>Pondera<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Diametri<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Tempora <lb></lb> cadendi.<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Pondera<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Diametri<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Tempora <lb></lb> cadendi.<emph.end type="italics"></emph.end></cell></row><row><cell>908 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end></cell><cell>0,8 <emph type="italics"></emph>digit.<emph.end type="italics"></emph.end></cell><cell>4″</cell><cell>510 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end></cell><cell>5,1 <emph type="italics"></emph>digit.<emph.end type="italics"></emph.end></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 obſervata corrigi debent. </s> <s>Nam globi mer<lb></lb>curiales (per Theoriam <emph type="italics"></emph>Galilæi<emph.end type="italics"></emph.end>) minutis quatuor ſecundis deſcribent <lb></lb>pedes <emph type="italics"></emph>Londinenſes<emph.end type="italics"></emph.end>257, & pedes 220 minutis tantum 3″ 42′. </s> <s>Ta<lb></lb>bula lignea utique, detracto peſſulo, tardius devolvebatur quam par <lb></lb>erat, & tarda ſua devolutione impediebat deſcenſum globorum <lb></lb>ſub initio. </s> <s>Nam globi incumbebant Tabulæ prope medium ejus, <lb></lb>& paulo quidem propiores erant axi ejus quam peſſulo. </s> <s>Et hinc <lb></lb>tempora cadendi prorogata fuerunt minutis tertiis octodecim cir<lb></lb>citer, & jam corrigi debent detrahendo illa minuta, præſertim in <lb></lb>globis majoribus qui Tabulæ devolventi paulo diutius incumbe<lb></lb>bant propter magnitudinem diametrorum. </s> <s>Quo facto, tempora <lb></lb>quibus globi ſex majores cecidere, evadent, 8″, 12′, 7″ 42′, 7″ 42′, <lb></lb>7″ 57′, 8″ 12′, & 7″ 42′. <pb xlink:href="039/01/354.jpg" pagenum="326"></pb><arrow.to.target n="note334"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note334"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Globorum igitur aere plenorum quintus, diametro digitorum <lb></lb>quinque pondere granorum 483 conſtructus, cecidit tempore <lb></lb>8″ 12′, deſcribendo altitudinem pedum 220. Pondus aquæ huic <lb></lb>globo æqualis, eſt 16600 granorum; & pondus aeris eidem æqualis <lb></lb>eſt (16600/860) <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>ſeu (19 3/10) <emph type="italics"></emph>gran<emph.end type="italics"></emph.end>; ideoque pondus globi in vacuo eſt <lb></lb>(502 3/10) <emph type="italics"></emph>gran<emph.end type="italics"></emph.end>; & hoc pondus eſt ad pondus aeris globo æqualis, ut <lb></lb>(502 3/10) ad (19 3/10), & ita ſunt 2 F ad octo tertias partes diametri glo<lb></lb>bi, id eſt, ad (13 1/3) digitos. </s> <s>Unde 2 F prodeunt 28 <emph type="italics"></emph>ped.<emph.end type="italics"></emph.end>11 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>Glo<lb></lb>bus cadendo in vacuo, toto ſuo pondere (502 3/10) granorum, tempore <lb></lb>minuti unius ſecundi deſcribit digitos 193 1/3 ut ſupra, & pondere <lb></lb>483 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end>deſcribit digitos 185,905, & eodem pondere 483 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end><lb></lb>etiam in vacuo deſcribit ſpatium F ſeu 14 <emph type="italics"></emph>ped.<emph.end type="italics"></emph.end>5 1/2 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>tempore <lb></lb>57′ 58′, & velocitatem maximam acquirit quacum poſſit in aere <lb></lb>deſcendere. </s> <s>Hac velocitate globus, tempore 8″ 12′, deſcribet ſpa<lb></lb>tium pedum 245 & digitorum 5 1/3. Aufer 1,3863 F ſeu 20 <emph type="italics"></emph>ped.<emph.end type="italics"></emph.end><lb></lb>0 1/2 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>& manebunt 225 <emph type="italics"></emph>ped.<emph.end type="italics"></emph.end>5 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end>Hoc ſpatium igitur globus, <lb></lb>tempore 8″ 12′, cadendo deſcribere debuit per Theoriam. </s> <s>De<lb></lb>ſcripſit vero ſpatium 220 pedum per Experimentum. </s> <s>Differentia <lb></lb>inſenſibilis eſt. </s></p> <p type="main"> <s>Similibus computis ad reliquos etiam globos aere plenos appli<lb></lb>catis, confeci Tabulam ſequentem. <lb></lb><arrow.to.target n="table4"></arrow.to.target> </s></p><table><table.target id="table4"></table.target><row><cell><emph type="italics"></emph>Globorum <lb></lb> pondera<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Dia<lb></lb>metri<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Tempora ca<lb></lb>dendi ab al<lb></lb>titudine pe<lb></lb>dum<emph.end type="italics"></emph.end>220.</cell><cell><emph type="italics"></emph>Spatia deſcriben<lb></lb>da per Theoriam.<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Exceſſus<emph.end type="italics"></emph.end></cell></row><row><cell>510 <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end></cell><cell>5,1 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end></cell><cell>8″</cell><cell>12′</cell><cell>226 <emph type="italics"></emph>ped.<emph.end type="italics"></emph.end></cell><cell>11 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end></cell><cell>6 <emph type="italics"></emph>ped.<emph.end type="italics"></emph.end></cell><cell>11 <emph type="italics"></emph>dig.<emph.end type="italics"></emph.end></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 reſi<lb></lb>ſtentia prope omnis per Theoriam noſtram recte exhibetur, ac <lb></lb>denſitati fluidorum, paribus globorum velocitatibus ac magnitudi<lb></lb>nibus, proportionalis eſt. </s></p><pb xlink:href="039/01/355.jpg" pagenum="327"></pb> <p type="main"> <s>In Scholio quod Sectioni ſextæ ſubjunctum eſt, oſtendimus per </s></p> <p type="main"> <s><arrow.to.target n="note335"></arrow.to.target>experimenta pendulorum quod globorum æqualium & æquivelo<lb></lb>cium in Aere, Aqua, & Argento vivo motorum reſiſtentiæ ſunt ut <lb></lb>fluidorum denſitates. </s> <s>Idem hic oſtendimus magis accurate per <lb></lb>experimenta corporum cadentium in Aere & Aqua. </s> <s>Nam pendula <lb></lb>ſingulis oſcillationibus motum cient in fluido motui penduli re<lb></lb>deuntis ſemper contrarium, & reſiſtentia ab hoc motu oriunda, ut <lb></lb>& reſiſtentia fili quo pendulum ſuſpendebatur, totam Penduli re<lb></lb>ſiſtentiam majorem reddiderunt quam reſiſtentia quæ per experi<lb></lb>menta corporum cadentium prodiit. </s> <s>Etenim per experimenta <lb></lb>pendulorum in Scholio illo expoſita, globus ejuſdem denſitatis <lb></lb>cum Aqua, deſcribendo longitudinem ſemidiametri ſuæ in Aere, <lb></lb>amittere deberet motus ſui partem (1/3342). At per Theoriam in hac <lb></lb>ſeptima Sectione expoſitam & experimentis cadentium confirma<lb></lb>tam, globus idem deſcribendo longitudinem eandem, amittere de<lb></lb>beret motus ſui partem tantum (1/4586), poſito quod denſitas Aquæ ſit <lb></lb>ad denſitatem Aeris ut 860 ad 1. Reſiſtentiæ igitur per experi<lb></lb>menta pendulorum majores prodiere (ob cauſas jam deſcriptas) <lb></lb>quam per experimenta globorum cadentium, idQ.E.I. ratione 4 ad <lb></lb>3 circiter. </s> <s>Attamen cum pendulorum in Aere, Aqua, & Argento <lb></lb>vivo oſcillantium reſiſtentiæ a cauſis ſimilibus ſimiliter augeantur, <lb></lb>proportio reſiſtentiarum in his Mediis, tam per experimenta pen<lb></lb>dulorum, quam per experimenta corporum cadentium, ſatis recte <lb></lb>exhibebitur. </s> <s>Et inde concludi poteſt quod corporum in fluidis <lb></lb>quibuſcunque fluidiſſimis motorum reſiſtentiæ, cæteris paribus, <lb></lb>ſunt ut denſitates fluidorum. </s></p> <p type="margin"> <s><margin.target id="note335"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>His ita ſtabilitis, dicere jam licet quamnam motus ſui partem <lb></lb>globus quilibet, in fluido quocunque projectus, dato tempore amit<lb></lb>tet quamproxime. </s> <s>Sit D diameter globi, & V velocitas ejus ſub <lb></lb>initio motus, & T tempus quo globus velocitate V in vacuo de<lb></lb>ſcribet ſpatium quod ſit ad ſpatium 2/3D ut denſitas globi ad denſi<lb></lb>tatem fluidi: & globus in fluido illo projectus, tempore quovis <lb></lb>alio <emph type="italics"></emph>t,<emph.end type="italics"></emph.end>amittet velocitatis ſuæ partem (<emph type="italics"></emph>t<emph.end type="italics"></emph.end>V/T+<emph type="italics"></emph>t<emph.end type="italics"></emph.end>), manente parte (TV/T+<emph type="italics"></emph>t<emph.end type="italics"></emph.end>), <lb></lb>& deſcribet ſpatium quod ſit ad ſpatium uniformi velocitate V eo<lb></lb>dem tempore deſcriptum in vacuo, ut logarithmus numeri (T+<emph type="italics"></emph>t<emph.end type="italics"></emph.end>/T) <lb></lb>multiplicatus per numerum 2,302585093 eſt ad numerum <emph type="italics"></emph>t<emph.end type="italics"></emph.end>/T, per <pb xlink:href="039/01/356.jpg" pagenum="328"></pb><arrow.to.target n="note336"></arrow.to.target>Corol. </s> <s>7, Prop.XXXV. </s> <s>In motibus tardis reſiſtentia poteſt eſſe pau<lb></lb>lo minor, propterea quod figura Globi paulo aptior ſit ad motum <lb></lb>quam figura Cylindri eadem diametro deſcripti. </s> <s>In motibus ve<lb></lb>locibus reſiſtentia poteſt eſſe paulo major, propterea quod elaſti<lb></lb>citas & compreſſio fluidi non augeantur in duplicata ratione ve<lb></lb>locitatis. </s> <s>Sed hujuſmodi minutias hic non expendo. </s></p> <p type="margin"> <s><margin.target id="note336"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Et quamvis Aer, Aqua, Argentum vivum & ſimilia fluida, per <lb></lb>diviſionem partium in infinitum, ſubtiliarentur & fierent Media in<lb></lb>finite fluida; tamen globis projectis haud minus reſiſterent. </s> <s>Nam <lb></lb>reſiſtentia, de qua agitur in Propoſitionibus præcedentibus, oritur <lb></lb>ab inertia materiæ; & inertia materiæ corporibus eſſentialis eſt & <lb></lb>quantitati materiæ ſemper proportionalis. </s> <s>Per diviſionem partium <lb></lb>fluidi, reſiſtentia quæ oritur a tenacitate & frictione partium, di<lb></lb>minui quidem poteſt: ſed quantitas materiæ per diviſionem par<lb></lb>tium ejus non diminuitur; & manente quantitate materiæ, manet <lb></lb>ejus vis inertiæ cui reſiſtentia, de qua hic agitur, ſemper proportio<lb></lb>nalis eſt. </s> <s>Ut hæc reſiſtentia diminuatur, diminui debet quantitas <lb></lb>materiæ in ſpatiis per quæ corpora moventur. </s> <s>Et propterea ſpa<lb></lb>tia Cœleſtia, per quæ globi Planetarum & Cometarum in omnes <lb></lb>partes liberrime & abſque omni motus diminutione ſenſibili per<lb></lb>petuo moventur, fluido omni corporeo deſtituuntur, ſi forte vapo<lb></lb>res longe tenuiſſimos & trajectos lucis radios excipias. </s></p> <p type="main"> <s>Projectilia utique motum cient in fluidis progrediendo, & hic <lb></lb>motus oritur ab exceſſu preſſionis fluidi ad projectilis partes anti<lb></lb>cas ſupra preſſionem ad ejus partes poſticas, & non minor eſſe po<lb></lb>teſt in Mediis infinite fluidis quam in Aere, Aqua, & Argento vivo <lb></lb>pro denſitate materiæ in ſingulis. </s> <s>Hic autem preſſionis exceſſus, <lb></lb>pro quantitate ſua, non tantum motum ciet in fluido, ſed etiam agit <lb></lb>in projectile ad motum ejus retardandum: & propterea reſi<lb></lb>ſtentia in omni fluido, eſt ut motus in fluido a projectili excita<lb></lb>tus, nec minor eſſe poteſt in Æthere ſubtiliſſimo pro denſitate <lb></lb>Ætheris, quam in Aere, Aqua, & Argento vivo pro denſitatibus <lb></lb>horum fluidorum. <pb xlink:href="039/01/357.jpg" pagenum="329"></pb><arrow.to.target n="note337"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note337"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De Motu per Fluida propagato.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLI. THEOREMA XXXII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Preſſio non propagatur per Fluidum ſecundum lineas rectas, niſi <lb></lb>ubi particulæ Fluidi in directum jacent.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Si jaceant particulæ <emph type="italics"></emph>a, b, c, d, e<emph.end type="italics"></emph.end>in linea recta, poteſt quidem <lb></lb>preſſio directe propagari ab <emph type="italics"></emph>a<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>e<emph.end type="italics"></emph.end>; at <lb></lb><figure id="id.039.01.357.1.jpg" xlink:href="039/01/357/1.jpg"></figure><lb></lb>particula <emph type="italics"></emph>e<emph.end type="italics"></emph.end>urgebit particulas oblique po<lb></lb>ſitas <emph type="italics"></emph>f<emph.end type="italics"></emph.end>& <emph type="italics"></emph>g<emph.end type="italics"></emph.end>oblique, & particulæ illæ <emph type="italics"></emph>f<emph.end type="italics"></emph.end>& <emph type="italics"></emph>g<emph.end type="italics"></emph.end><lb></lb>non ſuſtinebunt preſſionem illatam, niſi <lb></lb>fulciantur a particulis ulterioribus <emph type="italics"></emph>h<emph.end type="italics"></emph.end>& <emph type="italics"></emph>k<emph.end type="italics"></emph.end>; <lb></lb>quatenus autem fulciuntur, premunt par<lb></lb>ticulas fulcientes; & hæ non ſuſtinebunt <lb></lb>preſſionem niſi fulciantur ab ulterioribus <lb></lb><emph type="italics"></emph>l<emph.end type="italics"></emph.end>& <emph type="italics"></emph>m<emph.end type="italics"></emph.end>eaſque premant, & ſic deinceps in infinitum. </s> <s>Preſſio igi<lb></lb>tur, quam primum propagatur ad particulas quæ non in directum <lb></lb>jacent, divaricare incipiet & oblique propagabitur in infinitum; <lb></lb>& poſtquam incipit oblique propagari, ſi inciderit in particulas <lb></lb>ulteriores, quæ non in directum jacent, iterum divaricabit; id<lb></lb>que toties, quoties in particulas non accurate in directum ja<lb></lb>centes inciderit. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Si preſſionis, a dato puncto per Fluidum propagatæ, pars <lb></lb>aliqua obſtaculo intercipiatur; pars reliqua, quæ non intercipitur, <lb></lb>divaricabit in ſpatia pone obſtaculum. </s> <s>Id quod ſic etiam de<lb></lb>monſtrari poteſt. </s> <s>A puncto <emph type="italics"></emph>A<emph.end type="italics"></emph.end>propagetur preſſio quaquaver<lb></lb>ſum, idque ſi fieri poteſt ſecundum lineas rectas, & obſtaculo <lb></lb><emph type="italics"></emph>NBCK<emph.end type="italics"></emph.end>perforato in <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>intercipiatur ea omnis, præter par<lb></lb>tem Coniformem <emph type="italics"></emph>APQ,<emph.end type="italics"></emph.end>quæ per foramen circulare <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>tranſit. </s> <s><lb></lb>Planis tranſverſis <emph type="italics"></emph>de, fg, hi<emph.end type="italics"></emph.end>diſtinguatur conus <emph type="italics"></emph>APQ<emph.end type="italics"></emph.end>in fruſta; <lb></lb>& interea dum conus <emph type="italics"></emph>ABC,<emph.end type="italics"></emph.end>preſſionem propagando, urget fru-<pb xlink:href="039/01/358.jpg" pagenum="330"></pb><arrow.to.target n="note338"></arrow.to.target>ſtum conicum ulterius <emph type="italics"></emph>degf<emph.end type="italics"></emph.end>in ſuperficie <emph type="italics"></emph>de,<emph.end type="italics"></emph.end>& hoc fruſtum <lb></lb>urget fruſtum proximum <emph type="italics"></emph>fgih<emph.end type="italics"></emph.end>in ſuperficie <emph type="italics"></emph>fg,<emph.end type="italics"></emph.end>& fruſtum illud <lb></lb>urget fruſtum tertium, & ſic deinceps in infinitum; manifeſtum <lb></lb>eſt (per motus Legem tertiam) quod fruſtum primum <emph type="italics"></emph>defg,<emph.end type="italics"></emph.end>re<lb></lb>actione fruſti ſecundi <emph type="italics"></emph>fghi,<emph.end type="italics"></emph.end>tantum urgebitur & premetur in ſu<lb></lb>perficie <emph type="italics"></emph>fg,<emph.end type="italics"></emph.end>quantum urget & premit fruſtum illud ſecundum. </s> <s><lb></lb>Fruſtum igitur <emph type="italics"></emph>degf<emph.end type="italics"></emph.end>inter conum <emph type="italics"></emph>Ade<emph.end type="italics"></emph.end>& fruſtum <emph type="italics"></emph>fhig<emph.end type="italics"></emph.end>com<lb></lb>primitur utrinque, & propterea (per Corol. </s> <s>6. Prop. </s> <s>XIX.) figu<lb></lb>ram ſuam ſervare nequit, niſi vi eadem comprimatur undique. <lb></lb><figure id="id.039.01.358.1.jpg" xlink:href="039/01/358/1.jpg"></figure><lb></lb>Eodem igitur impetu quo premitur in ſuperficiebus <emph type="italics"></emph>de, fg,<emph.end type="italics"></emph.end>cona<lb></lb>bitur cedere ad latera <emph type="italics"></emph>df, eg<emph.end type="italics"></emph.end>; ibique (cum rigidum non ſit, ſed <lb></lb>omnimodo Fluidum) excurret ac dilatabitur, niſi Fluidum am<lb></lb>biens adſit, quo conatus iſte cohibeatur. </s> <s>Proinde conatu excur<lb></lb>rendi, premet tam Fluidum ambiens ad latera <emph type="italics"></emph>df, eg<emph.end type="italics"></emph.end>quam fruſtum <lb></lb><emph type="italics"></emph>fghi<emph.end type="italics"></emph.end>eodem impetu; & propterea preſſio non minus propagabi<lb></lb>tur a lateribus <emph type="italics"></emph>df, eg<emph.end type="italics"></emph.end>in ſpatia <emph type="italics"></emph>NO, KL<emph.end type="italics"></emph.end>hinc inde, quam pro<lb></lb>pagatur a ſuperficie <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>PQ. Q.E.D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/359.jpg" pagenum="331"></pb><arrow.to.target n="note339"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note338"></margin.target>DE MOTU <lb></lb>CORPORUM.</s></p> <p type="margin"> <s><margin.target id="note339"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLII. THEOREMA XXXIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Motus omnis per Fluidum propagatus divergit a recto tramite <lb></lb>in ſpatia immota.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Propagetur motus a puncto <emph type="italics"></emph>A<emph.end type="italics"></emph.end>per foramen <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>per<lb></lb>gatque (ſi fieri poteſt) in ſpatio conico <emph type="italics"></emph>BCQP,<emph.end type="italics"></emph.end>ſecundum li<lb></lb>neas rectas divergentes a puncto <emph type="italics"></emph>C.<emph.end type="italics"></emph.end>Et ponamus primo quod <lb></lb>motus iſte ſit undarum in ſuperficie ſtagnantis aquæ. </s> <s>Sintque <lb></lb><emph type="italics"></emph>de, fg, hi, kl,<emph.end type="italics"></emph.end>&c. </s> <s>undarum ſingularum partes altiſſimæ, valli<lb></lb>bus totidem intermediis ab invicem diſtinctæ. </s> <s>Igitur quoniam <lb></lb>aqua in undarum jugis altior eſt quam in Fluidi partibus immo<lb></lb>tis <emph type="italics"></emph>LK, NO,<emph.end type="italics"></emph.end>defluet eadem de jugorum terminis <emph type="italics"></emph>e, g, i, l,<emph.end type="italics"></emph.end>&c. <lb></lb><emph type="italics"></emph>d, f, h, k,<emph.end type="italics"></emph.end>&c. </s> <s>hinc inde, verſus <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NO<emph.end type="italics"></emph.end>: & quoniam in un<lb></lb>darum vallibus depreſſior eſt quam in Fluidi partibus immotis <lb></lb><emph type="italics"></emph>KL, NO<emph.end type="italics"></emph.end>; defluet eadem de partibus illis immotis in undarum <lb></lb>valles. </s> <s>Defluxu priore undarum juga, poſteriore valles hinc <lb></lb>inde dilatantur & propagantur verſus <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NO.<emph.end type="italics"></emph.end>Et quo<lb></lb>niam motus undarum ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>fit per continuum de<lb></lb>fluxum jugorum in valles proximos, adeoque celerior non eſt <lb></lb>quam pro celeritate deſcenſus; & deſcenſus aquæ, hinc inde, ver<lb></lb>ſus <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NO<emph.end type="italics"></emph.end>eadem velocitate peragi debet; propagabitur <lb></lb>dilatatio undarum, hinc inde, verſus <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NO,<emph.end type="italics"></emph.end>eadem velo<lb></lb>citate qua undæ ipſæ ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>recta progrediuntur. </s> <s><lb></lb>Proindeque ſpatium totum hinc inde, verſus <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NO,<emph.end type="italics"></emph.end>ab <lb></lb>undis dilatatis <emph type="italics"></emph>rfgr, shis, tklt, vmnv,<emph.end type="italics"></emph.end>&c. </s> <s>occupabitur. <lb></lb><emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end>Hæc ita ſe habere quilibet in aqua ſtagnante expe<lb></lb>riri poteſt. </s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Ponamus jam quod <emph type="italics"></emph>de, fg, hi, kl, mn<emph.end type="italics"></emph.end>deſignent pul<lb></lb>ſus a puncto <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>per Medium Elaſticum, ſucceſſive propagatos. </s> <s><lb></lb>Pulſus propagari concipe per ſucceſſivas condenſationes & rare<lb></lb>factiones Medii, ſic ut pulſus cujuſque pars denſiſſima ſphæricam <lb></lb>occupet ſuperficiem circa centrum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>deſcriptam, & inter pulſus <lb></lb>ſucceſſivos æqualia intercedant intervalla. </s> <s>Deſignent autem lineæ <lb></lb><emph type="italics"></emph>de, fg, hi, kl,<emph.end type="italics"></emph.end>&c. </s> <s>denſiſſimas pulſuum partes, per foramen <emph type="italics"></emph>BC<emph.end type="italics"></emph.end><lb></lb>propagatas. </s> <s>Et quoniam Medium ibi denſius eſt quam in ſpatiis <lb></lb>hinc inde verſus <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NO,<emph.end type="italics"></emph.end>dilatabit ſeſe tam verſus ſpatia illa <lb></lb><emph type="italics"></emph>KL, NO<emph.end type="italics"></emph.end>utrinque ſita, quam verſus pulſuum rariora intervalla; <pb xlink:href="039/01/360.jpg" pagenum="332"></pb><arrow.to.target n="note340"></arrow.to.target>eoque pacto rarius ſemper evadens e regione intervallorum ac <lb></lb>denſius e regione pulſuum, participabit eorundem motum. </s> <s>Et <lb></lb>quoniam pulſuum progreſſivus motus oritur a perpetua relaxa<lb></lb>tione partium denſiorum verſus antecedentia intervalla rariora; <lb></lb>& pulſus eadem fere celeritate ſeſe in Medii partes quieſcentes <lb></lb><emph type="italics"></emph>KL, NO<emph.end type="italics"></emph.end>hinc inde relaxare debent; pulſus illi eadem fere cele<lb></lb>ritate ſeſe dilatabunt undiQ.E.I. ſpatia immota <emph type="italics"></emph>KL, NO,<emph.end type="italics"></emph.end>qua <lb></lb>propagantur directe a centro <emph type="italics"></emph>A<emph.end type="italics"></emph.end>; adeoque ſpatium totum <emph type="italics"></emph>KLON<emph.end type="italics"></emph.end><lb></lb>occupabunt. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end>Hoc experimur in Sonis, qui vel monte <lb></lb>interpoſito audiuntur, vel in cubiculum per feneſtram admiſſi ſeſe <lb></lb>in omnes cubiculi partes dilatant, inque angulis omnibus audiun<lb></lb>tur, non tam reflexi a parietibus oppoſitis, quam a feneſtra directe <lb></lb>propagati, quantum ex ſenſu judicare licet. </s></p> <p type="margin"> <s><margin.target id="note340"></margin.target>DE MOTU <lb></lb>CORPORUM</s><figure id="id.039.01.360.1.jpg" xlink:href="039/01/360/1.jpg"></figure></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Ponamus denique quod motus cujuſcunque generis <lb></lb>propagetur ab <emph type="italics"></emph>A<emph.end type="italics"></emph.end>per foramen <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>: & quoniam propagatio iſta <lb></lb>non fit, niſi quatenus partes Medii centro <emph type="italics"></emph>A<emph.end type="italics"></emph.end>propiores urgent <lb></lb>commoventque partes ulteriores; & partes quæ urgentur fluidæ <lb></lb>ſunt, ideoque recedunt quaquaverſum in regiones ubi minus pre-<pb xlink:href="039/01/361.jpg" pagenum="333"></pb>muntur: recedent eædem verſus Medii partes omnes quieſcentes, <lb></lb><arrow.to.target n="note341"></arrow.to.target>tam laterales <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>NO,<emph.end type="italics"></emph.end>quam anteriores <emph type="italics"></emph>PQ,<emph.end type="italics"></emph.end>eoque pacto <lb></lb>motus omnis, quam primum per foramen <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>tranſiit, dilatari in<lb></lb>cipiet & abinde, tanquam a principio & centro, in partes omnes <lb></lb>directe propagari. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note341"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLIII. THEOREMA XXXIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corpus omne tremulum in Medio Elaſtico propagabit motum pul<lb></lb>ſuum undiQ.E.I. directum; in Medio vero non Elaſtico motum <lb></lb>circularem excitabit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Nam partes corporis tremuli vicibus alternis eundo & <lb></lb>redeundo, itu ſuo urgebunt & propellent partes Medii ſibi proxi<lb></lb>mas, & urgendo compriment eaſdem & condenſabunt; dein re<lb></lb>ditu ſuo ſinent partes compreſſas recedere & ſeſe expandere. </s> <s>Igi<lb></lb>tur partes Medii corpori tremulo proximæ ibunt & redibunt per <lb></lb>vices, ad inſtar partium corporis illius tremuli: & qua ratione <lb></lb>partes corporis hujus agitabant haſce Medii partes, hæ ſimilibus <lb></lb>tremoribus agitatæ agitabunt partes ſibi proximas, eæque ſimiliter <lb></lb>agitatæ agitabunt ulteriores, & ſic deinceps in infinitum. </s> <s>Et <lb></lb>quemadmodum Medii partes primæ eundo condenſantur & re<lb></lb>deundo relaxantur, ſic partes reliquæ quoties eunt condenſabun<lb></lb>tur, & quoties redeunt ſeſe expandent. </s> <s>Et propterea non omnes <lb></lb>ibunt & ſimul redibunt (ſic enim determinatas ab invicem diſtan<lb></lb>tias ſervando, non rarefierent & condenſarentur per vices) ſed ac<lb></lb>cedendo ad invicem ubi condenſantur, & recedendo ubi rarefiunt, <lb></lb>aliquæ earum ibunt dum aliæ redeunt; idque vicibus alternis in <lb></lb>infinitum. </s> <s>Partes autem euntes & eundo condenſatæ, ob motum <lb></lb>ſuum progreſſivum quo feriunt obſtacula, ſunt pulſus; & propte<lb></lb>rea pulſus ſucceſſivi a corpore omni tremulo in directum propaga<lb></lb>buntur; idque æqualibus circiter ab invicem diſtantiis, ob æqua<lb></lb>lia temporis intervalla, quibus corpus tremoribus ſuis ſingulis <lb></lb>ſingulos pulſus excitat. </s> <s>Et quanquam corporis tremuli par<lb></lb>tes eant & redeant ſecundum plagam aliquam certam & determi<lb></lb>natam, tamen pulſus inde per Medium propagati ſeſe dilatabunt <lb></lb>ad latera, per Propoſitionem præcedentem; & a corpore illo tre<lb></lb>mulo tanquam centro communi, ſecundum ſuperficies propemo<lb></lb>dum ſphæricas & concentricas, undique propagabuntur. </s> <s>Cujus <pb xlink:href="039/01/362.jpg" pagenum="334"></pb><arrow.to.target n="note342"></arrow.to.target>rei exemplum aliquod habemus in Undis, quæ ſi digito tremulo <lb></lb>excitentur, non ſolum pergent hinc inde ſecundum plagam motus <lb></lb>digiti, ſed, in modum circulorum concentrieorum, digitum ſtatim <lb></lb>cingent & undique propagabuntur. </s> <s>Nam gravitas Undarum ſup<lb></lb>plet locum vis Elaſticæ. </s></p> <p type="margin"> <s><margin.target id="note342"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. Quod ſi Medium non ſit Elaſticum: quoniam ejus partes a <lb></lb>corporis tremuli partibus vibratis preſſæ condenſari nequeunt, pro<lb></lb>pagabitur motus in inſtanti ad partes ubi Medium facillime ce<lb></lb>dit, hoc eſt, ad partes quas corpus tremulum alioqui vacuas a <lb></lb>tergo relinqueret. </s> <s>Idem eſt caſus cum caſu corporis in Medio <lb></lb>quocunque projecti. </s> <s>Medium cedendo projectilibus, non rece<lb></lb>dit in infinitum; ſed in circulum eundo, pergit ad ſpatia quæ <lb></lb>corpus relinquit a tergo. </s> <s>Igitur quoties corpus tremulum per<lb></lb>git in partem quamcunque, Medium cedendo perget per circu<lb></lb>lum ad partes quas corpus relinquit; & quoties corpus regredi<lb></lb>tur ad locum priorem, Medium inde repelletur & ad locum ſuum <lb></lb>priorem redibit. </s> <s>Et quamvis corpus tremulum non ſit firmum, <lb></lb>ſed modis omnibus flexile, ſi tamen magnitudine datum maneat, <lb></lb>quoniam tremoribus ſuis nequit Medium ubivis urgere, quin alibi <lb></lb>eidem ſimul cedat; efficiet ut Medium, recedendo a partibus <lb></lb>ubi premitur, pergat ſemper in orbem ad partes quæ eidem ce<lb></lb>dunt <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hallucinantur igitur qui credunt agitationem partium <lb></lb>Flammæ ad preſſionem, per Medium ambiens, ſecundum lineas <lb></lb>rectas propagandam conducere. </s> <s>Debebit ejuſmodi preſſio non <lb></lb>ab agitatione ſola partium Flammæ, ſed a totius dilatatione deri<lb></lb>vari. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLIV. THEOREMA XXXV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si aqua in Canalis cruribus erectis<emph.end type="italics"></emph.end>KL, MN <emph type="italics"></emph>vicibus alternis <lb></lb>aſcendat & deſcendat; conſtruatur autem Pendulum cujus <lb></lb>longitudo inter punctum ſuſpenſionis & centrum oſcillationis <lb></lb>æquetur ſemiſſi longitudinis aquæ in Canali: dico quod aqua <lb></lb>aſcendet & deſcendet iiſdem temporibus quibus Pendulum <lb></lb>oſcillatur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Longitudinem aquæ menſuro ſecundum axes canalis & crurum, <lb></lb>eandem ſummæ horum axium æquando; & reſiſtentiam aquæ quæ <pb xlink:href="039/01/363.jpg" pagenum="335"></pb>oritur ab attritu canalis, hic non conſidero. </s> <s>Deſignent igitur <emph type="italics"></emph>AB, <lb></lb><arrow.to.target n="note343"></arrow.to.target>CD<emph.end type="italics"></emph.end>mediocrem altitudinem aquæ in crure utroque; & ubi aqua <lb></lb>in crure <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>aſcendit ad altitudinem <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>deſcenderit aqua in <lb></lb>crure <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>ad altitudinem <emph type="italics"></emph>GH.<emph.end type="italics"></emph.end>Sit autem <emph type="italics"></emph>P<emph.end type="italics"></emph.end>corpus pendulum, <lb></lb><emph type="italics"></emph>VP<emph.end type="italics"></emph.end>filum, <emph type="italics"></emph>V<emph.end type="italics"></emph.end>punctum ſuſpenſionis, <emph type="italics"></emph>SPQR<emph.end type="italics"></emph.end>Cyclois quam Pen<lb></lb>dulum deſcribat, <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ejus punctum infimum, <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>arcus altitudini <lb></lb><emph type="italics"></emph>AE<emph.end type="italics"></emph.end>æqualis. </s> <s>Vis, qua motus aquæ alternis vicibus acceleratur <lb></lb><figure id="id.039.01.363.1.jpg" xlink:href="039/01/363/1.jpg"></figure><lb></lb>& retardatur, eſt exceſſus ponderis aquæ in alterutro crure ſupra <lb></lb>pondus in altero, ideoque, ubi aqua in crure <emph type="italics"></emph>KL<emph.end type="italics"></emph.end>aſcendit ad <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end><lb></lb>& in crure altero deſcendit ad <emph type="italics"></emph>GH,<emph.end type="italics"></emph.end>vis illa eſt pondus duplica<lb></lb>tum aquæ <emph type="italics"></emph>EABF,<emph.end type="italics"></emph.end>& propterea eſt ad pondus aquæ totius ut <lb></lb><emph type="italics"></emph>AE<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>VP<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>PR.<emph.end type="italics"></emph.end>Vis etiam, qua pondus <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in <lb></lb>loco quovis <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>acceleratur & retardatur in Cycloide, (per Corol. </s> <s><lb></lb>Prop. </s> <s>LI.) eſt ad ejus pondus totum, ut ejus diſtantia <emph type="italics"></emph>YQ<emph.end type="italics"></emph.end>a loco <lb></lb>infimo <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>ad Cycloidis longitudinem <emph type="italics"></emph>PR.<emph.end type="italics"></emph.end>Quare aquæ & pen<lb></lb>duli, æqualia ſpatia <emph type="italics"></emph>AE, PQ<emph.end type="italics"></emph.end>deſcribentium, vires motrices ſunt <lb></lb>ut pondera movenda; ideoque, ſi aqua & pendulum in princi<lb></lb>pio quieſcunt, vires illæ movebunt eadem æqualiter tempori<lb></lb>bus æqualibus, efficientque ut motu reciproco ſimul eant & re<lb></lb>deant. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note343"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur aquæ aſcendentis & deſcendentis, ſive motus in<lb></lb>tenſior ſit ſive remiſſior, vices omnes ſunt Iſochronæ. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si longitudo aquæ totius in canali ſit pedum <emph type="italics"></emph>Pariſien<lb></lb>ſium<emph.end type="italics"></emph.end>6 1/9: aqua tempore minuti unius ſecundi deſcendet, & tem<lb></lb>pore minuti alterius ſecundi aſcendet; & ſic deinceps vicibus al<lb></lb>ternis in infinitum. </s> <s>Nam pendulum pedum (3 1/18) longitudinis, <lb></lb>tempore minuti unius ſecundi oſcillatur. <pb xlink:href="039/01/364.jpg" pagenum="336"></pb><arrow.to.target n="note344"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note344"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Aucta autem vel diminuta longitudine aquæ, auge<lb></lb>tur vel diminuitur tempus reciprocationis in longitudinis ratione <lb></lb>ſubduplicata. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLV. THEOREMA XXXVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Undarum velocitas eſt in ſubduplicata ratione latitudinum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Conſequitur ex conſtructione Propoſitionis ſequentis. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLVI. PROBLEMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire velocitatem Undarum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Conſtituatur Pendulum cujus longitudo, inter punctum ſuſpen<lb></lb>ſionis & centrum oſcillationis, æquetur latitudini Undarum: & quo <lb></lb>tempore pendulum illud oſcillationes ſingulas peragit, eodem Un<lb></lb>dæ progrediendo latitudinem ſuam propemodum conficient. </s></p> <p type="main"> <s>Undarum latitudinem voco menſuram tranſverſam, quæ vel val<lb></lb>libus imis, vel ſummis culminibus interjacet. </s> <s>Deſignet <emph type="italics"></emph>ABCDEF<emph.end type="italics"></emph.end><lb></lb>ſuperficiem aquæ ſtagnantis, undis ſucceſſivis aſcendentem ac deſ<lb></lb>cendentem; ſintque <emph type="italics"></emph>A, C, E,<emph.end type="italics"></emph.end>&c. </s> <s>undarum culmina, & <emph type="italics"></emph>B, D, F,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>valles intermedii. </s> <s>Et quoniam motus undarum fit per aquæ ſuc<lb></lb>ceſſivum aſcenſum & deſcenſum, ſic ut ejus partes <emph type="italics"></emph>A, C, E,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>quæ nunc altiſſimæ ſunt, mox fiant infimæ; & vis motrix, qua <lb></lb>partes altiſſimæ deſcendunt & infimæ aſcendunt, eſt pondus aquæ <lb></lb>elevatæ; alternus ille aſcenſus & deſcenſus analogus erit motui re<lb></lb>ciproco aquæ in canali, eaſdemque temporis leges obſervabit: & <lb></lb>propterea (per Prop. </s> <s>XLIV) ſi diſtantiæ inter undarum loca altiſ<lb></lb>ſima <emph type="italics"></emph>A, C, E<emph.end type="italics"></emph.end>& infima <emph type="italics"></emph>B, D, F<emph.end type="italics"></emph.end>æquentur duplæ penduli longitu<lb></lb>dini; partes altiſſimæ <emph type="italics"></emph>A, C, E,<emph.end type="italics"></emph.end>tempore oſcillationis unius evadent <lb></lb>infimæ, & tempore oſcillationis alterius denuo aſcendent. </s> <s>Igitur <lb></lb>inter tranſitum Undarum ſingularum tempus erit oſcillationum <lb></lb>duarum; hoc eſt, Unda deſcribet latitudinem ſuam, quo tempore <lb></lb>pendulum illud bis oſcillatur; ſed eodem tempore pendulum, cu<lb></lb>jus longitudo quadrupla eſt, adeoque æquat undarum latitudinem, <lb></lb>oſcillabitur ſemel. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Igitur Undæ, quæ pedes <emph type="italics"></emph>Pariſienſes<emph.end type="italics"></emph.end>(3 1/18) latæ ſunt, <lb></lb>tempore minuti unius ſecundi progrediendo latitudinem ſuam con<lb></lb>ficient; adeoque tempore minuti unius primi percurrent pedes <lb></lb>183 1/3, & horæ ſpatio pedes 11000 quamproxime. </s><pb xlink:href="039/01/365.jpg" pagenum="337"></pb><figure id="id.039.01.365.1.jpg" xlink:href="039/01/365/1.jpg"></figure></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et undarum majorum vel minorum ve</s></p> <p type="main"> <s><arrow.to.target n="note345"></arrow.to.target>locitas augebitur vel diminuetur in ſubduplicata <lb></lb>ratione latitudinis. </s></p> <p type="margin"> <s><margin.target id="note345"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Hæc ita ſe habent ex Hypotheſi quod partes <lb></lb>aquæ recta aſcendunt vel recta deſcendunt; ſed <lb></lb>aſcenſus & deſcenſus ille verius fit per circulum, <lb></lb>ideoque tempus hac Propoſitione non niſi quam<lb></lb>proxime definitum eſſe affirmo. </s></p> <p type="main"> <s><emph type="center"></emph>PROP. XLVII. THEOR. XXXVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Pulſibus per Fluidum propagatis, ſingulæ Fluidi <lb></lb>particulæ, motu reciproco breviſſimo euntes & <lb></lb>redeuntes, accelerantur ſemper & retardantur <lb></lb>pro lego oſcillantis Penduli.<emph.end type="italics"></emph.end></s><figure id="id.039.01.365.2.jpg" xlink:href="039/01/365/2.jpg"></figure></p> <p type="main"> <s>Deſignent <emph type="italics"></emph>AB, BC, CD,<emph.end type="italics"></emph.end><lb></lb>&c. </s> <s>pulſuum ſucceſſivorum <lb></lb>æquales diſtantias; <emph type="italics"></emph>ABC<emph.end type="italics"></emph.end><lb></lb>plagam motus pulſuum ab <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>propagati; <emph type="italics"></emph>E, <lb></lb>F, G<emph.end type="italics"></emph.end>puncta tria Phyſica Me<lb></lb>dii quieſcentis, in recta <emph type="italics"></emph>AC<emph.end type="italics"></emph.end><lb></lb>ad æquales ab invicem di<lb></lb>ſtantias ſita; <emph type="italics"></emph>Ee, Ff, Gg,<emph.end type="italics"></emph.end><lb></lb>ſpatia æqualia perbrevia per <lb></lb>quæ puncta illa motu reciproco ſingulis vibratio<lb></lb>nibus eunt & redeunt; <foreign lang="grc">ε, φ, γ</foreign> loca quævis inter<lb></lb>media eorundem punctorum; & <emph type="italics"></emph>EF, FG<emph.end type="italics"></emph.end>lineolas <lb></lb>Phyſicas ſeu Medii partes lineares punctis illis in<lb></lb>terjectas, & ſucceſſive tranſlatas in loca <foreign lang="grc">εφ, φγ</foreign> & <lb></lb><emph type="italics"></emph>ef, fg.<emph.end type="italics"></emph.end>Rectæ <emph type="italics"></emph>Ee<emph.end type="italics"></emph.end>æqualis ducatur recta <emph type="italics"></emph>PS.<emph.end type="italics"></emph.end><lb></lb>Biſecetur eadem in <emph type="italics"></emph>O,<emph.end type="italics"></emph.end>centroque <emph type="italics"></emph>O<emph.end type="italics"></emph.end>& intervallo <lb></lb><emph type="italics"></emph>OP<emph.end type="italics"></emph.end>deſcribatur circulus <emph type="italics"></emph>SIPi.<emph.end type="italics"></emph.end>Per hujus cir<lb></lb>cumferentiam totam cum partibus ſuis exponatur <lb></lb>tempus totum vibrationis unius cum ipſius parti<lb></lb>bus proportionalibus; ſic ut completo tempore <lb></lb>quovis <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>PHSh,<emph.end type="italics"></emph.end>ſi demittatur ad <emph type="italics"></emph>PS<emph.end type="italics"></emph.end><lb></lb>perpendiculum <emph type="italics"></emph>HL<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>hl,<emph.end type="italics"></emph.end>& capiatur <emph type="italics"></emph>E<emph.end type="italics"></emph.end><foreign lang="grc">ε</foreign> æqua<lb></lb>lis <emph type="italics"></emph>PL<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>Pl,<emph.end type="italics"></emph.end>punctum Phyſicum <emph type="italics"></emph>E<emph.end type="italics"></emph.end>reperiatur <pb xlink:href="039/01/366.jpg" pagenum="338"></pb><arrow.to.target n="note346"></arrow.to.target>in <foreign lang="grc">ε. </foreign></s> <s>Hac lege punctum quodvis <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>eundo ab <emph type="italics"></emph>E<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.366.1.jpg" xlink:href="039/01/366/1.jpg"></figure><lb></lb>per <foreign lang="grc">ε</foreign> ad <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>& inde redeundo per <foreign lang="grc">ε</foreign> ad <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>iiſdem <lb></lb>accelerationis ac retardationis gradibus vibratio<lb></lb>nes ſingulas peraget cum oſcillante Pendulo. </s> <s>Pro<lb></lb>bandum eſt quod ſingula Medii puncta Phyſica <lb></lb>tali motu agitari debeant. </s> <s>Fingamus igitur Me<lb></lb>dium tali motu a cauſa quacunque cieri, & videa<lb></lb>mus quid inde ſequatur. </s></p> <p type="margin"> <s><margin.target id="note346"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>In circumferentia <emph type="italics"></emph>PHSh<emph.end type="italics"></emph.end>capiantur æquales ar<lb></lb>cus <emph type="italics"></emph>HI, IK<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>hi, ik,<emph.end type="italics"></emph.end>eam habentes rationem <lb></lb>ad circumferentiam totam quam habent æquales <lb></lb>rectæ <emph type="italics"></emph>EF, FG<emph.end type="italics"></emph.end>ad pulſuum intervallum totum <lb></lb><emph type="italics"></emph>BC.<emph.end type="italics"></emph.end>Et demiſſis perpendiculis <emph type="italics"></emph>IM, KN<emph.end type="italics"></emph.end>vel <lb></lb><emph type="italics"></emph>im, kn<emph.end type="italics"></emph.end>; quoniam puncta <emph type="italics"></emph>E, F, G<emph.end type="italics"></emph.end>motibus ſimili<lb></lb>bus ſucceſſive agitantur, & vibrationes ſuas integras <lb></lb>ex itu & reditu compoſitas interea peragunt dum <lb></lb><figure id="id.039.01.366.2.jpg" xlink:href="039/01/366/2.jpg"></figure><lb></lb>pulſus transfertur a <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>C<emph.end type="italics"></emph.end>; <lb></lb>ſi <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>PHSh<emph.end type="italics"></emph.end>ſit tem<lb></lb>pus ab initio motus puncti <lb></lb><emph type="italics"></emph>E,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>PHSi<emph.end type="italics"></emph.end>tem<lb></lb>pus ab initio motus puncti <lb></lb><emph type="italics"></emph>F,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>PHSk<emph.end type="italics"></emph.end>tem<lb></lb>pus ab initio motus puncti <lb></lb><emph type="italics"></emph>G<emph.end type="italics"></emph.end>; & propterea <emph type="italics"></emph>E<foreign lang="grc">ε</foreign>, F<foreign lang="grc">φ</foreign>, <lb></lb>G<emph.end type="italics"></emph.end><foreign lang="grc">γ</foreign> erunt ipſis <emph type="italics"></emph>PL, PM, <lb></lb>PN<emph.end type="italics"></emph.end>in itu punctorum, vel <lb></lb>ipſis <emph type="italics"></emph>Pl, Pm, Pn<emph.end type="italics"></emph.end>in punctorum reditu, æqua<lb></lb>les reſpective. </s> <s>Unde <foreign lang="grc">εγ</foreign> ſeu <emph type="italics"></emph>EG+G<foreign lang="grc">γ</foreign>-E<emph.end type="italics"></emph.end><foreign lang="grc">ε</foreign><lb></lb>in itu punctorum æqualis erit <emph type="italics"></emph>EG-LN,<emph.end type="italics"></emph.end>in re<lb></lb>ditu autem æqualis <emph type="italics"></emph>EG+ln.<emph.end type="italics"></emph.end>Sed <foreign lang="grc">εγ</foreign> latitudo eſt <lb></lb>ſeu expanſio partis Medii <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>in loco <foreign lang="grc">εγ</foreign>; & <lb></lb>propterea expanſio partis illius in itu, eſt ad ejus <lb></lb>expanſionem mediocrem, ut <emph type="italics"></emph>EG-LN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>; <lb></lb>in reditu autem ut <emph type="italics"></emph>EG+ln<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>EG+LN<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>EG.<emph.end type="italics"></emph.end>Quare cum ſit <emph type="italics"></emph>LN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>KH<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>IM<emph.end type="italics"></emph.end>ad <lb></lb>radium <emph type="italics"></emph>OP,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>KH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>ut circumferentia <lb></lb><emph type="italics"></emph>PHShP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>id eſt (ſi ponatur V pro ra<lb></lb>dio circuli circumferentiam habentis æqualem in<lb></lb>tervallo pulſuum <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>OP<emph.end type="italics"></emph.end>ad V; & ex æ<lb></lb>quo <emph type="italics"></emph>LN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>EG,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>IM<emph.end type="italics"></emph.end>ad V: erit expanſio <lb></lb>partis <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>punctive Phyſici <emph type="italics"></emph>F<emph.end type="italics"></emph.end>in loco <foreign lang="grc">εγ</foreign>, ad ex-<pb xlink:href="039/01/367.jpg" pagenum="339"></pb>panſionem mediocrem quam pars illa habet in loco ſuo primo <lb></lb><arrow.to.target n="note347"></arrow.to.target><emph type="italics"></emph>EG,<emph.end type="italics"></emph.end>ut V-<emph type="italics"></emph>IM<emph.end type="italics"></emph.end>ad V in itu, utque V+<emph type="italics"></emph>im<emph.end type="italics"></emph.end>ad V in reditu. </s> <s>Unde <lb></lb>vis elaſtica puncti <emph type="italics"></emph>F<emph.end type="italics"></emph.end>in loco <foreign lang="grc">εγ</foreign>, eſt ad vim ejus elaſticam medio<lb></lb>crem in loco <emph type="italics"></emph>EG,<emph.end type="italics"></emph.end>ut (I/V-<emph type="italics"></emph>IM<emph.end type="italics"></emph.end>) ad I/V in itu, in reditu vero ut <lb></lb>(I/V+<emph type="italics"></emph>im<emph.end type="italics"></emph.end>) ad I/V. </s> <s>Et eodem argumento vires elaſticæ punctorum <lb></lb>Phyſieorum <emph type="italics"></emph>E<emph.end type="italics"></emph.end>& <emph type="italics"></emph>G<emph.end type="italics"></emph.end>in itu, ſunt ut (I/V-<emph type="italics"></emph>HL<emph.end type="italics"></emph.end>) & (I/V-<emph type="italics"></emph>KN<emph.end type="italics"></emph.end>) <lb></lb>ad I/V; & virium differentia ad Medii vim elaſticam mediocrem, <lb></lb>ut (<emph type="italics"></emph>HL-KN<emph.end type="italics"></emph.end>/VV-VX<emph type="italics"></emph>HL<emph.end type="italics"></emph.end>-VX<emph type="italics"></emph>KN+HLXKN<emph.end type="italics"></emph.end>) ad I/V. </s> <s>Hoc eſt, ut <lb></lb>(<emph type="italics"></emph>HL-KN<emph.end type="italics"></emph.end>/VV) ad I/V, ſive ut <emph type="italics"></emph>HL-KN<emph.end type="italics"></emph.end>ad V, ſi modo (ob angu<lb></lb>ſtos limites vibrationum) ſupponamus <emph type="italics"></emph>HL<emph.end type="italics"></emph.end>& <emph type="italics"></emph>KN<emph.end type="italics"></emph.end>indefinite <lb></lb>minores eſſe quantitate V. </s> <s>Quare cum quantitas V detur, diffe<lb></lb>rentia virium eſt ut <emph type="italics"></emph>HL-KN,<emph.end type="italics"></emph.end>hoc eſt (ob proportionales <lb></lb><emph type="italics"></emph>HL-KN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>HK,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>OM<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>OI<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>OP,<emph.end type="italics"></emph.end>dataſque <emph type="italics"></emph>HK<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>OP<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>OM<emph.end type="italics"></emph.end>; id eſt, ſi <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end>biſecetur in <foreign lang="grc">Ω</foreign>, ut <foreign lang="grc">Ωφ. </foreign></s> <s>Et eodem <lb></lb>argumento differentia virium elaſticarum punctorum Phyſieorum <lb></lb><foreign lang="grc">ε</foreign> & <foreign lang="grc">γ</foreign>, in reditu lineolæ Phyſicæ <foreign lang="grc">εγ</foreign> eſt ut <foreign lang="grc">Ωφ. </foreign></s> <s>Sed differentia <lb></lb>illa (id eſt, exceſſus vis elaſticæ puncti <foreign lang="grc">ε</foreign> ſupra vim elaſticam pun<lb></lb>cti <foreign lang="grc">γ</foreign>,) eſt vis qua interjecta Medii lineola Phyſica <foreign lang="grc">εγ</foreign> acceleratur; <lb></lb>& propterea vis acceleratrix lineolæ Phyſicæ <foreign lang="grc">εγ</foreign>, eſt ut ipſius di<lb></lb>ſtantia a medio vibrationis loco <foreign lang="grc">Ω. </foreign></s> <s>Proinde tempus (per Prop. </s> <s><lb></lb>XXXVIII. Lib. </s> <s>1.) recte exponitur per arcum <emph type="italics"></emph>PI<emph.end type="italics"></emph.end>; & Medii pars <lb></lb>linearis <foreign lang="grc">εγ</foreign> lege præſcripta movetur, id eſt, lege oſcillantis Pen<lb></lb>duli: eſtque par ratio partium omnium linearium ex quibus Me<lb></lb>dium totum componitur. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note347"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc patet quod numerus pulſuum propagatorum idem <lb></lb>ſit cum numero vibrationum corporis tremuli, neque multiplica<lb></lb>tur in eorum progreſſu. </s> <s>Nam lineola Phyſica <foreign lang="grc">εγ</foreign>, quamprimum <lb></lb>ad locum ſuum primum redierit, quieſcet; neQ.E.D.inceps move<lb></lb>bitur, niſi vel ab impetu corporis tremuli, vel ab impetu pulſuum <lb></lb>qui a corpore tremulo propagantur, motu novo cieatur. </s> <s>Quie<lb></lb>ſcet igitur quamprimum pulſus a corpore tremulo propagari <lb></lb>deſinunt. <pb xlink:href="039/01/368.jpg" pagenum="340"></pb><arrow.to.target n="note348"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note348"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLVIII. THEOREMA XXXVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Pulſuum in Fluido Elaſtico propagatorum velocitates, ſunt in ra<lb></lb>tione compoſita ex ſubduplicata ratione vis Elaſticæ directe & <lb></lb>ſubduplicata ratione denſitatis inverſe; ſi modo Fluidi vis <lb></lb>Elaſtica ejuſdem condenſationi proportionalis eſſe ſupponatur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>1. Si Media ſint homogenea, & pulſuum diſtantiæ in his <lb></lb>Mediis æquentur inter ſe, ſed motus in uno Medio intenſior ſit: <lb></lb>contractiones & dilatationes partium analogarum erunt ut iidem <lb></lb>motus. </s> <s>Accurata quidem non eſt hæc proportio. </s> <s>Verum tamen <lb></lb>niſi contractiones & dilatationes ſint valde intenſæ, non errabit <lb></lb>ſenſibiliter, ideoque pro Phyſice accurata haberi poteſt. </s> <s>Sunt <lb></lb>autem vires Elaſticæ motrices ut contractiones & dilatationes; & <lb></lb>velocitates partium æqualium ſimul genitæ ſunt ut vires. </s> <s>Ideoque <lb></lb>æquales & correſpondentes pulſuum correſpondentium partes, <lb></lb>itus & reditus ſuos per ſpatia contractionibus & dilatationibus <lb></lb>proportionalia, cum velocitatibus quæ ſunt ut ſpatia, ſimul pera<lb></lb>gent: & propterea pulſus, qui tempore itus & reditus unius lati<lb></lb>tudinem ſuam progrediendo conficiunt, & in loca pulſuum pro<lb></lb>xime præcedentium ſemper ſuccedunt, ob æqualitatem diſtantia<lb></lb>rum, æquali cum velocitate in Medio utroque progredientur. </s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>2. Sin pulſuum diſtantiæ ſeu longitudines ſint majores in <lb></lb>uno Medio quam in altero; ponamus quod partes correſponden<lb></lb>tes ſpatia latitudinibus pulſuum proportionalia ſingulis vicibus <lb></lb>eundo & redeundo deſcribant: & æquales erunt earum contra<lb></lb>ctiones & dilatationes. </s> <s>Ideoque ſi Media ſint homogenea, æqua<lb></lb>les erunt etiam vires illæ Elaſticæ motrices quibus reciproco motu <lb></lb>agitantur. </s> <s>Materia autem his viribus movenda, eſt ut pulſuum <lb></lb>latitudo; & in eadem ratione eſt ſpatium per quod ſingulis vici<lb></lb>bus eundo & redeundo moveri debent. </s> <s>Eſtque tempus itus & <lb></lb>reditus unius in ratione compoſita ex ratione ſubduplicata mate<lb></lb>riæ & ratione ſubduplicata ſpatii, atque adeo ut ſpatium. </s> <s>Pulſus <lb></lb>autem temporibus itus & reditus unius eundo latitudines ſuas <lb></lb>conficiunt, hoc eſt, ſpatia temporibus proportionalia percurrunt; <lb></lb>& propterea ſunt æquiveloces. </s></p> <p type="main"> <s><emph type="italics"></emph>Caſ<emph.end type="italics"></emph.end>3 In Mediis igitur denſitate & vi Elaſtica paribus, pulſus <lb></lb>omnes ſunt æquiveloces. </s> <s>Quod ſi Medii vel denſitas vel vis Ela<lb></lb>ſtica intendatur, quoniam vis motrix in ratione vis Elaſticæ, & <lb></lb>materia movenda in ratione denſitatis augetur; tempus quo mo-<pb xlink:href="039/01/369.jpg" pagenum="341"></pb>tus iidem peragantur ac prius, augebitur in ſubduplicata ratione <lb></lb><arrow.to.target n="note349"></arrow.to.target>denſitatis, ac diminuetur in ſubduplicata ratione vis Elaſticæ. </s> <s>Et <lb></lb>propterea velocitas pulſuum erit in ratione compoſita ex ratione <lb></lb>ſubduplicata denſitatis Medii inverſe & ratione ſubduplicata vis <lb></lb>Elaſticæ directe. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note349"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Hæc Propoſitio ulterius patebit ex conſtructione ſequentis. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLIX. PROBLEMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Datis Medii denſitate & vi Elaſtica, invenire velocitatem pul<lb></lb>ſuum.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Fingamus Medium ab incumbente pondere, pro more Aeris <lb></lb>noſtri comprimi; ſitque A altitudo Medii homogenei, cujus pon<lb></lb>dus adæquet pondus incumbens, & cujus denſitas eadem ſit cum <lb></lb>denſitate Medii compreſſi, in quo pulſus propagantur. </s> <s>Conſti<lb></lb>tui autem intelligatur Pendulum, cujus longitudo inter punctum <lb></lb>ſuſpenſionis & centrum oſcillationis ſit A: & quo tempore Pen<lb></lb>dulum illud oſcillationem integram ex itu & reditu compoſitam <lb></lb>peragit, eodem pulſus eundo conficiet ſpatium circumferentiæ <lb></lb>circuli radio A deſcripti æquale. </s></p> <p type="main"> <s>Nam ſtantibus quæ in Propoſitione XLVII conſtructa ſunt, <lb></lb>ſi linea quævis Phyſica <emph type="italics"></emph>EF,<emph.end type="italics"></emph.end>ſingulis vibrationibus deſcribendo <lb></lb>ſpatium <emph type="italics"></emph>PS,<emph.end type="italics"></emph.end>urgeatur in extremis itus & reditus cujuſque locis <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>a vi Elaſtica quæ ipſius ponderi æquetur; peraget hæc <lb></lb>vibrationes ſingulas quo tempore eadem in Cycloide, cujus peri<lb></lb>meter tota longitudini <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>æqualis eſt, oſcillari poſſet: id adeo <lb></lb>quia vires æquales æqualia corpuſcula per æqualia ſpatia ſimul im<lb></lb>pellent. </s> <s>Quare cum oſcillationum tempora ſint in ſubduplicata <lb></lb>ratione longitudinis Pendulorum, & longitudo Penduli æquetur <lb></lb>dimidio arcui Cycloidis totius; foret tempus vibrationis unius ad <lb></lb>tempus oſcillationis Penduli cujus longitudo eſt A, in ſubdupli<lb></lb>cata ratione longitudinis 1/2 <emph type="italics"></emph>PS<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>PO<emph.end type="italics"></emph.end>ad longitudinem A. </s> <s>Sed <lb></lb>vis Elaſtica qua lineola Phyſica <emph type="italics"></emph>EG,<emph.end type="italics"></emph.end>in locis ſuis extremis <emph type="italics"></emph>P, S<emph.end type="italics"></emph.end><lb></lb>exiſtens, urgetur, erat (in demonſtratione Propoſitionis XLVII) <lb></lb>ad ejus vim totam Elaſticam ut <emph type="italics"></emph>HL-KN<emph.end type="italics"></emph.end>ad V, hoc eſt <lb></lb>(cum punctum <emph type="italics"></emph>K<emph.end type="italics"></emph.end>jam incidat in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>HK<emph.end type="italics"></emph.end>ad V: & vis illa <lb></lb>tota, hoc eſt pondus incumbens, quo lineola <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>comprimitur, <lb></lb>eſt ad pondus lineolæ ut ponderis incumbentis altitudo A ad line<lb></lb>olæ longitudinem <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>; adeoque ex æquo, vis qua lineola <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>in <lb></lb>locis ſuis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph.end type="italics"></emph.end>urgetur, eſt ad lineolæ illius pondus ut <emph type="italics"></emph>HK<emph.end type="italics"></emph.end>XA <lb></lb>ad VX<emph type="italics"></emph>EG,<emph.end type="italics"></emph.end>ſive ut <emph type="italics"></emph>PO<emph.end type="italics"></emph.end>XA ad VV, nam <emph type="italics"></emph>HK<emph.end type="italics"></emph.end>erat ad <emph type="italics"></emph>EG<emph.end type="italics"></emph.end>ut <pb xlink:href="039/01/370.jpg" pagenum="342"></pb><arrow.to.target n="note350"></arrow.to.target><emph type="italics"></emph>PO<emph.end type="italics"></emph.end>ad V. </s> <s>Quare cum tempora, quibus æqualia corpora per <lb></lb>æqualia ſpatia impelluntur, ſint reciproce in ſubduplicata ratione <lb></lb>virium, erit tempus vibrationis unius urgente vi illa Elaſtica, ad <lb></lb>tempus vibrationis urgente vi ponderis, in ſubduplicata ratione <lb></lb>VV ad <emph type="italics"></emph>PO<emph.end type="italics"></emph.end>XA, atque adeo ad tempus oſcillationis Penduli cu<lb></lb>jus longitudo eſt A, in ſubduplicata ratione VV ad <emph type="italics"></emph>PO<emph.end type="italics"></emph.end>XA, & <lb></lb>ſubduplicata ratione <emph type="italics"></emph>PO<emph.end type="italics"></emph.end>ad A conjunctim; id eſt, in ratione in<lb></lb>tegra V ad A. </s> <s>Sed tempore vibrationis unius ex itu & reditu com<lb></lb>poſitæ, pulſus progrediendo conficit latitudinem ſuam <emph type="italics"></emph>BC.<emph.end type="italics"></emph.end>Ergo <lb></lb>tempus quo pulſus percurrit ſpatium <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>eſt ad tempus oſcillati<lb></lb>onis unius ex itu & reditu compoſitæ, ut V ad A, id eſt, ut <emph type="italics"></emph>BC<emph.end type="italics"></emph.end><lb></lb>ad circumferentiam circuli cujus radius eſt A. </s> <s>Tempus autem, <lb></lb>quo pulſus percurret ſpatium <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end>eſt ad tempus quo percurret <lb></lb>longitudinem huic circumferentiæ æqualem, in eadem ratione; <lb></lb>ideoque tempore talis oſcillationis pulſus percurret longitudinem <lb></lb>huic circumferentiæ æqualem. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note350"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Velocitas pulſuum ea eſt quam acquirunt Gravia, æqua<lb></lb>liter accelerato motu cadendo, & caſu ſuo deſcribendo dimidium <lb></lb>altitudinis A. </s> <s>Nam tempore caſus hujus, cum velocitate cadendo <lb></lb>acquiſita, pulſus percurret ſpatium quod erit æquale toti altitu<lb></lb>dini A, adeoque tempore oſcillationis unius ex itu & reditu com<lb></lb>poſitæ, percurret ſpatium æquale circumferentiæ circuli radio A <lb></lb>deſcripti: eſt enim tempus caſus ad tempus oſcillationis ut radius <lb></lb>circuli ad ejuſdem circumferentiam. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Unde cum altitudo illa A ſit ut Fluidi vis Elaſtica di<lb></lb>recte & denſitas ejuſdem inverſe; velocitas pulſuum erit in ratione <lb></lb>compoſita ex ſubduplicata ratione denſitatis inverſe & ſubdupli<lb></lb>cata ratione vis Elaſticæ directe. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO L. PROBLEMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire pulſuum diſtantias.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Corporis, cujus tremore pulſus excitantur, inveniatur numerus <lb></lb>Vibrationum dato tempore. </s> <s>Per numerum illum dividatur ſpa<lb></lb>tium quod pulſus eodem tempore percurrere poſſit, & pars in<lb></lb>venta erit pulſus unius latitudo. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Spectant Propoſitiones noviſſimæ ad motum Lucis & Sonorum. </s> <s><lb></lb>Lux enim cum propagetur ſecundum lineas rectas, in actione ſola <pb xlink:href="039/01/371.jpg" pagenum="343"></pb>(per Prop. </s> <s>XLI. & XLII.) conſiſtere nequit. </s> <s>Soni vero propterea <lb></lb><arrow.to.target n="note351"></arrow.to.target>quod a corporibus tremulis oriantur, nihil aliud ſunt quam aeris <lb></lb>pulſus propagati, per Prop. </s> <s>XLIII. </s> <s>Confirmatur id ex tremoribus <lb></lb>quos excitant in corporibus objectis, ſi modo vehementes ſint & <lb></lb>graves, quales ſunt ſoni Tympanorum. </s> <s>Nam tremores celeriores <lb></lb>& breviores difficilius excitantur. </s> <s>Sed & ſonos quoſvis, in chor<lb></lb>das corporibus ſonoris uniſonas impactos, exeſtare tremores notiſ<lb></lb>ſimum eſt. </s> <s>Confirmatur etiam ex velocitate ſonorum. </s> <s>Nam cum <lb></lb>pondera ſpecifica Aquæ pluvialis & Argenti vivi ſint ad invicem <lb></lb>ut 1 ad 13 2/3 circiter, & ubi Mercurius in <emph type="italics"></emph>Barometro<emph.end type="italics"></emph.end>altitudinem <lb></lb>attingit digitorum <emph type="italics"></emph>Anglieorum<emph.end type="italics"></emph.end>30, pondus ſpecificum Aeris & <lb></lb>aquæ pluvialis ſint ad invicem ut 1 ad 870 circiter: erunt pon<lb></lb>dera ſpecifica aeris & argenti vivi ut 1 ad 11890. Proinde cum <lb></lb>altitudo argenti vivi ſit 30 digitorum, altitudo aeris uniformis, <lb></lb>cujus pondus aerem noſtrum ſubjectum comprimere poſſet, erit <lb></lb>356700 digitorum, ſeu pedum <emph type="italics"></emph>Anglieorum<emph.end type="italics"></emph.end>29725. Eſtque hæc <lb></lb>altitudo illa ipſa quam in conſtructione ſuperioris Problematis no<lb></lb>minavimus A. </s> <s>Circuli radio 29725 pedum deſcripti circumferen<lb></lb>tia eſt pedum 186768. Et cum Pendulum digitos 39 1/5 longum, <lb></lb>oſcillationem ex itu & reditu compoſitam, tempore minutorum <lb></lb>duorum ſecundorum, uti notum eſt, abſolvat; Pendulum pedes <lb></lb>29725, ſeu digitos 356700 longum, oſcillationem conſimilem tem<lb></lb>pore minutorum ſecundorum 190 3/4 abſolvere debebit. </s> <s>Eo igitur <lb></lb>tempore ſonus progrediendo conſiciet pedes 186768, adeoque <lb></lb>tempore minuti unius ſecundi pedes 979. </s></p> <p type="margin"> <s><margin.target id="note351"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s>Cæterum in hoc computo nulla habetur ratio craſſitudinis ſoli<lb></lb>darum particularum aeris, per quam ſonus utique propagatur in <lb></lb>inſtanti. </s> <s>Cum pondus aeris ſit ad pondus aquæ ut 1 ad 870, & <lb></lb>ſales ſint fere duplo denſiores quam aqua; ſi particulæ aeris po<lb></lb>nantur eſſe ejuſdem circiter denſitatis cum particulis vel aquæ <lb></lb>vel ſalium, & raritas aeris oriatur ab intervallis particularum: <lb></lb>diameter particulæ aeris erit ad intervallum inter centra parti<lb></lb>cularum, ut 1 ad 9 vel 10 circiter, & ad intervallum inter par<lb></lb>ticulas ut 1 ad 8 vel 9. Proinde ad pedes 979 quos ſonus tem<lb></lb>pore minuti unius ſecundi juxta calculum ſuperiorem conficiet, <lb></lb>addere licet pedes (979/9) ſeu 109 circiter, ob craſſitudinem particu<lb></lb>larum aeris: & ſie ſonus tempore minuti unius ſecundi conficiet <lb></lb>pedes 1088 circiter. </s></p> <p type="main"> <s>His adde quod vapores in aere latentes, cum ſint alterius ela<lb></lb>teris & alterius toni, vix aut ne vix quidem participant motum <lb></lb>aeris veri quo ſoni propagantur. </s> <s>His autem quieſcentibus, mo-<pb xlink:href="039/01/372.jpg" pagenum="344"></pb><arrow.to.target n="note352"></arrow.to.target>tus ille celerius propagabitur per ſolum aerem verum, idQ.E.I. <lb></lb>ſubduplicata ratione minoris materiæ. </s> <s>Ut ſi Atmoſphæra con<lb></lb>ſtet ex decem partibus aeris veri & una parte vaporum, motus <lb></lb>ſonorum celerior erit in ſubduplicata ratione 11 ad 10, vel in in<lb></lb>tegra circiter ratione 21 ad 20, quam ſi propagaretur per undecim <lb></lb>partes aeris veri: ideoque motus ſonorum ſupra inventus, augen<lb></lb>dus erit in hac ratione. </s> <s>Quo pacto ſonus, tempore minuti unius <lb></lb>ſecundi, conficiet pedes 1142. </s></p> <p type="margin"> <s><margin.target id="note352"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s>Hæc ita ſe habere debent tempore verno & autumnali, ubi aer <lb></lb>per calorem temperatum rareſcit & ejus vis elaſtica nonnihil in<lb></lb>tenditur. </s> <s>At hyberno tempore, ubi aer per frigus condenſatur, <lb></lb>& ejus vis elaſtica remittitur, motus ſonorum tardior eſſe debet in <lb></lb>ſubduplicata ratione denſitatis; & viciſſim æſtivo tempore debet <lb></lb>eſſe velocior. </s></p> <p type="main"> <s>Conſtat autem per experimenta quod ſoni tempore minuti uNI<lb></lb>us ſecundi eundo, conficiunt pedes <emph type="italics"></emph>Londinenſes<emph.end type="italics"></emph.end>plus minus 1142, <lb></lb><emph type="italics"></emph>Pariſienſes<emph.end type="italics"></emph.end>vero 1070. </s></p> <p type="main"> <s>Cognita ſonorum velocitate innoteſcunt etiam intervalla pul<lb></lb>ſuum. </s> <s>Invenit utique <emph type="italics"></emph>D. Sauveur<emph.end type="italics"></emph.end>(factis a ſe experimentis) quod <lb></lb>fiſtula aperta, cujus longitudo eſt pedum <emph type="italics"></emph>Pariſienſium<emph.end type="italics"></emph.end>plus minus <lb></lb>quinque, ſonum edit ejuſdem toni cum ſono chordæ quæ tempore <lb></lb>minuti unius ſecundi centies recurrit. </s> <s>Sunt igitur pulſus plus mi<lb></lb>nus centum in ſpatio pedum <emph type="italics"></emph>Pariſienſium<emph.end type="italics"></emph.end>1070, quos ſonus tem<lb></lb>pore minuti unius ſecundi percurrit; adeoque pulſus unus occu<lb></lb>pat ſpatium pedum <emph type="italics"></emph>Pariſienſium<emph.end type="italics"></emph.end>quaſi 10 (7/10), id eſt, duplam circi<lb></lb>ter longitudinem fiſtulæ. </s> <s>Unde verſimile eſt quod latitudines <lb></lb>pulſuum, in omnium apertarum fiſtularum ſonis, æquentur duplis <lb></lb>longitudinibus fiſtularum. </s></p> <p type="main"> <s>Porro cur ſoni ceſſante motu corporis ſonori ſtatim ceſſant, ne<lb></lb>Q.E.D.utius audiuntur ubi longiſſime diſtamus a corporibus ſono<lb></lb>ris, quam cum proxime abſumus, patet ex Corollario Propoſitio<lb></lb>nis XLVII Libri hujus. </s> <s>Sed & cur ſoni in Tubis ſtenterophoNI<lb></lb>cis valde augentur, ex allatis principiis manifeſtum eſt. </s> <s>Motus <lb></lb>enim omnis reciprocus ſingulis recurſibus a cauſa generante augeri <lb></lb>ſolet. </s> <s>Motus autem in Tubis dilatationem ſonorum impedienti<lb></lb>bus, tardius amittitur & fortius recurrit, & propterea a motu <lb></lb>novo ſingulis recurſibus impreſſo, magis augetur. </s> <s>Et hæc ſunt <lb></lb>præcipua Phænomena Sonorum. <pb xlink:href="039/01/373.jpg" pagenum="345"></pb><arrow.to.target n="note353"></arrow.to.target></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note353"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>SECTIO IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>De motu Circulari Fluidorum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>HYPOTHESIS.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>REſiſtentiam, quæ oritur ex defectu lubricitatis partium Fluidi, <lb></lb>cæteris paribus, proportionalem eſſe velocitati, qua partes <lb></lb>Fluidi ſeparantur ab invicem.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITION LI. THEOREMA XXXIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Cylindrus ſolidus infinite longus in Fluido uniformi & infinito <lb></lb>circa axem poſitione datum uniformi cum motu revolvatur, & <lb></lb>ab hujus impulſu ſolo agatur Fluidum in orbem, perſeveret <lb></lb>autera Fluidi pars unaquæque uniformiter in motu ſuo; dico <lb></lb>quod tempora periodica partium Fluidi ſunt ut ipſarum diſtantiæ <lb></lb>ab axe Cylindri.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>AFL<emph.end type="italics"></emph.end>Cylindrus uNI<lb></lb><figure id="id.039.01.373.1.jpg" xlink:href="039/01/373/1.jpg"></figure><lb></lb>formiter circa axem <emph type="italics"></emph>S<emph.end type="italics"></emph.end>in or<lb></lb>bem actus, & circulis con<lb></lb>centricis <emph type="italics"></emph>BGM, CHN, <lb></lb>DIO, EKP,<emph.end type="italics"></emph.end>&c. </s> <s>diſtin<lb></lb>guatur Fluidum in Orbes cy<lb></lb>lindricos innumeros concen<lb></lb>tricos ſolidos ejuſdem craſſi<lb></lb>tudinis. </s> <s>Et quoniam homo<lb></lb>geneum eſt Fluidum, im<lb></lb>preſſiones contiguorum Or<lb></lb>bium in ſe mutuo factæ, <lb></lb>erunt (per Hypotheſin) ut <lb></lb>eorum tranſlationes ab invicem & ſuperficies contiguæ in quibus <lb></lb>impreſſiones fiunt. </s> <s>Si impreſſio in Orbem aliquem major eſt vel <pb xlink:href="039/01/374.jpg" pagenum="346"></pb><arrow.to.target n="note354"></arrow.to.target>minor ex parte concava quam ex parte convexa; prævalebit im<lb></lb>preſſio fortior, & motum Orbis vel accelerabit vel retardabit, <lb></lb>prout in eandem regionem cum ipſius motu vel in contrariam di<lb></lb>rigitur. </s> <s>Proinde ut Orbis unuſquiſQ.E.I. motu ſuo uniformiter <lb></lb>perſeveret, debent impreſſiones ex parte utraque ſibi invicem æqua<lb></lb>ri, & fieri in regiones contrarias. </s> <s>Unde cum impreſſiones ſunt ut <lb></lb>contiguæ ſuperficies & harum tranſlationes ab invicem, erunt tran<lb></lb>ſlationes inverſe ut ſuperficies, hoc eſt, inverſe ut ſuperficierum di<lb></lb>ſtantiæ ab axe. </s> <s>Sunt autem differentiæ motuum angularium circa <lb></lb>axem ut hæ tranſlationes applicatæ ad diſtantias, ſive ut tranſlati<lb></lb>ones directe & diſtantiæ inverſe; hoc eſt (conjunctis rationibus) <lb></lb>ut quadrata diſtantiarum inverſe. </s> <s>Quare ſi ad infinitæ rectæ <lb></lb><emph type="italics"></emph>SABCDEQ<emph.end type="italics"></emph.end>partes ſin<lb></lb><figure id="id.039.01.374.1.jpg" xlink:href="039/01/374/1.jpg"></figure><lb></lb>gulas erigantur perpendicula <lb></lb><emph type="italics"></emph>Aa, Bb, Cc, Dd, Ee,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>ipſarum <emph type="italics"></emph>SA, SB, SC, SD, <lb></lb>SE,<emph.end type="italics"></emph.end>&c. </s> <s>quadratis reciproce <lb></lb>proportionalia, & per ter<lb></lb>minos perpendicularium du<lb></lb>ci intelligatur linea curva <lb></lb>Hyperbolica; erunt ſummæ <lb></lb>differentiarum, hoc eſt, mo<lb></lb>tus toti angulares, ut re<lb></lb>ſpondentes ſummæ linearum <lb></lb><emph type="italics"></emph>Aa, Bb, Cc, Dd, Ee<emph.end type="italics"></emph.end>: id <lb></lb>eſt, ſi ad conſtituendum Me<lb></lb>dium uniformiter fluidum, Orbium numerus augeatur & latitudo <lb></lb>minuatur in infinitum, ut areæ Hyperbolicæ his ſummis analogæ <lb></lb><emph type="italics"></emph>AaQ, BbQ, CcQ, DdQ, EeQ,<emph.end type="italics"></emph.end>&c. </s> <s>Et tempora motibus an<lb></lb>gularibus reciproce proportionalia, erunt etiam his areis reciproce <lb></lb>proportionalia. </s> <s>Eſt igitur tempus periodicum particulæ cujuſvis <lb></lb><emph type="italics"></emph>D<emph.end type="italics"></emph.end>reciproce ut area <emph type="italics"></emph>DdQ,<emph.end type="italics"></emph.end>hoc eſt, (per notas Curvarum qua<lb></lb>draturas) directe ut diſtantia <emph type="italics"></emph>SD. Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note354"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc motus angulares particularum fluidi ſunt reci<lb></lb>proce ut ipſarum diſtantiæ ab axe cylindri, & velocitates abſo<lb></lb>lutæ ſunt æquales. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si fluidum in vaſe cylindrico longitudinis infinitæ con<lb></lb>tineatur, & cylindrum alium interiorem contineat, revolvatur <lb></lb>autem cylindrus uterque circa axem communem, ſintque revolu-<pb xlink:href="039/01/375.jpg" pagenum="347"></pb>tionum tempora ut ipſorum ſemidiametri, & perſeveret fluidi pars <lb></lb><arrow.to.target n="note355"></arrow.to.target>unaquæQ.E.I. motu ſuo: erunt partium ſingularum tempora peri<lb></lb>odica ut ipſarum diſtantiæ ab axe cylindrorum. </s></p> <p type="margin"> <s><margin.target id="note355"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Si cylindro & fluido ad hunc modum motis addatur <lb></lb>vel auferatur communis quilibet motus angularis; quoniam hoc <lb></lb>novo motu non mutatur attritus mutuus partium fluidi, non mu<lb></lb>tabuntur motus partium inter ſe. </s> <s>Nam tranſlationes partium ab <lb></lb>invicem pendent ab attritu. </s> <s>Pars quælibet in eo perſeverabit <lb></lb>motu, qui, attritu utrinQ.E.I. contrarias partes facto, non magis <lb></lb>acceleratur quam retardatur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Unde ſi toti cylindrorum & fluidi Syſtemati auferatur <lb></lb>motus omnis angularis cylindri exterioris, habebitur motus fluidi <lb></lb>in cylindro quieſcente. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Igitur ſi fluido & cylindro exteriore quieſcentibus, re<lb></lb>volvatur cylindrus interior uniformiter; communicabitur motus <lb></lb>circularis fluido, & paulatim per totum fluidum propagabitur; <lb></lb>nec prius deſinet augeri quam fluidi partes ſingulæ motum Corol<lb></lb>lario quarto definitum acquirant. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Et quoniam fluidum conatur motum ſuum adhuc latius <lb></lb>propagare, hujus impetu circumagetur etiam cylindrus exterior <lb></lb>niſi violenter detentus; & accelerabitur ejus motus quoad uſque <lb></lb>tempora periodica cylindri utriuſque æquentur inter ſe. </s> <s>Quod ſi <lb></lb>cylindrus exterior violenter detineatur, conabitur is motum fluidi <lb></lb>retardare; & niſi cylindrus interior vi aliqua extrinſecus impreſſa <lb></lb>motum illum conſervet, efficiet ut idem paulatim ceſſet. </s></p> <p type="main"> <s>Quæ omnia in Aqua profunda ſtagnante experiri licet. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LII. THEOREMA XL.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Sphæra ſolida, in Fluido uniformi & infinito, circa axem poſi<lb></lb>tione datum uniformi cum motu revolvatur, & ab hujus im<lb></lb>pulſu ſolo agatur Fluidum in orbem; perſeveret autem Fluidi <lb></lb>pars unaquæque uniformiter in motu ſuo: dico quod tem<lb></lb>pora periodica partium Fluidi erunt ut quadrata diſtantiarum <lb></lb>à centro Sphæræ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>1. Sit <emph type="italics"></emph>AFL<emph.end type="italics"></emph.end>Sphæra uniformiter circa axem <emph type="italics"></emph>S<emph.end type="italics"></emph.end>in orbem <lb></lb>acta, & circulis concentricis <emph type="italics"></emph>BGM, CHN, DIO, EKP,<emph.end type="italics"></emph.end>&c. <pb xlink:href="039/01/376.jpg" pagenum="348"></pb><arrow.to.target n="note356"></arrow.to.target>diſtinguatur Fluidum in Orbes innumeros concentricos ejuſdem <lb></lb>craſſitudinis. </s> <s>Finge autem Orbes illos eſſe ſolidos; & quoniam <lb></lb>homogeneum eſt Fluidum, impreſſiones contiguorum Orbium in <lb></lb>ſe mutuo factæ, erunt (per Hypotheſin) ut eorum tranſlationes <lb></lb>ab invicem & ſuperficies contiguæ in quibus impreſſiones fiunt. </s> <s><lb></lb>Si impreſſio in Orbem aliquem major eſt vel minor ex parte con<lb></lb>cava quam ex parte convexa; prævalebit impeſſio fortior, & velo<lb></lb>citatem Orbis vel accelerabit vel retardabit, prout in eandem regi<lb></lb>onem cum ipſius motu vel in contrariam dirigitur. </s> <s>Proinde ut <lb></lb>Orbis unuſquiſQ.E.I. motu ſuo perſeveret uniformiter, debebunt <lb></lb>impreſſiones ex parte utraque ſibi invicem æquari, & fieri in re<lb></lb>giones contrarias. </s> <s>Unde cum impreſſiones ſint ut contiguæ ſu<lb></lb>perficies & harum tranſlationes ab invicem; erunt tranſlationes <lb></lb>inverſe ut ſuperficies, hoc eſt, inverſe ut quadrata diſtantiarum ſu<lb></lb>perficierum à centro. </s> <s>Sunt autem differentiæ motuum angularium <lb></lb>circa axem ut hæ tranſlationes applicatæ ad diſtantias, ſive ut <lb></lb>tranſlationes directe & diſtantiæ inverſe; hoc eſt (conjunctis ra<lb></lb>tionibus) ut cubi diſtantiarum inverſe. </s> <s>Quare ſi ad rectæ infi<lb></lb>nitæ <emph type="italics"></emph>SABCDEQ<emph.end type="italics"></emph.end>partes ſingulas erigantur perpendicula <emph type="italics"></emph>Aa, <lb></lb>Bb, Cc, Dd, Ee,<emph.end type="italics"></emph.end>&c. </s> <s>ipſarum <emph type="italics"></emph>SA, SB, SC, SD, SE,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>cubis reciproce proportionalia, erunt ſummæ differentiarum, hoc <lb></lb>eſt, motus toti angulares, ut reſpondentes ſummæ linearum <emph type="italics"></emph>Aa, <lb></lb>Bb, Cc, Dd, Ee<emph.end type="italics"></emph.end>: id eſt (ſi ad conſtituendum Medium uniformi<lb></lb>ter fluidum, numerus Orbium augeatur & latitudo minuatur in in<lb></lb>finitum) ut areæ Hyperbolicæ his ſummis analogæ <emph type="italics"></emph>AaQ, BbQ, <lb></lb>CcQ, DdQ, EeQ,<emph.end type="italics"></emph.end>&c. </s> <s>Et tempora periodica motibus angu<lb></lb>laribus reciproce proportionalia, erunt etiam his areis reciproce <lb></lb>proportionalia. </s> <s>Eſt igitur tempus periodicum Orbis cujuſvis <lb></lb><emph type="italics"></emph>DIO<emph.end type="italics"></emph.end>reciproce ut area <emph type="italics"></emph>DdQ,<emph.end type="italics"></emph.end>hoc eſt, (per notas Curvarum <lb></lb>quadraturas) directe ut quadratum diſtantiæ <emph type="italics"></emph>SD.<emph.end type="italics"></emph.end>Id quod vo<lb></lb>lui primo demonſtrare. </s></p> <p type="margin"> <s><margin.target id="note356"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>2. A centro Sphæræ ducantur infinitæ rectæ quam pluri<lb></lb>mæ, quæ cum axe datos contineant angulos, æqualibus differen<lb></lb>tiis ſe mutuo ſuperantes; & his rectis circa axem revolutis concipe <lb></lb>Orbes in annulos innumeros ſecari; & annulus unuſquiſque habe<lb></lb>bit annulos quatuor ſibi contiguos, unum interiorem, alterum ex<lb></lb>teriorem & duos laterales. </s> <s>Attritu interioris & exterioris non <lb></lb>poteſt annulus unuſquiſque, niſi in motu juxta legem caſus primi <lb></lb>facto, æqualiter & in partes contrarias urgeri. </s> <s>Patet hoc ex de<lb></lb>monſtratione caſus primi. </s> <s>Et propterea annulorum ſeries quælibet <pb xlink:href="039/01/377.jpg" pagenum="349"></pb>a Globo in infinitum recta pergens, movebitur pro lege caſus pri<lb></lb><arrow.to.target n="note357"></arrow.to.target>mi, niſi quatenus impeditur ab attritu annulorum ad latera. </s> <s>At <lb></lb>in motu hac lege facto, attritus annulorum ad latera nullus eſt; <lb></lb>neque adeo motum, quo minus hac lege fiat, impediet. </s> <s>Si an<lb></lb>nuli, qui a centro æqualiter diſtant, vel citius revolverentur vel <lb></lb>tardius juxta polos quam juxta æquatorem; tardiores accelera<lb></lb>rentur, & velociores retardarentur ab attritu mutuo, & ſic verge<lb></lb>rent ſemper tempora periodica ad æqualitatem, pro lege caſus <lb></lb>primi. </s> <s>Non impedit igitur hic attritus quo minus motus fiat ſe<lb></lb>cundum legem caſus primi, & propterea lex illa obtinebit: hoc <lb></lb>eſt, annulorum ſingulorum tempora periodica erunt ut quadrata <lb></lb>diſtantiarum ipſorum à centro Globi. </s> <s>Quod volui ſecundo de<lb></lb>monſtrare. </s></p> <p type="margin"> <s><margin.target id="note357"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cas.<emph.end type="italics"></emph.end>3. Dividatur jam annulus unuſquiſque ſectionibus tranſ<lb></lb>verſis in particulas innumeras conſtituentes ſubſtantiam abſolute <lb></lb>& uniformiter fluidam; & quoniam hæ ſectiones non ſpectant ad <lb></lb>legem motus circularis, ſed ad conſtitutionem Fluidi ſolummodo <lb></lb>conducunt, perſeverabit motus circularis ut prius. </s> <s>His ſectionibus <lb></lb>annuli omnes quam minimi aſperitatem & vim attritus mutui aut <lb></lb>non mutabunt aut mutabunt æqualiter. </s> <s>Et manente cauſarum <lb></lb>proportione manebit effectuum proportio, hoc eſt, proportio mo<lb></lb>tuum & periodieorum temporum. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end>Cæterum cum motus <lb></lb>circularis, & abinde orta vis centrifuga, major ſit ad Eclipticam <lb></lb>quam ad Polos; debebit cauſa aliqua adeſſe qua particulæ ſingulæ <lb></lb>in circulis ſuis retineantur; ne materia quæ ad Eclipticam eſt, rece<lb></lb>dat ſemper à centro & per exteriora Vorticis migret ad Polos, in<lb></lb>deque per axem ad Eclipticam circulatione perpetua revertatur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc motus angulares partium fluidi circa axem globi, <lb></lb>ſunt reciproce ut quadrata diſtantiarum à centro globi, & veloci<lb></lb>tates abſolutæ reciproce ut eadem quadrata applicata ad diſtantias <lb></lb>ab axe. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Si globus in fluido quieſcente ſimilari & infinito circa <lb></lb>axem poſitione datum uniformi cum motu revolvatur, commuNI<lb></lb>cabitur motus fluido in morem Vorticis, & motus iſte paulatim <lb></lb>propagabitur in infin tum; neque prius ceſſabit in ſingulis fluidi <lb></lb>partibus accelerari, quam tempora periodica ſingularum partium <lb></lb>ſint ut quadrata diſtantiarum à centro globi. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Quoniam Vorticis partes interiores ob majorem ſuam <lb></lb>velocitatem atterunt & urgent exteriores, motumQ.E.I.ſis ea acti-<pb xlink:href="039/01/378.jpg" pagenum="350"></pb><arrow.to.target n="note358"></arrow.to.target>one perpetuo communicant, & exteriores illi eandem motus quan<lb></lb>titatem in alios adhuc exteriores ſimul tranſferunt, eaque actione <lb></lb>ſervant quantitatem motus ſui plane invariatam; patet quod mo<lb></lb>tus perpetuo transfertur à centro ad circumferentiam Vorticis, & <lb></lb>per infinitatem circumferentiæ abſorbetur. </s> <s>Materia inter ſphæri<lb></lb>cas duas quaſvis ſuperficies Vortici concentricas nunquam accele<lb></lb>rabitur, eo quod motum omnem à materia interiore acceptum <lb></lb>transfert ſemper in exteriorem. </s></p> <p type="margin"> <s><margin.target id="note358"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Proinde ad conſervationem Vorticis conſtanter in eo<lb></lb>dem movendi ſtatu, requiritur principium aliquod activum, à quo <lb></lb>globus eandem ſemper quantitatem motus accipiat, quam imprimit <lb></lb>in materiam Vorticis. </s> <s>Abſque tali principio neceſſe eſt ut globus <lb></lb>& Vorticis partes interiores, propagantes ſemper motum ſuum in <lb></lb>exteriores, neque novum aliquem motum recipientes, tardeſcant <lb></lb>paulatim & in orbem agi definant. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Si globus alter huic Vortici ad certam ab ipſius centro <lb></lb>diſtantiam innataret, & interea circa axem inclinatione datum vi <lb></lb>aliqua conſtanter revolveretur; hujus motu raperetur fluidum in <lb></lb>Vorticem: & primo revolveretur hic Vortex novus & exiguus una <lb></lb>cum globo circa centrum alterius, & interea latius ſerperet ipſius <lb></lb>motus, & paulatim propagaretur in infinitum, ad modum Vorticis <lb></lb>primi. </s> <s>Et eadem ratione qua hujus globus raperetur motu Vorti<lb></lb>cis alterius, raperetur etiam globus alterius motu hujus, ſic ut <lb></lb>globi duo circa intermedium aliquod punctum revolverentur, ſe<lb></lb>que mutuo ob motum illum circularem fugerent, niſi per vim <lb></lb>aliquam cohibiti. </s> <s>Poſtea ſi vires conſtanter impreſſæ, quibus <lb></lb>globi in motibus ſuis perſeverant, ceſſarent, & omnia legibus Me<lb></lb>chanicis permitterentur, langueſceret paulatim motus globorum <lb></lb>(ob rationem in Corol. </s> <s>3. & 4. aſſignatam) & Vortices tandem <lb></lb>conquieſcerent. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Si globi plures datis in locis circum axes poſitione da<lb></lb>tos certis cum velocitatibus conſtanter revolverentur, fierent Vor<lb></lb>tices totidem in infinitum pergentes. </s> <s>Nam globi ſinguli, eadem <lb></lb>ratione qua unus aliquis motum ſuum propagat in infinitum, pro<lb></lb>pagabunt etiam motus ſuos in infinitum, adeo ut fluidi infiniti <lb></lb>pars unaquæque eo agitetur motu qui ex omnium globorum acti<lb></lb>onibus reſultat. </s> <s>Unde Vortices non definientur certis limitibus, <lb></lb>ſed in ſe mutuo paulatim excurrent; globique per actiones Vorti<lb></lb>cum in ſe mutuo, perpetuo movebuntur de locis ſuis, uti in <lb></lb>Corollario ſuperiore expoſitum eſt; neque certam quamvis inter ſe<pb xlink:href="039/01/379.jpg" pagenum="351"></pb>poſitionem ſervabunt, niſi per vim aliquam retenti. </s> <s>Ceſſantibus <lb></lb><arrow.to.target n="note359"></arrow.to.target>autem viribus illis quæ in globos conſtanter impreſſæ conſervant <lb></lb>hoſce motus, materia ob rationem in Corollario tertio & quarto <lb></lb>aſſignatam, paulatim requieſcet & in Vortices agi deſinet. </s></p> <p type="margin"> <s><margin.target id="note359"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Si fluidum ſimilare claudatur in vaſe ſphærico, ac <lb></lb>globi in centro conſiſtentis uniformi rotatione agatur in Vorticem, <lb></lb>globus autem & vas in eandem partem circa axem eundem revol<lb></lb>vantur, ſintque eorum tempora periodica ut quadrata ſemidiame<lb></lb>trorum: partes fluidi non prius perſeverabunt in motibus ſuis ſine <lb></lb>acceleratione & retardatione, quam ſint eorum tempora periodica <lb></lb>ut quadrata diſtantiarum à centro Vorticis. </s> <s>Alia nulla Vorticis <lb></lb>conſtitutio poteſt eſſe permanens. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Si vas, fluidum incluſum & globus ſervent hunc mo<lb></lb>tum, & motu præterea communi angulari circa axem quemvis da<lb></lb>tum revolvantur; quoniam hoc motu novo non mutatur attritus <lb></lb>partium fluidi in ſe invicem, non mutabuntur motus partium in<lb></lb>ter ſe. </s> <s>Nam tranſlationes partium inter ſe pendent ab attritu. </s> <s><lb></lb>Pars quælibet in eo perſeverabit motu, quo fit ut attritu ex uno <lb></lb>latere non magis tardetur quam acceleretur attritu ex altero. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>9. Unde ſi vas quieſcat ac detur motus globi, dabitur <lb></lb>motus fluidi. </s> <s>Nam concipe planum tranſire per axem globi & <lb></lb>motu contrario revolvi; & pone ſummam temporis revolutionis <lb></lb>hujus & revolutionis globi eſſe ad tempus revolutionis globi, ut <lb></lb>quadratum ſemidiametri vaſis ad quadratum ſemidiametri globi: <lb></lb>& tempora periodica partium fluidi reſpectu plani hujus, erunt ut <lb></lb>quadrata diſtantiarum ſuarum à centro globi. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>10. Proinde ſi vas vel circa axem eundem cum globo, vel <lb></lb>circa diverſum aliquem, data cum velocitate quacunque movea<lb></lb>tur, dabitur motus fluidi. </s> <s>Nam ſi Syſtemati toti auferatur v ſis <lb></lb>motus angularis, manebunt motus omnes iidem inter ſe qui prius, <lb></lb>per Corol. </s> <s>8. Et motus iſti per Corol. </s> <s>9. dabuntur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>11. Si vas & fluidum quieſcant & globus uniformi cum <lb></lb>motu revolvatur, propagabitur motus paulatim per fluidum torum <lb></lb>in vas, & circumagetur vas niſi violenter detentum, neque prius <lb></lb>definent fluidum & vas accelerari, quam ſint eorum tempora peri<lb></lb>odica æqualia temporibus periodicis globi. </s> <s>Quod ſi vas vi aliqua <lb></lb>detineatur vel revolvatur motu quovis conſtanti & uniformi, de<lb></lb>vemet Medium paulatim ad ſtatum motus in Corollariis 8. 9 & 10. <lb></lb>definiti, nes in alio unquam ſtatu quocunque perſeverabit. </s> <s>De<lb></lb>inde vero ſi, viribus illis ceſſantibus quibus vas & globus certis <pb xlink:href="039/01/380.jpg" pagenum="352"></pb><arrow.to.target n="note360"></arrow.to.target>motibus revolvebantur, permittatur Syſtema totum Legibus Me<lb></lb>chanicis; vas & globus in ſe invicem agent mediante fluido, ne<lb></lb>que motus ſuos in ſe mutuo per fluidum propagare prius ceſſa<lb></lb>bunt, quam eorum tempora periodica æquentur inter ſe, & Syſte<lb></lb>ma totum ad inſtar corporis unius ſolidi ſimul revolvatur. </s></p> <p type="margin"> <s><margin.target id="note360"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In his omnibus ſuppono fluidum ex materia quoad denſitatem <lb></lb>& fluiditatem uniformi conſtare. </s> <s>Tale eſt in quo globus idem <lb></lb>codem cum motu, in eodem temporis intervallo, motus ſimiles & <lb></lb>æquales, ad æquales ſemper à ſe diſtantias, ubivis in fluido conſti<lb></lb>tutus, propagare poſſit. </s> <s>Conatur quidem materia per motum <lb></lb>ſuum circularem recedere ab axe Vorticis, & propterea premit <lb></lb>materiam omnem ulteriorem. </s> <s>Ex hac preſſione fit attritus par<lb></lb>tium fortior & ſeparatio ab invicem difficilior; & per conſequens <lb></lb>diminuitur materiæ fluiditas. </s> <s>Rurſus ſi partes fluidi ſunt alicubi <lb></lb>craſſiores ſeu majores, fluiditas ibi minor erit, ob pauciores ſuper<lb></lb>ficies in quibus partes ſeparentur ab invicem. </s> <s>In hujuſmodi caſi<lb></lb>bus deficientem fluiditatem vel lubricitate partium vel lentore alia<lb></lb>ve aliqua conditione reſtitui ſuppono. </s> <s>Hoc niſi fiat, materia ubi <lb></lb>minus fluida eſt magis cohærebit & ſegnior erit, adeoque motum <lb></lb>tardius recipiet & longius propagabit quam pro ratione ſuperius <lb></lb>aſſignata. </s> <s>Si figura vaſis non ſit Sphærica, movebuntur particulæ <lb></lb>in lineis non circularibus ſed conformibus eidem vaſis figuræ, & <lb></lb>tempora periodica erunt ut quadrata mediocrium diſtantiarum à <lb></lb>centro quamproxime. </s> <s>In partibus inter centrum & circumferen<lb></lb>tiam, ubi latiora ſunt ſpatia, tardiores erunt motus, ubi anguſtiora <lb></lb>velociores, neque tamen particulæ velociores petent circumferen<lb></lb>tiam. </s> <s>Arcus enim deſcribent minus curvos, & conatus recedendi <lb></lb>à centro non minus diminuetur per decrementum hujus curva<lb></lb>turæ, quam augebitur per incrementum velocitatis. </s> <s>Pergendo a <lb></lb>ſpatiis anguſtioribus in latiora recedent paulo longius a centro, <lb></lb>ſed iſto receſſu tardeſcent; & accedendo poſtea de latioribus ad <lb></lb>anguſtiora accelerabuntur, & ſic per vices tardeſcent & accelera<lb></lb>buntur particulæ ſingulæ in perpetuum. </s> <s>Hæc ita ſe habebunt in <lb></lb>vaſe rigido. </s> <s>Nam in fluido infinito conſtitutio Vorticum innote<lb></lb>ſcit per Propoſitionis hujus Corollarium ſextum. </s></p> <p type="main"> <s>Proprietates autem Vorticum hac Propoſitione inveſtigare co<lb></lb>natus ſum, ut pertentarem ſiqua ratione Phænomena cœleſtia per <pb xlink:href="039/01/381.jpg" pagenum="353"></pb>Vortices explicari poſſint. </s> <s>Nam Phænomenon eſt, quod Planeta<lb></lb><arrow.to.target n="note361"></arrow.to.target>rum circa Jovem revolventium tempora periodica ſunt in ratione <lb></lb>ſeſquiplicata diſtantiarum a centro Jovis; & eadem Regula obti<lb></lb>net in Planetis qui circa Solem revolvuntur. </s> <s>Obtinent autem hæ <lb></lb>Regulæ in Planetis utriſque quam accuratiſſime, quatenus obſer<lb></lb>vationes Aſtronomicæ hactenus prodidere. </s> <s>Ideoque ſi Planetæ <lb></lb>illi à Vorticibus circa Jovem & Solem revolventibus deferantur, <lb></lb>debebunt etiam hi Vortices eadem lege revolvi. </s> <s>Verum tempora <lb></lb>periodica partium Vorticis prodierunt in ratione duplicata diſtan<lb></lb>tiarum a centro motus: neque poteſt ratio illa diminui & ad ra<lb></lb>tionem ſeſquiplicatam reduci, niſi vel materia Vorticis eo fluidior <lb></lb>ſit quo longius diſtat a centro, vel reſiſtentia, quæ oritur ex de<lb></lb>fectu lubricitatis partium fluidi, ex aucta velocitate qua partes <lb></lb>fluidi ſeparantur ab invicem, augeatur in majori ratione quam ea <lb></lb>eſt in qua velocitas augetur. </s> <s>Quorum tamen neutrum rationi <lb></lb>conſentaneum videtur. </s> <s>Partes craſſiores & minus fluidæ (niſi gra<lb></lb>ves ſint in centrum) circumferentiam petent; & veriſimile eſt <lb></lb>quod, etiamſi Demonſtrationum gratia Hypotheſin talem initio <lb></lb>Sectionis hujus propoſuerim ut Reſiſtentia velocitati proportiona<lb></lb>lis eſſet, tamen Reſiſtentia in minori ſit ratione quam ea velocita<lb></lb>tis eſt. </s> <s>Quo conceſſo, tempora periodica partium Vorticis erunt <lb></lb>in majori quam duplicata ratione diſtantiarum ab ipſius centro. </s> <s><lb></lb>Quod ſi Vortices (uti aliquorum eſt opinio) celerius moveantur <lb></lb>prope centrum, dein tardius uſque ad certum limitem, tum denuo <lb></lb>celerius juxta circumferentiam; certe nec ratio ſeſquiplicata neque <lb></lb>alia quævis certa ac determinata obtinere poteſt. </s> <s>Viderint itaque <lb></lb>Philoſophi quo pacto Phænomenon illud rationis ſeſquiplicatæ per <lb></lb>Vortices explicari poſſit. </s></p> <p type="margin"> <s><margin.target id="note361"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO LIII. THEOREMA XLI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corpora quæ in Vortice delata in orbem redeunt, ejuſdem ſunt den<lb></lb>ſitatis cum Vortice, & eadem lege cum ipſius partibus (quoad <lb></lb>velocitatem & curſus determinationem) moventur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi Vorticis pars aliqua exigua, cujus particulæ ſeu puncta <lb></lb>phyſica datum ſervant ſitum inter ſe, congelari ſupponatur: hæc, <lb></lb>quoniam neque quoad denſitatem ſuam, neque quoad vim inſitam <lb></lb>aut figuram ſuam mutatur, movebitur eadem lege ac prius: & <pb xlink:href="039/01/382.jpg" pagenum="354"></pb><arrow.to.target n="note362"></arrow.to.target>contra, ſi Vorticis pars congelata & ſolida ejuſdem ſit denſitatis <lb></lb>cum reliquo Vortice, & reſolvatur in fluidum; movebitur hæc ea<lb></lb>dem lege ac prius, niſi quatenus ipſius particulæ jam fluidæ factæ <lb></lb>moveantur inter ſe. </s> <s>Negligatur igitur motus particularum inter <lb></lb>ſe, tanquam ad totius motum progreſſivum nil ſpectans, & motus <lb></lb>totius idem erit ac prius. </s> <s>Motus autem idem erit cum motu alia<lb></lb>rum Vorticis partium a centro æqualiter diſtantium, propterea <lb></lb>quod ſolidum in Fluidum reſolutum fit pars Vorticis cæteris parti<lb></lb>bus conſimilis. </s> <s>Ergo ſolidum, ſi ſit ejuſdem denſitatis cum ma<lb></lb>teria Vorticis, eodem motu cum ipſius partibus movebitur, in ma<lb></lb>teria proxime ambiente relative quieſcens. </s> <s>Sin denſius ſit, jam <lb></lb>magis conabitur recedere à centro Vorticis quam prius; adeoque <lb></lb>Vorticis vim illam, qua prius in Orbita ſua tanquam in æquilibrio <lb></lb>conſtitutum retinebatur, jam ſuperans, recedet a centro & revol<lb></lb>vendo deſcribet Spiralem, non amplius in eundem Orbem rediens <lb></lb>Et eodem argumento ſi rarius ſit, accedet ad centrum. </s> <s>Igitur non <lb></lb>redibit in eundem Orbem niſi ſit ejuſdem denſitatis cum fluido <lb></lb>Eo autem in caſu oſtenſum eſt, quod revolveretur eadem lege cum <lb></lb>partibus fluidi à centro Vorticis æqualiter diſtantibus. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note362"></margin.target>DE MOTU <lb></lb>CORPORUM</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Ergo ſolidum quod in Vortice revolvitur & in eundem <lb></lb>Orbem ſemper redit, relative quieſcit in fluido cui innatat. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Et ſi Vortex ſit quoad denſitatem uniformis, corpus <lb></lb>idem ad quamlibet a centro Vorticis diſtantiam revolvi poteſt. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hinc liquet Planetas à Vorticibus corporeis non deferri. </s> <s>Nam<lb></lb>Planetæ ſecundum Hypotheſin <emph type="italics"></emph>Copernicæam<emph.end type="italics"></emph.end>circa Solem delati re<lb></lb>volvuntur in Ellipſibus umbilicum habentibus in Sole, & radiis ad<lb></lb>Solem ductis areas deſcribunt temporibus proportionales. </s> <s>At par<lb></lb>tes Vorticis tali motu revolvi nequeunt. </s> <s>Deſignent <emph type="italics"></emph>AD, BE, CF<emph.end type="italics"></emph.end>,<lb></lb>Orbes tres circa Solem <emph type="italics"></emph>S<emph.end type="italics"></emph.end>deſcriptos, quorum extimus <emph type="italics"></emph>CF<emph.end type="italics"></emph.end>circulus<lb></lb>ſit Soli concentricus, & interiorum duorum Aphelia ſint <emph type="italics"></emph>A, B<emph.end type="italics"></emph.end>&<lb></lb>Perihelia <emph type="italics"></emph>D, E.<emph.end type="italics"></emph.end>Ergo corpus quod revolvitur in Orbe <emph type="italics"></emph>CF,<emph.end type="italics"></emph.end>radio<lb></lb>ad Solem ducto areas temporibus proportionales deſcribendo, mo<lb></lb>vebitur uniformi cum motu. </s> <s>Corpus autem quod revolvitur in<lb></lb>Orbe <emph type="italics"></emph>BE,<emph.end type="italics"></emph.end>tardius movebitur in Aphelio <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& velocius in Peri<lb></lb>helio <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>ſecundum leges Aſtronomicas; cum tamen ſecundum le<lb></lb>ges Mechanicas materia Vorticis in ſpatio anguſtiore inter <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& C<pb xlink:href="039/01/383.jpg" pagenum="355"></pb>velocius moveri debeat quam in ſpatio latiore inter <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>F<emph.end type="italics"></emph.end>; id <lb></lb><arrow.to.target n="note363"></arrow.to.target>eſt, in Aphelio velocius quam in Perihelio. </s> <s>Quæ duo repugnant <lb></lb>inter ſe. </s> <s>Sic in principio Signi <lb></lb><figure id="id.039.01.383.1.jpg" xlink:href="039/01/383/1.jpg"></figure><lb></lb>Virginis, ubi Aphelium Martis <lb></lb>jam verſatur, diſtantia inter or<lb></lb>bes Martis & Veneris eſt ad di<lb></lb>ſtantiam eorundem orbium in <lb></lb>principio Signi Piſcium ut tria <lb></lb>ad duo circiter, & propterea <lb></lb>materia Vorticis inter Orbes il<lb></lb>los in principio Piſcium debet <lb></lb>eſſe velocior quam in principio <lb></lb>Virginis in ratione trium ad duo. </s> <s><lb></lb>Nam quo anguſtius eſt ſpatium <lb></lb>per quod eadem Materiæ quan<lb></lb>titas eodem revolutionis unius <lb></lb>tempore tranſit, eo majori cum <lb></lb>velocitate tranſire debet. </s> <s>Igitur ſi Terra in hac Materia cœſe<lb></lb>ſti relative quieſcens ab ea deferretur, & una circa Solem re<lb></lb>volveretur, foret hujus velocitas in principio Piſcium ad ejuſdem <lb></lb>velocitatem in principio Virginis in ratione ſeſquialtera. </s> <s>Unde <lb></lb>Solis motus diurnus apparens in principio Virginis major eſſet <lb></lb>quam minutorum primorum ſeptuaginta, & in principio Piſcium <lb></lb>minor quam minutorum quadraginta & octo: cum tamen (expe<lb></lb>rientia teſte) apparens iſte Solis motus major ſit in principio Pi<lb></lb>ſcium quam in principio Virginis, & propterea Terra velocior in <lb></lb>principio Virginis quam in principio Piſcium. </s> <s>Itaque Hypotheſis <lb></lb>Vorticum cum Phænomenis Aſtronomicis omnino pugnat, & non <lb></lb>tam ad explicandos quam ad perturbandos motus cœleſtes, con<lb></lb>ducit. </s> <s>Quomodo vero motus iſti in ſpatiis liberis abſque Vorti<lb></lb>cibus peraguntur intelligi poteſt ex Libro primo, & in Mundi <lb></lb>Syſtemate plenius docebitur. </s></p><pb xlink:href="039/01/384.jpg" pagenum="356"></pb></subchap2></subchap1><subchap1><subchap2> <p type="margin"> <s><margin.target id="note363"></margin.target>LIBER <lb></lb>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"></emph>DE <lb></lb>MUNDI <lb></lb>SYSTEMATE <lb></lb>LIBER TERTIUS.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>IN Libris præcedentibus principia Philoſophiæ tradidi, non ta<lb></lb>men Philoſophica ſed Mathematica tantum, ex quibus vide<lb></lb>licet in rebus Philoſophicis diſputari poſſit. </s> <s>Hæc ſunt mo<lb></lb>tuum & virium leges & conditiones, quæ ad Philoſophiam ma<lb></lb>xime ſpectant. </s> <s>Eadem tamen, ne ſterilia videantur, illuſtravi <lb></lb>Scholiis quibuſdam Philoſophicis, ea tractans quæ generalia ſunt, <lb></lb>& in quibus Philoſophia maxime fundari videtur, uti corporum <lb></lb>denſitatem & reſiſtentiam, ſpatia corporibus vacua, motumque <lb></lb>Lucis & Sonorum. </s> <s>Supereſt ut ex iiſdem principiis doceamus con<lb></lb>ſtitutionem Syſtematis Mundani. </s> <s>De hoc argumento compoſue<lb></lb>ram Librum tertium methodo populari, ut a pluribus legeretur. </s> <s><lb></lb>Sed quibus Principia poſita ſatis intellecta non fuerint, ii vim con<lb></lb>ſequentiarum minime percipient, neque præjudicia deponent qui<lb></lb>bus a multis retro annis inſueverunt: & propterea ne res in diſpu<lb></lb>tationes trahatur, ſummam libri illius tranſtuli in Propoſitiones, <lb></lb>more Mathematico, ut ab iis ſolis legantur qui Principia prius <lb></lb>evolverint. </s> <s>Veruntamen quoniam Propoſitiones ibi quam pluri<lb></lb>mæ occurrant, quæ Lectoribus etiam Mathematice doctis moram <lb></lb>nimiam injicere poſſint, author eſſe nolo ut quiſquam eas omnes <lb></lb>evolvat; ſuffecerit ſiquis Definitiones, Leges motuum & ſectiones <lb></lb>tres priores Libri primi ſedulo legat, dein tranſeat ad hunc Li<lb></lb>brum de Mundi Syſtemate, & reliquas Librorum priorum Propo<lb></lb>ſitiones hic citatas pro lubitu conſulat. </s></p><pb xlink:href="039/01/385.jpg" pagenum="357"></pb> <p type="main"> <s><emph type="center"></emph>REGULÆ <lb></lb>PHILOSOPHANDI.<emph.end type="center"></emph.end><lb></lb><gap desc="hr tag"></gap></s></p> <p type="main"> <s><emph type="center"></emph>REGULA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Cauſas rerum naturalium non plures admitti debere, quam quæ <lb></lb>& veræ ſint & earum Phænomenis explicandis ſufficiant.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>DIcunt utique Philoſophi: Natura nihil agit fruſtra, & fruſtra <lb></lb>fit per plura quod fieri poteſt per pauciora. </s> <s>Natura enim <lb></lb>ſimplex eſt & rerum cauſis ſuperfluis non luxuriat. </s></p> <p type="main"> <s><emph type="center"></emph>REGULA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Ideoque Effectuum naturalium ejuſdem generis eædem ſunt <lb></lb>Cauſæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Uti reſpirationis in Homine & in Beſtia; deſcenſus lapidum in <lb></lb><emph type="italics"></emph>Europa<emph.end type="italics"></emph.end>& in <emph type="italics"></emph>America<emph.end type="italics"></emph.end>; Lucis in Igne culinari & in Sole; reflexi<lb></lb>onis Lucis in Terra & in Planetis. </s></p> <p type="main"> <s><emph type="center"></emph>REGULA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Qualitates corporum quæ intendi & remitti nequeunt, quæque <lb></lb>corporibus omnibus competunt in quibus experimenta inſtituere <lb></lb>licet, pro qualitatibus corporum univerſorum habendæ ſunt.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam qualitates corporum non niſi per experimenta innoteſcunt; <lb></lb>ideoque generales ſtatuendæ ſunt quotquot cum experimentis ge<lb></lb>neraliter quadrant; & quæ minui non poſſunt, non poſſunt au<lb></lb>ferri. </s> <s>Certe contra experimentorum tenorem ſomnia temere con<lb></lb>fingenda non ſunt, nec a Naturæ ana logia recedendum eſt, cum <pb xlink:href="039/01/386.jpg" pagenum="358"></pb><arrow.to.target n="note364"></arrow.to.target>ea ſimplex eſſe ſoleat & ſibi ſemper conſona. </s> <s>Extenſio corporum <lb></lb>non niſi per ſenſus innoteſcit, nec in omnibus ſentitur: ſed quia <lb></lb>ſenſibilibus omnibus competit, de univerſis affirmatur, Corpora <lb></lb>plura dura eſſe experimur. </s> <s>Oritur autem durities totius a duritie <lb></lb>partium, & inde non horum tantum corporum quæ ſentiuntur, <lb></lb>ſed aliorum etiam omnium particulas indiviſas eſſe duras merito <lb></lb>concludimus. </s> <s>Corpora omnia impenetrabilia eſſe non ratione ſed <lb></lb>ſenſu colligimus. </s> <s>Quæ tractamus, impenetrabilia inveniuntur, & <lb></lb>inde concludimus impenetrabilitatem eſſe proprietatem corporum <lb></lb>univerſorum. </s> <s>Corpora omnia mobilia oſſe, & viribus quibuſdam <lb></lb>(quas vires inertiæ vocamus) perſeverare in motu vel quiete, ex <lb></lb>hiſce corporum viſorum proprietatibus colligimus. </s> <s>Extenſio, du<lb></lb>rities, impenetrabilitas, mobilitas & vis inertiæ totius, oritur ab <lb></lb>extenſione, duritie, impenetrabilitate, mobilitate & viribus iner<lb></lb>tiæ partium: & inde concludimus omnes omnium corporum par<lb></lb>tes minimas extendi & duras eſſe & impenetrabiles & mobiles &<lb></lb>viribus inertiæ præditas. </s> <s>Et hoc eſt fundamentum Philoſophiæ <lb></lb>totius. </s> <s>Porro corporum partes diviſas & ſibi mutuo contiguas ab <lb></lb>invicem ſeparari poſſe, ex Phænomenis novimus, & partes indi<lb></lb>viſas in partes minores ratione diſtingui poſſe ex Mathematica <lb></lb>certum eſt. </s> <s>Utrum vero partes illæ diſtinctæ & nondum diviſæ <lb></lb>per vires Naturæ dividi & ab invicem ſeparari poſſint, incertum <lb></lb>eſt. </s> <s>At ſi vel unico conſtaret experimento quod particula aliqua <lb></lb>indiviſa, frangendo corpus durum & ſolidum, diviſionem patere<lb></lb>tur: concluderemus vi hujus Regulæ, quod non ſolum partes di<lb></lb>viſæ ſeparabiles eſſent, ſed etiam quod indiviſæ in infinitum dividi <lb></lb>poſſent. </s></p> <p type="margin"> <s><margin.target id="note364"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Denique ſi corpora omnia in circuitu Terræ gravia eſſe in Ter<lb></lb>ram, idque pro quantitate materiæ in ſingulis, & Lunam gravem <lb></lb>eſſe in Terram pro quantitate materiæ ſuæ, & viciſſim mare no<lb></lb>ſtrum grave eſſe in Lunam, & Planetas omnes graves eſſe in ſe <lb></lb>mutuo, & Cometarum ſimilem eſſe gravitatem, per experimenta <lb></lb>& obſervationes Aſtronomicas univerſaliter conſtet: dicendum erit <lb></lb>per hanc Regulam quod corpora omnia in ſe mutuo gravitant. </s> <s><lb></lb>Nam & fortius erit argumentum ex Phænomenis de gravitate uNI<lb></lb>verſali, quam de corporum impenetrabilitate: de qua utiQ.E.I. <lb></lb>corporibus Cœleſtibus nullum experimentum, nullam prorſus ob<lb></lb>ſervationem habemus. </s></p><pb xlink:href="039/01/387.jpg" pagenum="359"></pb></subchap2><subchap2> <p type="main"> <s><emph type="center"></emph>PHÆNOMENA.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note365"></arrow.to.target><gap desc="hr tag"></gap></s></p> <p type="margin"> <s><margin.target id="note365"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph>PHÆNOMENON I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Planetas Circumjoviales, radiis ad centrum Jovis ductis, areas <lb></lb>deſcribere temporibus proportionales, eorumque tempora periodica <lb></lb>eſſe in ratione ſeſquiplicata diſtantiarum ab ipſius centro.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>COnſtat ex obſervationibus Aſtronomicis. </s> <s>Orbes horum Pla<lb></lb>netarum non differunt ſenſibiliter a circulis Jovi concentri<lb></lb>cis, & motus eorum in his circulis uniformes deprehenduntur. </s> <s><lb></lb>Tempora vero periodica eſſe in ſeſquiplicata ratione ſemidiame<lb></lb>trorum Orbium conſentiunt Aſtronomi; & idem ex Tabula ſe<lb></lb>quente manifeſtum eſt. <lb></lb><emph type="italics"></emph>Satellitum Jovialium tempora periodica.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="table5"></arrow.to.target><arrow.to.target n="table6"></arrow.to.target></s></p><table><table.target id="table5"></table.target><row><cell>1<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.18<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.27′.34″.</cell><cell>3<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.13<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.13′.42″.</cell><cell>7<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.3<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.42′.36″.</cell><cell>16<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.16<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.32′.9″.</cell></row></table><table><row><cell><emph type="italics"></emph>Diſtantiæ Satellitum a centro Jovis.<emph.end type="italics"></emph.end><lb></lb></cell></row><row><cell><emph type="italics"></emph>Ex obſervationibus<emph.end type="italics"></emph.end></cell><cell>1</cell><cell>2</cell><cell>3</cell><cell>4</cell><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></lb> Jovis</cell></row><row><cell>Townlei <emph type="italics"></emph>per Microm.<emph.end type="italics"></emph.end></cell><cell>5,52</cell><cell>8,78</cell><cell>13,47</cell><cell>24,72</cell></row><row><cell>Caſſini <emph type="italics"></emph>per Teleſcop.<emph.end type="italics"></emph.end></cell><cell>5</cell><cell>8</cell><cell>13</cell><cell>23</cell></row><row><cell>Caſſini <emph type="italics"></emph>per Eclipſ. Satell.<emph.end type="italics"></emph.end></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"></emph>Ex temporibus periodicis.<emph.end type="italics"></emph.end></cell><cell>5,667</cell><cell>9,017</cell><cell>14,384</cell><cell>25,299</cell></row></table><table><table.target id="table6"></table.target><row><cell><emph type="italics"></emph>Satellitum Jovialium tempora periodica.<emph.end type="italics"></emph.end><lb></lb></cell></row></table> <p type="main"> <s><emph type="center"></emph>PHÆNOMENON II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Planetas Circumſaturnios, radiis ad Saturnum ductis, areas deſcri<lb></lb>bere temporibus proportionales, & eorum tempora periodica <lb></lb>eſſe in ratione ſeſquiplicata diſtantiarum ab ipſius centro.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſſinus<emph.end type="italics"></emph.end>utique ex obſervationibus ſuis diſtantias eorum a centro <lb></lb>Saturni & periodica tempora hujuſmodi eſſe ſtatuit. <pb xlink:href="039/01/388.jpg" pagenum="360"></pb><arrow.to.target n="note366"></arrow.to.target><arrow.to.target n="table7"></arrow.to.target><arrow.to.target n="table8"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note366"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p><table><table.target id="table7"></table.target><row><cell><emph type="italics"></emph>Satellitum Saturniorum tempora periodica.<emph.end type="italics"></emph.end><lb></lb></cell></row><row><cell>1<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.21<emph type="sup"></emph><emph.end type="sup"></emph.end>.19′.</cell><cell>2<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.17<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.41′.</cell><cell>4<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.13<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.47′.</cell><cell>15<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.22<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.41′.</cell><cell>79<emph type="sup"></emph>d<emph.end type="sup"></emph.end>.22<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.4′.</cell></row><row><cell><emph type="italics"></emph>Diſtantiæ Satellitum a centro Saturni in ſemidiametris Annuli<emph.end type="italics"></emph.end></cell></row><row><cell><emph type="italics"></emph>Ex obſervationibus<emph.end type="italics"></emph.end></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"></emph>Ex temporibus periodicis<emph.end type="italics"></emph.end></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></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Planetas quinque primarios Mercurium, Venerem, Martem, Jo<lb></lb>vem & Saturnum Orbibus ſuis Solem cingere.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Mercurium & Venerem circa Solem revolvi ex eorum phaſibus <lb></lb> lunaribus demonſtratur. </s> <s>Plena facie lucentes ultra Solem ſiti ſunt, <lb></lb> dimidiata è regione Solis, falcata cis Solem; per diſcum ejus ad <lb></lb> modum macularum nonnunquam tranſeuntes. </s> <s>Ex Martis quoque <lb></lb> plena facie prope Solis conjunctionem, & gibboſa in quadraturis, <lb></lb> certum eſt quod is Solem ambit. </s> <s>De Jove etiam & Saturno idem <lb></lb> ex eorum phaſibus ſemper plenis demonſtratur. <lb></lb> PHÆNOMENON IV.<lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Planetarum quinque primariorum, & (vel Solis circa Terram vel) <lb></lb> Terræ circa Solem tempora periodica eſſe in ratione ſeſquipli<lb></lb>cata mediocrium diſtantiarum à Sole.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Hæc à <emph type="italics"></emph>Keplero<emph.end type="italics"></emph.end>inventa ratio in confeſſo eſt apud omnes. </s> <s>Ea<lb></lb>dem utique ſunt tempora periodica, eædemque orbium dimen<lb></lb>ſiones, ſive Sol circa Terram, ſive Terra circa Solem revolvatur. <lb></lb> Ac de menſura quidem temporum periodieorum convenit inter <lb></lb> Aſtronomos univerſos. </s> <s>Magnitudines autem Orbium <emph type="italics"></emph>Keplerus<emph.end type="italics"></emph.end>& <lb></lb> <emph type="italics"></emph>Bullialdus<emph.end type="italics"></emph.end>omnium diligentiſſime ex Obſervationibus determina<lb></lb>verunt: & diſtantiæ mediocres, quæ temporibus periodicis reſpon<lb></lb>dent, non differunt ſenſibiliter à diſtantiis quas illi invenerunt, <lb></lb> ſuntQ.E.I.ter ipſas ut plurimum intermediæ; uti in Tabula ſe<lb></lb>quente videre licet. <lb></lb> <pb xlink:href="039/01/389.jpg" pagenum="361"></pb><lb></lb><arrow.to.target n="note367"></arrow.to.target></s></p><table><row><cell><emph type="italics"></emph>Planetarum ac Telluris diſtantiæ mediocres à Sole.<emph.end type="italics"></emph.end></cell></row><row><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell>Secundum <emph type="italics"></emph>Keplerum<emph.end type="italics"></emph.end></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"></emph>Bullialdum<emph.end type="italics"></emph.end></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"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s>De diſtantiis Mercurii & Veneris a Sole diſputandi non eſt locus, <lb></lb> cum hæ per eorum Elongationes à Sole determinentur.</s> <s> De di<lb></lb>ſtantiis etiam ſuperiorum Planetarum à Sole tollitur omnis diſpu<lb></lb>tatio per Eclipſes Satellitum Jovis.</s> <s> Etenim per Eclipſes illas de<lb></lb>terminatur poſitio umbræ quam Jupiter projicit, & eo nomine <lb></lb> habetur Jovis longitudo Heliocentrica.</s> <s> Ex longitudinibus autem <lb></lb> Heliocentrica & Geocentrica inter ſe collatis determinatur diſtan<lb></lb>tia Jovis.</s> <s> <lb></lb> PHÆNOMENON V.</s></p> <p type="main"> <s><emph type="italics"></emph>Planetas primarios, radiis ad Terram ductis, areas deſcribere tem<lb></lb>poribus minime proportionales; at radiis ad Solem ductis, areas <lb></lb> temporibus proportionales percurrere.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam reſpectu Terræ nunc progrediuntur, nunc ſtationarii ſunt, <lb></lb> nunc etiam regrediuntur: At Solis reſpectu ſemper progrediuntur, <lb></lb> idque propemodum uniformi cum motu, ſed paulo celerius tamen <lb></lb> in Periheliis ac tardius in Apheliis, ſic ut arearum æquabilis ſit de<lb></lb>ſcriptio. </s> <s>Propoſitio eſt Aſtronomis notiſſima, & in Jove apprime <lb></lb> demonſtratur per Eclipſes Satellitum, quibus Eclipſibus Helio<lb></lb>centricas Planetæ hujus longitudines & diſtantias à Sole determi<lb></lb>nari diximus. <lb></lb> PHÆNOMENON VI.</s></p> <p type="main"> <s><emph type="italics"></emph>Lunam radio ad centrum Terræ ducto, aream tempori proporti<lb></lb>onalem deſcribere.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Patet ex Lunæ motu apparente cum ipſius diametro apparente <lb></lb> collato. </s> <s>Perturbatur autem motus Lunaris aliquantulum à vi So<lb></lb>lis, ſed errorum inſenſibiles minutias in hiſce Phænomenis negligo. <lb></lb> <pb xlink:href="039/01/390.jpg" pagenum="362"></pb><lb></lb></s></p></subchap2><subchap2><p> <s><arrow.to.target n="note368"></arrow.to.target>PROPOSITIONES.<lb></lb><gap desc="hr tag"></gap><lb></lb>PROPOSITIO I. THEOREMA I.<lb></lb></s></p> <p type="margin"> <s><margin.target id="note368"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Vires, quibus Planetæ Circumjoviales perpetuo retrahuntur à me<lb></lb>tibus rectilineis & in Orbibus ſuis retinentur, reſpicere cen<lb></lb>trum Jovis, & eſſe reciproce ut quadrata diſtantiarum loco<lb></lb>rum ab eodem centro.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>PAtet pars prior Propoſitionis per Phænomenon primum, & <lb></lb> Propoſitionem ſecundam vel tertiam Libri primi: & pars <lb></lb> poſterior per Phænomenon primum, & Corollarium ſextum Pro<lb></lb>poſitionis quartæ ejuſdem Libri. <lb></lb></s> </p> <p type="main"> <s>Idem intellige de Planetis qui Saturnum comitantur, per Phæ<lb></lb>nomenon ſecundum. <lb></lb> PROPOSITIO II. THEOREMA II.<lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Vires, quibus Planetæ primarii perpetuo retrahuntur à motibus <lb></lb> rectilineis, & in Orbibus ſuis retinentur, reſpicere Solem, & <lb></lb> eſſe reciproce ut quadrata diſtantiarum ab ipſius centro.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Patet pars prior Propoſitionis per Phænomenon quintum, & <lb></lb> Propoſitionem ſecundam Libri primi: & pars poſterior per Phæ<lb></lb>nomenon quartum, & Propoſitionem quartam ejuſdem Libri. <lb></lb> Accuratiſſime autem demonſtratur hæc pars Propoſitionis per <lb></lb> quietem Apheliorum. </s> <s>Nam aberratio quam minima à ratione <lb></lb> duplicata (per Corol. 1. Prop. XLV. Lib. I.) motum Apſidum in <lb></lb> ſingulis revolutionibus notabilem, in plunibus enormem efficere <lb></lb> deberet. <lb></lb> <pb xlink:href="039/01/391.jpg" pagenum="363"></pb><lb></lb>PROPOSITIO III. THEOREMA III.<lb></lb><arrow.to.target n="note369"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note369"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Vim qua Luna retinetur in Orbe ſuo reſpicere Terram, & eſſe re<lb></lb>citroce ut quadratum diſtantiæ loeorum ab ipſius centro.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Patet aſſertionis pars prior per Phænomenon ſextum, & Propo<lb></lb>poſitionem ſecundam vel tertiam Libri primi: & pars poſterior <lb></lb> per motum tardiſſimum Lunaris Apogæi. </s> <s>Nam motus ille, qui <lb></lb> ſingulis revolutionibus eſt graduum tantum trium & minutorum <lb></lb> trium in conſequentia, contemni poteſt. </s> <s>Patet enim (per Corol. 1. <lb></lb> Prop. XLV. Lib.I.) quod ſi diſtantia Lunæ a centro Terræ ſit ad <lb></lb> ſemidiametrum Terræ ut D ad 1; vis a qua motus talis oriatur ſit <lb></lb> reciproce ut D (2 4/243), id eſt, reciproce ut ea ipſius D dignitas cu<lb></lb>jus index eſt (2 4/243), hoc eſt, in ratione diſtantiæ paulo majore quam <lb></lb> duplicata inverſe, ſed quæ partibus 59 1/4 propius ad duplicatam <lb></lb> quam ad triplicatam accedit. </s> <s>Oritur vero ab actione Solis (uti <lb></lb> poſthac dicetur) & propterea hic negligendus eſt. </s> <s>Actio Solis <lb></lb> quatenus Lunam diſtrahit a Terra, eſt ut diſtantia Lunæ a Terra <lb></lb> quamproxime; ideoque (per ea quæ dicuntur in Corol. 2. Prop. <lb></lb> XLV. Lib. I.) eſt ad Lunæ vim centripetam ut 2 ad 357,45 circi<lb></lb>ter, ſeu 1 ad (178 29/40). Et neglecta Solis vi tantilla, vis reliqua qua <lb></lb> Luna retinetur in Orbe erit reciproce ut D<emph type="sup"></emph>2<emph.end type="sup"></emph.end>. Id quod etiam <lb></lb> plenius conſtabit conferendo hanc vim cum vi gravitatis, ut fit <lb></lb> in Propoſitione ſequente. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Si vis centripeta mediocris qua Luna retinetur in Orbe, <lb></lb> augeatur primo in ratione (177 29/40) ad (178 29/40), deinde etiam in rati<lb></lb>one duplicata ſemidiametri Terræ ad mediocrem diſtantiam centri <lb></lb> Lunæ a centro Terræ: habebitur vis centripeta Lunaris ad ſuper<lb></lb>ficiem Terræ, poſito quod vis illa deſcendendo ad ſuperficiem <lb></lb> Terræ, perpetuo augeatur in reciproca altitudinis ratione du<lb></lb>plicata. <lb></lb> PROPOSITIO IV. THEOREMA IV.<lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Lunam gravitare in Terram, & vi gravitatis retrahi ſemper a <lb></lb> motu rectilineo, & in Orbe ſuo retineri.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Lunæ diſtantia mediocris a Terra in Syzygiis eſt ſemidiametro<lb></lb>rum terreſtrium, ſecundum pleroſque Aſtronomorum 59, ſecun<lb></lb>dum <emph type="italics"></emph>Vendelinum<emph.end type="italics"></emph.end>60, ſecundum <emph type="italics"></emph>Copernicum<emph.end type="italics"></emph.end>60 1/3, & ſecundum <emph type="italics"></emph>Ty-<emph.end type="italics"></emph.end><lb></lb><pb xlink:href="039/01/392.jpg" pagenum="364"></pb><lb></lb><arrow.to.target n="note370"></arrow.to.target><emph type="italics"></emph>chonem<emph.end type="italics"></emph.end>56 1/2. Aſt <emph type="italics"></emph>Tycho,<emph.end type="italics"></emph.end>& quotquot ejus Tabulas refractionum <lb></lb> ſequuntur, conſtituendo refractiones Solis & Lunæ (omnino con<lb></lb>tra naturam Lucis) majores quam Fixarum, idque ſcrupulis quaſi <lb></lb> quatuor vel quinque, auxerunt parallaxin Lunæ ſcrupulis totidem, <lb></lb> hoc eſt, quaſi duodecima vel decima quinta parte totius paralla<lb></lb>xeos. </s> <s>Corrigatur iſte error, & diſtantia evadet quaſi 60 1/2 ſemi<lb></lb>diametrorum terreſtrium, fere ut ab aliis aſſignatum eſt. </s> <s>Aſſuma<lb></lb>mus diſtantiam mediocrem ſexaginta ſemidiametrorum; & Luna<lb></lb>rem periodum reſpectu Fixarum compleri diebus 27, horis 7, mi<lb></lb>nutis primis 43, ut ab Aſtronomis ſtatuitur; atque ambitum Terræ <lb></lb> eſſe pedum Pariſienſium 123249600, uti a <emph type="italics"></emph>Gallis<emph.end type="italics"></emph.end>menſurantibus de<lb></lb>finitum eſt: Et ſi Luna motu omni privari fingatur ac dimitti ut, <lb></lb> urgente vi illa omni qua in Orbe ſuo retinetur, deſcendat in Ter<lb></lb>ram; hæc ſpatio minuti unius primi cadendo deſcribet pedes Pari<lb></lb>ſienſes (15 1/12). Colligitur hoc ex calculo vel per Propoſitionem <lb></lb> XXXVI. Libri primi, vel (quod eodem recidit) per Corollarium <lb></lb> nonum Propoſitionis quartæ ejuſdem Libri, confecto. </s> <s>Nam ar<lb></lb>cus illius quem Luna tempore minuti unius primi, medio ſuo <lb></lb> motu, ad diſtantiam ſexaginta ſemidiametrorum terreſtrium de<lb></lb>ſcribat, ſinus verſus eſt pedum Pariſienſium (15 1/12) circiter. </s> <s>Unde <lb></lb> cum vis illa accedendo ad Terram augeatur in duplicata diſtantiæ <lb></lb> ratione inverſa, adeoque ad ſuperficiem Terræ major ſit partibus <lb></lb> 60X60 quam ad Lunam; corpus vi illa in regionibus noſtris ca<lb></lb>dendo, deſcribere deberet ſpatio minuti unius primi pedes Pari<lb></lb>ſienſes 60X60X(15 1/12), & ſpatio minuti unius ſecundi pedes (15 1/12). <lb></lb> Atqui corpora in regionibus noſtris vi gravitatis cadendo, deſcri<lb></lb>bunt tempore minuti unius ſecundi pedes Pariſienſes (15 1/12), uti <lb></lb> <emph type="italics"></emph>Hugenius<emph.end type="italics"></emph.end>factis pendulorum experimentis & computo inde inito, <lb></lb> demonſtravit: & propterea (per Reg. 1. & 11.) vis qua Luna in <lb></lb> Orbe ſuo retinetur, illa ipſa eſt quam nos Gravitatem dicere ſole<lb></lb>mus. </s> <s>Nam ſi Gravitas ab ea diverſa eſt, corpora viribus utriſque <lb></lb> conjunctis Terram petendo, duplo velocius deſcendent, & ſpatio <lb></lb> minuti unius ſecundi cadendo deſcribent pedes Pariſienſes 30 1/6: <lb></lb> omnino contra Experientiam. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note370"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s>Calculus hic fundatur in hypotheſi quod Terra quieſcit. </s> <s>Nam <lb></lb> ſi Terra & Luna circum Solem moveantur, & interea quoque cir<lb></lb>cum commune gravitatis centrum revolvantur: diſtantia centro<lb></lb>rum Lunæ ac Terræ ab invicem erit 60 1/2 ſemidiametrorum ter<lb></lb>reſtrium; uti computationem (per Prop. LX. Lib. I.) ineunti <lb></lb> patebit. <lb></lb> <pb xlink:href="039/01/393.jpg" pagenum="365"></pb><lb></lb>PROPOSITIO V. THEOREMA V.<lb></lb><arrow.to.target n="note371"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note371"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Planetas Circumjoviales gravitare in Jovem, Circumſaturnios in <lb></lb> Saturnum, & Circumſolares in Solem, & vi gravitatis ſuæ <lb></lb> retrahi ſemper à motibus rectilineis, & in Orbibus curvili<lb></lb>neis retineri.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam revolutiones Planetarum Circumjovialium circa Jovem, Cir<lb></lb>cumſaturniorum circa Saturnum, & Mercurii ac Veneris reliquo<lb></lb>rumque Circumſolarium circa Solem ſunt Phænomena ejuſdem ge<lb></lb>neris cum revolutione Lunæ circa Terram; & propterea per <lb></lb> Reg. 11. à cauſis ejuſdem generis dependent: præſertim cum de<lb></lb>monſtratum ſit quod vires, à quibus revolutiones illæ dependent, <lb></lb> reſpiciant centra Jovis, Saturni ac Solis, & recedendo à Jove, Sa<lb></lb>turno & Sole decreſcant eadem ratione ac lege, qua vis gravitatis <lb></lb> decreſcit in receſſu à Terra. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Gravitas igitur datur in Planetas univerſos. </s> <s>Nam Ve<lb></lb>nerem, Mercurium, cæteroſque eſſe corpora ejuſdem generis cum <lb></lb> Jove & Saturno, nemo dubitat. </s> <s>Et cum attractio omnis (per mo<lb></lb>tus Legem tertiam) mutua ſit, Jupiter in Satellites ſuos omnes, <lb></lb> Saturnus in ſuos, TerraQ.E.I. Lunam, & Sol in Planetas omnes <lb></lb> primarios gravitabit. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Gravitatem, quæ Planetam unumquemque reſpicit, eſſe <lb></lb> reciproce ut quadratum diſtantiæ loeorum ab ipſius centro. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Graves ſunt Planetæ omnes in ſe mutuo per Corol. 1. <lb></lb> & 2. Et hinc Jupiter & Saturnus prope conjunctionem ſe invicem <lb></lb> attrahendo, ſenſibiliter perturbant motus mutuos, Sol perturbat <lb></lb> motus Lunares, Sol & Luna perturbant Mare noſtrum, ut in <lb></lb> ſequentibus explicabitur. <lb></lb> PROPOSITIO VI. THEOREMA VI.<lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Corpora omnia in Planetas ſingulos gravitare, & pondera eorum <lb></lb> in eundem quemvis Planetam, paribus diſtantiis à centro Pla<lb></lb>netæ, proportionalia eſſe quantitati materiæ in ſingulis.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Deſcenſus gravium omnium in Terram (dempta ſaltem inæquali <lb></lb> retardatione quæ ex Aeris perexigua reſiſtentia oritur) æqualibus <lb></lb> <pb xlink:href="039/01/394.jpg" pagenum="366"></pb><lb></lb><arrow.to.target n="note372"></arrow.to.target>temporibus fieri, jamdudum obſervarunt alii; & accuratiſſime qui<lb></lb>dem notare licet æqualitatem temporum in Pendulis. </s> <s>Rem tentavi <lb></lb> in Auro, Argento, Plumbo, Vitro, Arena, Sale communi, Ligno, <lb></lb> Aqua, Tritico. </s> <s>Comparabam pyxides duas ligneas rotundas & <lb></lb> æquales. </s> <s>Unam implebam Ligno, & idem Auri pondus ſuſpende<lb></lb>bam (quam potui exacte) in alterius centro oſcillationis. </s> <s>Pyxides <lb></lb> ab æqualibus pedum undecim filis pendentes, conſtituebant Pen<lb></lb>dula, quoad pondus, figuram, & acris reſiſtentiam omnino paria: <lb></lb> Et paribus oſcillationibus, juxta poſitæ, ibant una & redibant di<lb></lb>utiſſime. </s> <s>Proinde copia materiæ in Auro (per Corol. 1. & 6. Prop. <lb></lb> XXIV. Lib. II.) erat ad copiam materiæ in Ligno, ut vis motricis <lb></lb> actio in totum Aurum ad ejuſdem actionem in totum Lignum; hoc <lb></lb> eſt, ut pondus ad pondus. </s> <s>Et ſic in cæteris. </s> <s>In corporibus ejuſ<lb></lb>dem ponderis differentia materiæ, quæ vel minor eſſet quam pars <lb></lb> milleſima materiæ totius, his experimentis manifeſto deprehendi <lb></lb> potuit. </s> <s>Jam vero naturam gravitatis in Planetas eandem eſſe atque <lb></lb> in Terram, non eſt dubium. </s> <s>Elevari enim fingantur corpora hæc <lb></lb> Terreſtria ad uſque Orbem Lunæ, & una cum Luna motu omni <lb></lb> privata demitti, ut in Terram ſimul cadant; & per jam ante oſtenſa <lb></lb> certum eſt quod temporibus æqualibus deſcribent æqualia ſpatia <lb></lb> cum Luna, adeoque quod ſunt ad quantitatem materiæ in Luna, ut <lb></lb> pondera ſua ad ipſius pondus. </s> <s>Porro quoniam Satellites Jovis <lb></lb> temporibus revolvuntur quæ ſunt in ratione ſeſquiplicata diſtanti<lb></lb>arum à centro Jovis, erunt eorum gravitates acceleratrices in Jo<lb></lb>vem reciproce ut quadrata diſtantiarum à centro Jovis; & prop<lb></lb>terea in æqualibus a Jove diſtantiis, eorum gravitates acceleratrices <lb></lb> evaderent æquales. </s> <s>Proinde temporibus æqualibus ab æqualibus <lb></lb> altitudinibus cadendo, deſcriberent æqualia ſpatia; perinde ut fit <lb></lb> in gravibus, in hac Terra noſtra. </s> <s>Et eodem argumento Planetæ <lb></lb> circumſolares ab æqualibus à Sole diſtantiis demiſſi, deſcenſu ſuo <lb></lb> in Solem æqualibus temporibus æqualia ſpatia deſcriberent. </s> <s>Vires <lb></lb> autem, quibus corpora inæqualia æqualiter accelerantur, ſunt ut <lb></lb> corpora; hoc eſt, pondera ut quantitates materiæ in Planetis. <lb></lb> Porro Jovis & ejus Satellitum pondera in Solem proportionalia <lb></lb> eſſe quantitatibus materiæ eorum, patet ex motu Satellitum quam <lb></lb> maxime regulari; per Corol. 3. Prop. LXV. Lib. I. Nam ſi ho<lb></lb>rum aliqui magis traherentur in Solem, pro quantitate materiæ <lb></lb> ſuæ, quam cæteri: motus Satellitum (per Corol. 2. Prop. LXV. <lb></lb> Lib. I.) ex inæqualitate attractionis perturbarentur. </s> <s>Si (paribus <lb></lb> à Sole diſtantiis) Satelles aliquis gravior eſſet in Solem pro quan<lb></lb><pb xlink:href="039/01/395.jpg" pagenum="367"></pb><lb></lb>titate materiæ ſuæ, quam Jupiter pro quantitate materiæ ſuæ, in <lb></lb> <arrow.to.target n="note373"></arrow.to.target>ratione quacunQ.E.D.ta, puta <emph type="italics"></emph>d<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>e<emph.end type="italics"></emph.end>: diſtantia inter centrum So<lb></lb>lis & centrum Orbis Satellitis, major ſemper foret quam diſtantia <lb></lb> inter centrum Solis & centrum Jovis in ratione ſubduplicata quam <lb></lb> proxime; uti calculis quibuſdam initis inveni. </s> <s>Et ſi Satelles mi<lb></lb>nus gravis eſſet in Solem in ratione illa <emph type="italics"></emph>d<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>diſtantia centri <lb></lb> Orbis Satellitis à Sole minor foret quam diſtantia centri Jovis à <lb></lb> Sole in ratione illa ſubduplicata. </s> <s>Igitur ſi in æqualibus à Sole <lb></lb> diſtantiis, gravitas acceleratrix Satellitis cujuſvis in Solem major <lb></lb> eſſet vel minor quam gravitas acceleratrix Jovis in Solem, parte <lb></lb> tantum milleſima gravitatis totius, foret diſtantia centri Orbis <lb></lb> Satellitis à Sole major vel minor quam diſtantia Jovis à Sole <lb></lb> parte (7/2000) diſtantiæ totius, id eſt, parte quinta diſtantiæ Satellitis <lb></lb> extimi à centro Jovis: Quæ quidem Orbis eccentricitas foret &c. valde <lb></lb> ſenſibilis. </s> <s>Sed Orbes Satellitum ſunt Jovi concentrici, & propte<lb></lb>rea gravitates acceleratrices Jovis & Satellitum in Solem æquantur <lb></lb> inter ſe. </s> <s>Et eodem argumento pondera Saturni & Comitum ejus <lb></lb> in Solem, in æqualibus à Sole diſtantiis, ſunt ut quantitates mate<lb></lb>riæ in ipſis: Et pondera Lunæ ac Terræ in Solem vel nulla ſunt, <lb></lb> vel earum maſſis accurate proportionalia. </s> <s>Aliqua autem ſunt per <lb></lb> Corol. 1. & 3. Prop. V. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note372"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="margin"> <s><margin.target id="note373"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s>Quinetiam pondera partium ſingularum Planetæ cujuſQ.E.I. <lb></lb> alium quemcunque, ſunt inter ſe ut materia in partibus ſingulis. <lb></lb> Nam ſi partes aliquæ plus gravitarent, aliæ minus, quam pro quan<lb></lb>titate materiæ: Planeta totus, pro genere partium quibus maxime <lb></lb> abundet, gravitaret magis vel minus quam pro quantitate materiæ <lb></lb> totius. </s> <s>Sed nec refert utrum partes illæ externæ ſint vel internæ. <lb></lb> Nam ſi verbi gratia corpora Terreſtria, quæ apud nos ſunt, in <lb></lb> Orbem Lunæ elevari fingantur, & conferantur cum corporo Lunæ: <lb></lb> Si horum pondera eſſent ad pondera partium externarum Lunæ <lb></lb> ut quantitates materiæ in iiſdem, ad pondera vero partium in<lb></lb>ternarum in majori vel minori ratione, forent eadem ad pondus <lb></lb> Lunæ totius in majori vel minori ratione: contra quam ſupra <lb></lb> oſtenſum eſt. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc pondera corporum non pendent ab eorum for<lb></lb>mis & texturis. </s> <s>Nam ſi cum formis variari poſſent; forent ma<lb></lb>jora vel minora, pro varietate formarum, in æquali materia: om<lb></lb>nino contra Experientiam. <lb></lb> <pb xlink:href="039/01/396.jpg" pagenum="368"></pb><lb></lb><arrow.to.target n="note374"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note374"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Corpora univerſa quæ circa Terram ſunt, gravia ſunt <lb></lb> in Terram; & pondera omnium, quæ æqualiter à centro Terræ <lb></lb> diſtant, ſunt ut quantitates materiæ in iiſdem. </s> <s>Hæc eſt qualitas <lb></lb> omnium in quibus experimenta inſtituere licet, & propterea per <lb></lb> Reg.111. de univerſis affirmanda eſt. </s> <s>Si Æther aut corpus aliud <lb></lb> quodcunque vel gravitate omnino deſtitueretur, vel pro quantitate <lb></lb> materiæ ſuæ minus gravitaret: quoniam id (ex mente <emph type="italics"></emph>Ariſtotelis, <lb></lb> Carteſii & aliorum<emph.end type="italics"></emph.end>non differet ab aliis corporibus niſi in forma<lb></lb>materiæ, poſſet idem per mutationem formæ gradatim tranſmutari <lb></lb> in corpus ejuſdem conditionis cum iis quæ, pro quantitate materiæ, <lb></lb> quam maxime gravitant, & viciſſim corpora maxime gravia, fer<lb></lb>mam illius gradatim induendo, poſſent gravitatem ſuam gradatim <lb></lb> amittere. </s> <s>Ac proinde pondera penderent à formis corporum, <lb></lb> poſſentque cum formis variari, contra quam probatum eſt in <lb></lb> Corollario ſuperiore. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Spatia omnia non ſunt æqualiter plena. </s> <s>Nam ſi ſpatia <lb></lb> omnia æqualiter plena eſſent, gravitas ſpecifica fluidi quo regio <lb></lb> aeris impleretur, ob ſummam denſitatem materiæ, nil cederet gra<lb></lb>vitati ſpecificæ argenti vivi, vel auri, vel corporis alterius cujuſ<lb></lb>cunQ.E.D.nſiſſimi; & propterea nec aurum neque aliud quod<lb></lb>cunque corpus in aere deſcendere poſſet. </s> <s>Nam corpora in flui<lb></lb>dis, niſi ſpecifice graviora ſint, minime deſcendunt. </s> <s>Quod ſi <lb></lb> quantitas materiæ in ſpatio dato per rarefactionem quamcunque <lb></lb> diminui poſſit, quidni diminui poſſit in infinitum? <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Si omnes omnium corporum particulæ ſolidæ ſint ejuſ<lb></lb>dem denſitatis, neque abſque poris rarefieri poſſint, Vacuum da<lb></lb>tur. </s> <s>Ejuſdem denſitatis eſſe dico, quarum vires inertiæ ſunt ut <lb></lb> magnitudines. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Vis gravitatis diverſi eſt generis à vi magnetica. </s> <s>Nam <lb></lb> attractio magnetica non eſt ut materia attracta. </s> <s>Corpora aliqua <lb></lb> magis trahuntur, alia minus, plurima non trahuntur. </s> <s>Et vis mag<lb></lb>netica in uno & eodem corpore intendi poteſt & remitti, eſtque <lb></lb> nonnunquam longe major pro quantitate materiæ quam vis gra<lb></lb>vitatis, & in receſſu à Magnete decreſcit in ratione diſtantiæ non <lb></lb> duplicata, ſed fere triplicata, quantum ex craſſis quibuſdam obſer<lb></lb>vationibus animadvertere potui. <lb></lb> <pb xlink:href="039/01/397.jpg" pagenum="369"></pb><lb></lb>PROPOSITIO VII. THEOREMA VII.<lb></lb><arrow.to.target n="note375"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note375"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Gravitatem in corpora univerſa fieri, eamque proportionalem eſſe <lb></lb> quantitati materiæ in ſingulis.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Planetas omnes in ſe mutuo graves eſſe jam ante probavimus, <lb></lb> ut & gravitatem in unumquemque ſeorſim ſpectatum eſſe reci<lb></lb>proce ut quadratum diſtantiæ loeorum à centro Planetæ. Et inde <lb></lb> conſequens eſt, (per Prop. LXIX. Lib. I. & ejus Corollaria) gra<lb></lb>vitatem in omnes proportionalem eſſe materiæ in iiſdem. <lb></lb> </s></p> <p type="main"> <s>Porro cum Pianetæ cujuſvis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>partes omnes graves ſint in Pla<lb></lb>netam quemvis <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>& gravitas partis cujuſque ſit ad gravitatem <lb></lb> totius, ut materia partis ad materiam totius, & actioni omni re<lb></lb>actio (per motus Legem tertiam) æqualis ſit; Planeta <emph type="italics"></emph>B<emph.end type="italics"></emph.end>in partes <lb></lb> omnes Planetæ <emph type="italics"></emph>A<emph.end type="italics"></emph.end>viciſſim gravitabit, & erit gravitas ſua in par<lb></lb>tem unamquamque ad gravitatem ſuam in totum, ut materia par<lb></lb>tis ad materiam totius. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Oritur igitur & componitur gravitas in Planetam to<lb></lb>tum ex gravitate in partes ſingulas. </s> <s>Cujus rei exempla habemus <lb></lb> in attractionibus Magneticis & Electricis. </s> <s>Oritur enim attractio <lb></lb> omnis in totum ex attractionibus in partes ſingulas. </s> <s>Res intelli<lb></lb>getur in gravitate, concipiendo Planetas plures minores in unum <lb></lb> Globum coire & Planetam majorem componere. </s> <s>Nam vis totius <lb></lb> ex viribus partium componentium oriri debebit. </s> <s>Siquis objiciat <lb></lb> quod corpora omnia, quæ apud nos ſunt, hac lege gravitare de<lb></lb>berent in ſe mutuo, cum tamen cjuſmodi gravitas neutiquam ſen<lb></lb>tiatur: Reſpondeo quod gravitas in hæc corpora, cum ſit ad gra<lb></lb>vitatem in Terram totam ut ſunt hæc corpora ad Terram totam, <lb></lb> longe minor eſt quam quæ ſentiri poſſit. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Gravitatio in ſingulas corporis particulas æquales eſt <lb></lb> reciproce ut quadratum diſtantiæ loeorum à particulis. </s> <s>Patet per <lb></lb> Corol. 3. Prop. LXXIV. Lib. I. <lb></lb> <pb xlink:href="039/01/398.jpg" pagenum="370"></pb><lb></lb><arrow.to.target n="note376"></arrow.to.target>PROPOSITIO VIII. THEOREMA VIII.<lb></lb></s></p> <p type="margin"> <s><margin.target id="note376"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Si Globorum duorum in ſe mutuo gravitantium materia undique, <lb></lb> in regionibus quæ à centris æqualiter diſtant, homogenea ſit: <lb></lb> erit pondus Globi alterutrius in alterum reciproce ut quadra<lb></lb>tum diſtantiæ inter centra.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Poſtquam inveniſſem gravitatem in Planetam totum oriri & <lb></lb> componi ex gravitatibus in partes; & eſſe in partes ſingulas reci<lb></lb>proce proportionalem quadratis diſtantiarum a partibus: dubita<lb></lb>bam an reciproca illa proportio duplicata obtineret accurate in vi <lb></lb> tota ex viribus pluribus compoſita, an vero quam proxime. </s> <s>Nam <lb></lb> fieri poſſet ut proportio, quæ in majoribus diſtantiis ſatis accu<lb></lb>rate obtineret, prope ſuperficiem Planetæ ob inæquales particu<lb></lb>larum diſtantias & ſitus diſſimiles, notabiliter erraret. </s> <s>Tandem <lb></lb> vero, per Prop. LXXV. & LXXVI. Libri primi & ipſarum Corol<lb></lb>laria, intellexi veritatem Propoſitionis de qua hic agitur. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc inveniri & inter ſe comparari poſſunt pondera <lb></lb> corporum in diverſos Planetas. </s> <s>Nam pondera corporum æqua<lb></lb>lium circum Planetas in circulis revolventium ſunt (per Corol. 2. <lb></lb> Prop. IV. Lib.I.) ut diametri circulorum directe & quadrata tem<lb></lb>porum periodieorum inverſe; & pondera ad ſuperficies Planeta<lb></lb>rum, aliaſve quaſvis a centro diſtantias, majora ſunt vel minora <lb></lb> (per hanc Propoſitionem) in duplicata ratione diſtantiarum in<lb></lb>verſa. Sic ex temporibus periodicis Veneris circum Solem die<lb></lb>rum 224 & horarum 16 1/4, Satellitis extimi circumjovialis circum <lb></lb> Jovem dierum 16 & horarum (16 1/15), Satellitis Hugeniani circum <lb></lb> Saturnum dierum 15 & horarum 22 2/3, & Lunæ circum Terram <lb></lb> dierum 27, hor. 7. min. 43, collatis cum diſtantia mediocri Vene<lb></lb>ris a Sole & cum elongationibus maximis heliocentricis Satellitis <lb></lb> extimi circumjovialis a centro Jovis 8′. 21 1/2″, Satellitis Hugeniani <lb></lb> a centro Saturni 3′. 20″, & Lunæ a Terra 10′, computum ineundo <lb></lb> inveni quod corporum æqualium & a Sole, Jove, Saturno ac Terra <lb></lb> æqualiter diſtantium pondera in Solem, Jovem, Saturnum ac Ter<lb></lb>ram forent ad invicem ut 1, (1/1033), (1/2411), & (1/227512) reſpective. </s> <s>Eſt enim <lb></lb> parallaxis Solis ex obſervationibus noviſſimis quaſi 10″, & <emph type="italics"></emph>Hal<lb></lb>leius<emph.end type="italics"></emph.end>noſter per emerſiones Jovis & Satellitum e parte obſcura <lb></lb> <pb xlink:href="039/01/399.jpg" pagenum="371"></pb><lb></lb>Lunæ, determinavit quod elongatio maxima heliocentrica Satelli<lb></lb><arrow.to.target n="note377"></arrow.to.target>tis extimi Jovialis a centro Jovis in mediocri Jovis a Sole diſtan<lb></lb>tia ſit 8′. 21 1/2″, & diameter Jovis 41″. Ex duratione Eclipſeon <lb></lb> Satellitum in umbram Jovis incidentium prodit hæc diameter <lb></lb> quaſi 40″, atque adeo ſemidiameter 20″. Menſuravit autem <emph type="italics"></emph>Hu<lb></lb>genius<emph.end type="italics"></emph.end>elongationem maximam heliocentricam Satellitis a ſe de<lb></lb>tecti 3′. 20″ a centro Saturni, & hujus elongationis pars quarta, <lb></lb> nempe 50″, eſt diameter annuli Saturni e Sole viſi, & diameter Sa<lb></lb>turni eſt ad diametrum annuli ut 4 ad 9, ideoque ſemidiameter <lb></lb> Saturni e Sole viſi eſt 11″. Subducatur lux erratica quæ haud <lb></lb> minor eſſe ſolet quam 2″ vel 3″: Et manebit ſemidiameter Saturni <lb></lb> quaſi 9″. Ex hiſce autem & Solis ſemidiametro mediocri 16′. 6″ <lb></lb> computum ineundo prodeunt veræ Solis, Jovis, Saturni ac Terræ <lb></lb> ſemidiametri ad invicem ut 10000, 1077, 889 & 104. Unde, <lb></lb> cum pondera æqualium corporum 2 centris Solis, Jovis, Saturni <lb></lb> ac Terræ æqualiter diſtantium, ſint in Solem, Jovem, Saturnum <lb></lb> ac Terram, ut 1, (1/1033), (1/2411), & (1/227512) reſpective, & auctis vel dimi<lb></lb>nutis diſtantiis pondera diminuantur vel augeantur in duplicata <lb></lb> ratione: pondera æqualium corporum in Solem, Jovem, Satur<lb></lb>num ac Terram in diſtantiis 10000, 1077, 889, & 104 ab eorum <lb></lb> centris, atque adeo in eorum ſuperficiebus, erunt ut 10000, 835, <lb></lb> 525, & 410 reſpective. </s> <s>Quanta ſint pondera corporum in ſuper<lb></lb>ficie Lunæ dicemus in ſequentibus. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note377"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Innoteſcit etiam quantitas materiæ in Planetis ſingulis. <lb></lb> Nam quantitates materiæ in Planetis ſunt ut eorum vires in æqua<lb></lb>libus diſtantiis ab eorum centris, id eſt, in Sole, Jove, Saturno ac <lb></lb> Terra ſunt ut 1, (1/1033), (1/2411), & (1/227512) reſpective. </s> <s>Si parallaxis Solis <lb></lb> ſtatuatur major vel minor quam 10″, debebit quantitas materiæ in <lb></lb> Terra augeri vel diminui in triplicata ratione. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Innoteſcunt etiam denſitates Planetarum. </s> <s>Nam pon<lb></lb>dera corporum æqualium & homogeneorum in Sphæras homoge<lb></lb>neas ſunt in ſuperficiebus Sphærarum ut Sphærarum diametri, per <lb></lb> Prop. LXXII. Lib. I. ideoque Sphærarum heterogenearum denſi<lb></lb>tates ſunt ut pondera illa applicata ad Sphærarum diametros. <lb></lb> Erant autem veræ Solis, Jovis, Saturni ac Terræ diametri ad invi<lb></lb>cem ut 10000, 1077, 889, & 104, & pondera in eoſdem ut 10000, <lb></lb> 835, 525, & 410, & propterea denſitates ſunt ut 100, 78, 59, <lb></lb> & 396. Denſitas Terræ quæ prodit ex hoc computo non pendet <lb></lb> a parallaxi Solis, ſed determinatur per parallaxin Lunæ, & prop<lb></lb><pb xlink:href="039/01/400.jpg" pagenum="372"></pb><lb></lb><arrow.to.target n="note378"></arrow.to.target>terea hic recte definitur. </s> <s>Eſt igitur Sol paulo denſior quam Jupi<lb></lb>ter, & Jupiter quam Saturnus, & Terra quadruplo denſior quam <lb></lb> Sol. </s> <s>Nam per ingentem ſuum calorem Sol rareſcit. </s> <s>Luna vero <lb></lb> denſior eſt quam Terra, ut in ſequentibus patebit. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note378"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Denſiores igitur ſunt Planetæ qui ſunt minores, cæ<lb></lb>teris paribus. </s> <s>Sic enim vis gravitatis in eorum ſuperficiebus ad <lb></lb> æqualitatem magis accedit. </s> <s>Sed & denſiores ſunt Planetæ, cæte<lb></lb>ris paribus, qui ſunt Soli propiores; ut Jupiter Saturno, & Terra <lb></lb> Jove. </s> <s>In diverſis utiQ.E.D.ſtantiis a Sole collocandi erant Planetæ <lb></lb> ut quilibet pro gradu denſitatis calore Solis majore vel minore <lb></lb> frueretur. </s> <s>Aqua noſtra, ſi Terra locaretur in orbe Saturni, rige<lb></lb>ſceret, ſi in orbe Mercurii in vapores ſtatim abiret. </s> <s>Nam lux <lb></lb> Solis, cui calor proportionalis eſt, ſeptuplo denſior eſt in orbe <lb></lb> Mercurii quam apud nos: & Thermometro expertus ſum quod <lb></lb> ſeptuplo Solis æſtivi calore aqua ebullit. </s> <s>Dubium vero non eſt <lb></lb> quin materia Mercurii ad calorem accommodetur, & propterea <lb></lb> denſior ſit hac noſtra; cum materia omnis denſior ad operationes <lb></lb> Naturales obeundas majorem calorem requirat. <lb></lb> PROPOSITIO IX. THEOREMA IX.<lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Gravitatem pergendo a ſuperficiebus Planetarum deorſum de<lb></lb>creſcere in ratione diſtantiarum a centro quam proxime.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Si materia Planetæ quoad denſitatem uniformis eſſet, obtineret <lb></lb> hæc Propoſitio accurate: per Prop. LXXIII. Lib. I. Error igitur <lb></lb> tantus eſt, quantus ab inæquabili denſitate oriri poſſit. <lb></lb> PROPOSITIO X. THEOREMA X.<lb></lb><emph type="italics"></emph>Motus Planetarum in Cœlis diutiſſime conſervari poſſe.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>In Scholio Propoſitionis XL. Lib. II. oſtenſum eſt quod globus <lb></lb> Aquæ congelatæ in Aere noſtro, libere movendo & longitudinem <lb></lb> ſemidiametri ſuæ deſcribendo, ex reſiſtentia Aeris amitteret motus <lb></lb> ſui partem (1/4586). Obtinet autem eadem proportio quam proxime <lb></lb> in globis utcunque magnis & velocibus. </s> <s>Jam vero Globum Terræ <lb></lb> noſtræ denſiorem eſſe quam ſi totus ex Aqua conſtaret, ſic colligo. <lb></lb> Si Globue hicce totus eſſet aqueus, quæcunque rariora eſſent quam <lb></lb> aqua, ob minorem ſpecificam gravitatem emergerent & ſupernata<lb></lb><pb xlink:href="039/01/401.jpg" pagenum="373"></pb><lb></lb>rent. Ea Q.E.D. cauſa Globus terreus aquis undique coopertus, <lb></lb> <arrow.to.target n="note379"></arrow.to.target>ſi rarior eſſet quam aqua, emergeret alicubi, & aqua omnis inde <lb></lb> defluens congregaretur in regione oppoſita. </s> <s>Et par eſt ratio <lb></lb> Terræ noſtræ maribus magna ex parte circumdatæ. Hæc ſi den<lb></lb>ſior non eſſet, emergeret ex maribus, & parte ſui pro gradu levi<lb></lb>tatis extaret ex Aqua, maribus omnibus in regionem oppoſitam <lb></lb> confluentibus. </s> <s>Eodem argumento maculæ Solares leviores ſunt. <lb></lb> quam materia lucida Solaris cui ſupernatant. </s> <s>Et in formatione <lb></lb> qualicunque Planetarum, materia omnis gravior, quo tempore <lb></lb> maſſa tota fluida erat, centrum petebat. </s> <s>Unde cum Terra com<lb></lb>munis ſuprema quaſi duplo gravior ſit quam aqua, & paulo infe<lb></lb>rius in fodinis quaſi triplo vel quadruplo aut etiam quintuplo gra<lb></lb>vior reperiatur: veriſimile eſt quod copia materiæ totius in Terra <lb></lb> quaſi quintuplo vel ſextuplo major ſit quam ſi tota ex aqua con<lb></lb>ſtaret; præſertim cum Terram quaſi quintuplo denſiorem eſſe <lb></lb> quam Jovem jam ante oſtenſum ſit. </s> <s>Igitur ſi Jupiter paulo den<lb></lb>ſior ſit quam aqua, hic ſpatio dierum triginta, quibus lon<lb></lb> gitudinem 459 ſemidiametrorum ſuarum deſcribit, amitteret in<lb></lb>Medio ejuſdem denſitatis cum Aere noſtro motus ſui partem fere<lb></lb>decimam. </s> <s>Verum cum reſiſtentia Mediorum minuatur in ratione<lb></lb>ponderis ac denſitatis, ſic ut aqua, quæ partibus 13 2/3 levior eſt <lb></lb> quam argentum vivum, minus reſiſtat in eadem ratione; & aer, <lb></lb> qui partibus 850 levior eſt quam aqua, minus reſiſtat in eadem <lb></lb> ratione: ſi aſcendatur in cœlos ubi pondus Medii, in quo Planetæ <lb></lb> moventur, diminuitur in immenſum, reſiſtentia prope ceſſabit. <lb></lb> HYPOTHESIS I.<lb></lb><emph type="italics"></emph>Centrum Syſtematis Mundani quieſcere.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note379"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s>Hoc ab omnibus conceſſum eſt, dum aliqui Terram alii Solem <lb></lb> in centro Syſtematis quieſcere contendant. </s> <s>Videamus quid inde <lb></lb> ſequatur. <lb></lb> PROPOSITIO XI. THEOREMA XI.<lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Commune centrum gravitatis Terræ, Solis & Planetarum om<lb></lb>nium quieſcere.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam centrum illud (per Legum Corol. 4.) vel quieſcet vel <lb></lb> progredietur uniformiter in directum. </s> <s>Sed centro illo ſemper <lb></lb> <pb xlink:href="039/01/402.jpg" pagenum="374"></pb><lb></lb><arrow.to.target n="note380"></arrow.to.target>progrediente, centrum Mundi quoque movebitur contra Hy<lb></lb>potheſin. <lb></lb> PROPOSITIO XII. THEOREMA XII.<lb></lb></s></p> <p type="margin"> <s><margin.target id="note380"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Solem motu perpetuo agitari, ſed nunquam longe recedere a com<lb></lb>muni gravitatis centro Planetarum omnium.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Nam cum (per Corol. 2. Prop. VIII.) materia in Sole ſit ad <lb></lb> materiam in Jove ut 1033 ad 1, & diſtantia Jovis a Sole ſit ad <lb></lb> ſemidiametrum Solis in ratione paulo majore; incidet commune <lb></lb> centrum gravitatis Jovis & Solis in punctum paulo ſupra ſuper<lb></lb>ficiem Solis. </s> <s>Eodem argumento cum materia in Sole ſit ad ma<lb></lb>teriam in Saturno ut 2411 ad 1, & diſtantia Saturni a Sole ſit ad <lb></lb> ſemidiametrum Solis in ratione paulo minore: incidet commune <lb></lb> centrum gravitatis Saturni & Solis in punctum paulo infra ſuper<lb></lb>ficiem Solis. </s> <s>Et ejuſdem calculi veſtigiis inſiſtendo ſi Terra & <lb></lb> Planetæ omnes ex una Solis parte conſiſterent, commune omnium <lb></lb> centrum gravitatis vix integra Solis diametro a centro Solis di<lb></lb>ſtaret. </s> <s>Aliis in caſibus diſtantia centrorum ſemper minor eſt. <lb></lb> Et propterea cum centrum illud gravitatis perpetuo quieſcit, Sol <lb></lb> pro vario Planetarum ſitu in omnes partes movebitur, ſed à cen<lb></lb>tro illo nunquam longe recedet. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc commune gravitatis centrum Terræ, Solis & Pla<lb></lb>netarum omnium pro centro Mundi habendum eſt. </s> <s>Nam cum <lb></lb> Terra, Sol & Planetæ omnes gravitent in ſe mutuo, & propte<lb></lb>rea, pro vi gravitatis ſuæ, ſecundum leges motus perpetuo agi<lb></lb>tentur: perſpicuum eſt quod horum centra mobilia pro Mundi <lb></lb> centro quieſcente haberi nequeunt. </s> <s>Si corpus illud in centro <lb></lb> locandum eſſet in quod corpora omnia maxime gravitant (uti <lb></lb> vulgi eſt opinio) privilegium iſtud concedendum eſſet Soli. <lb></lb> Cum autem Sol moveatur, eligendum erit punctum quieſcens, <lb></lb> a quo centrum Solis quam minime diſcedit, & a quo idem ad<lb></lb>huc minus diſcederet, ſi modo Sol denſior eſſet & major, ut <lb></lb> minus moveretur. <lb></lb> <pb xlink:href="039/01/403.jpg" pagenum="375"></pb><lb></lb><arrow.to.target n="note381"></arrow.to.target>PROPOSITIO XIII. THEOREMA XIII.<lb></lb></s></p> <p type="margin"> <s><margin.target id="note381"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Planetæ moventur in Ellipfibus umbilicum habentibus in centro <lb></lb> Solis, & radiis ad centrum illud ductis areas deſcribunt <lb></lb> temporibus proportionales.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Diſputavimus ſupra de his motibus ex Phænomenis. </s> <s>Jam cog<lb></lb>nitis motuum principiis, ex his colligimus motus cœleſtes a pri<lb></lb>ori. </s> <s>Quoniam pondera Planetarum in Solem ſunt reciproce ut <lb></lb> quadrata diſtantiarum a centro Solis; ſi Sol quieſceret & Planetæ <lb></lb> reliqui non agerent in ſe mutuo, forent orbes eorum Elliptici, <lb></lb> Solem in umbilico communi habentes, & areæ deſcriberentur tem<lb></lb>poribus proportionales (per Prop. I. & XI, & Corol. I. Prop. <lb></lb> XIII Lib. I.) Actiones autem Planetarum in ſe mutuo perexiguæ <lb></lb> ſunt (ut poſſint contemni) & motus Planetarum in Ellipſibus <lb></lb> circa Solem mobilem minus perturbant (per Prop. LXVI. Lib. I.) <lb></lb> quam ſi motus iſti circa Solem quieſcentem peragerentur. <lb></lb> </s></p> <p type="main"> <s>Actio quidem Jovis in Saturnum non eſt omnino contemnenda. <lb></lb> Nam gravitas in Jovem eſt ad gravitatem in Solem (paribus di<lb></lb>ſtantiis) ut 1 ad 1033; adeoQ.E.I. conjunctione Jovis & Saturni, <lb></lb> quoniam diſtantia Saturni a Jove eſt ad diſtantiam Saturni a Sole <lb></lb> fere ut 4 ad 9, erit gravitas Saturni in Jovem ad gravitatem Sa<lb></lb>turni in Solem ut 81 ad 16X1033 ſeu 1 ad 204 circiter. </s> <s>Et <lb></lb> hinc oritur perturbatio orbis Saturni in ſingulis Planetæ hujus <lb></lb> cum Jove conjunctionibus adeo ſenſibilis ut ad eandem Aſtronomi <lb></lb> hæreant. </s> <s>Pro vario ſitu Planetæ in his conjunctionibus, Eccen<lb></lb>tricitas ejus nunc augetur nunc diminuitur, Aphelium nunc pro<lb></lb>movetur nunc forte retrahitur, & medius motus per vices accele<lb></lb>ratur & retardatur. </s> <s>Error tamen omnis in motu ejus circum So<lb></lb>lem a tanta vi oriundus (præterquam in motu medio) evitari fere <lb></lb> poteſt conſtituendo umbilicum inferiorem Orbis ejus in communi <lb></lb> centro gravitatis Jovis & Solis (per Prop. LXVII. Lib. I.) & prop<lb></lb>terea ubi maximus eſt, vix ſuperat minuta duo prima. Et error <lb></lb> maximus in motu medio vix ſuperat minuta duo prima annuatim. <lb></lb> In conjunctione autem Jovis & Saturni gravitates acceleratrices <lb></lb> Solis in Saturnum, Jovis in Saturnum & Jovis in Solem ſunt fere <lb></lb> ut 16, 81 & (16X81X2411/25) ſeu 124986, adeoQ.E.D.fferentia gravi<lb></lb>tatum Solis in Saturnum & Jovis in Saturnum eſt ad gravitatem <lb></lb> <pb xlink:href="039/01/404.jpg" pagenum="376"></pb><lb></lb><arrow.to.target n="note382"></arrow.to.target>Jovis in Solem ut 65 ad 124986 ſeu 1 ad 1923. Huic autem dif<lb></lb>ferentiæ proportionalis eſt maxima Saturni efficacia ad perturban<lb></lb>dum motum Jovis, & propterea perturbatio orbis Jovialis longe <lb></lb> minor eſt quam ea Saturnii. </s> <s>Reliquorum orbium perturbationes <lb></lb> ſunt adhuc longe minores, præterquam quod Orbis Terræ ſenſi<lb></lb>biliter perturbatur a Luna. </s> <s>Commune centrum gravitatis Terræ <lb></lb> & Lunæ, Ellipſin circum Solem in umbilico poſitum percurrit, & <lb></lb> radio ad Solem ducto areas in eadem temporibus proportionales <lb></lb> deſcribit, Terra vero circum hoc centrum commune motu men<lb></lb>ſtruo revolvitur. <lb></lb> PROPOSITIO XIV. THEOREMA XIV.<lb></lb><emph type="italics"></emph>Orbium Aphelia & Nodi quieſcunt.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note382"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s>Aphelia quieſcunt, per Prop. XI. Lib. I. ut & Orbium plana, <lb></lb> per ejuſdem Libri Prop. 1. & quieſcentibus planis quieſcunt Nodi. <lb></lb> Attamen a Planetarum revolventium & Cometarum actionibus in <lb></lb> ſe invicem orientur inæqualitates aliquæ, ſed quæ ob parvitatem <lb></lb> hic contemni poſſunt. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Quieſcunt etiam Stellæ fixæ, propterea quod datas ad <lb></lb> Aphelia Nodoſque poſitiones ſervant. <lb></lb> </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Ideoque cum nulla ſit earum parallaxis ſenſibilis ex <lb></lb> Terræ motu annuo oriunda, vires earum ob immenſam corporum <lb></lb> diſtantiam nullos edent ſenſibiles effectus in regione Syſtematis <lb></lb> noſtri. </s> <s>Quinimo Fixæ in omnes cæli partes æqualiter diſperſæ <lb></lb> contrariis attractionibus vires mutuas deſtruunt, per Prop. LXX. <lb></lb> Lib. I. <lb></lb> <emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Cum Planetæ Soli propiores (nempe Mercurius, Venus, Terra, <lb></lb> & Mars) ob corporum parvitatem parum agant in ſe invicem: <lb></lb> horum Aphelia & Nodi quieſcent, niſi quatenus a viribus Jovis, <lb></lb> Saturni, & corporum ſuperiorum turbentur. </s> <s>Et inde colligi po<lb></lb>teſt per theoriam gravitatis, quod horum Aphelia moventur ali<lb></lb>quantulum in conſequentia reſpectu fixarum, idQ.E.I. proporti<lb></lb>one ſeſquiplicata diſtantiarum horum Planetarum a Sole. </s> <s>Ut ſi <lb></lb> Aphelium Martis in annis centum conficiat 35′ in conſequentia <lb></lb> reſpectu fixarum; Aphelia Terræ, Veneris, & Mercurii in annis <lb></lb> centum conficient 18′. 36″, 11′. 27″, & 4′. 29″ reſpective. </s> <s>Et hi <lb></lb> motus, ob parvitatem, negliguntur in hac Propoſitione. <lb></lb> <pb xlink:href="039/01/405.jpg" pagenum="377"></pb><lb></lb><arrow.to.target n="note383"></arrow.to.target>PROPOSITIO XV. PROBLEMA I.<lb></lb><emph type="italics"></emph>Invenire Orbium principales diametros.<emph.end type="italics"></emph.end><lb></lb></s> </p> <p type="margin"> <s><margin.target id="note383"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s>Capiendæ ſunt hæ in ratione ſubſeſquiplicata temporum perio<lb></lb>dieorum, per Prop. XV. Lib. I. deinde ſigillatim augendæ in rati<lb></lb>one ſummæ maſſarum Solis & Planetæ cujuſque revolventis ad <lb></lb> primam duarum medie proportionalium inter ſummam illam & <lb></lb> Solem, per Prop. LX. Lib. I. <lb></lb> PROPOSITIO XVI. PROBLEMA II.<lb></lb><emph type="italics"></emph>Invenire Orbium Eccentricitates & Aphelia.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Problema confit per Prop. XVIII. Lib. I. <lb></lb> PROPOSITIO XVII. THEOREMA XV.<lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Planetarum motus diurnos uniformes eſſe, & librationem Lunæ <lb></lb> ex ipſius motu diurno oriri.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Patet per motus Legem I, & Corol. 22. Prop. LXVI. Lib. I. <lb></lb> Quoniam vero Lunæ, circa axem ſuum uniformiter revolventis, <lb></lb> dies menſtruus eſt; hujus facies eadem ulteriorem umbilicum or<lb></lb>bis ipſius ſemper reſpiciet, & propterea pro ſitu umbilici illius <lb></lb> deviabit hinc inde a Terra. </s> <s>Hæc eſt libratio in longitudinem. <lb></lb> Nam libratio in latitudinem orta eſt ex inclinatione axis Lunaris <lb></lb> ad planum orbis. </s> <s>Porro hæc ita ſe habere, ex Phænomenis mani<lb></lb>feſtum eſt. <lb></lb> PROPOSITIO XVIII. THEOREMA XVI.<lb></lb></s></p> <p type="main"> <s><emph type="italics"></emph>Axes Planetarum diametris quæ ad eoſdem axes normaliter du<lb></lb>cuntur minores eſſe.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Planetæ ſublato omni motu circulari diurno figuram Sphæricam, <lb></lb> ob æqualem undique partium gravitatem, affectare deberent. </s> <s>Per <lb></lb> motum illum circularem fit ut partes ab axe recedentes juxta <lb></lb> æquatorem aſcendere conentur. </s> <s>Ideoque materia ſi fluida ſit <lb></lb> <pb xlink:href="039/01/406.jpg" pagenum="378"></pb><lb></lb><arrow.to.target n="note384"></arrow.to.target>aſcenſu ſuo ad æquatorem diametros adaugebit, axem vero de<lb></lb>ſcenſu ſuo ad polos diminuet. </s> <s>Sic Jovis diameter (conſentienti<lb></lb>bus Aſtronomorum obſervationibus) brevior deprehenditur inter <lb></lb> polos quam ab oriente in occidentem. </s> <s>Eodem argumento, niſi <lb></lb> Terra noſtra paulo altior eſſet ſub æquatore quam ad polos, Ma<lb></lb>ria ad polos ſubſiderent, & juxta æquatorem aſcendendo, ibi om<lb></lb>nia inundarent. <lb></lb> PROPOSITIO XIX. PROBLEMA III.<lb></lb><emph type="italics"></emph>Invenire proportionem axis Planetæ ad diametros eidem <lb></lb> perpendiculares.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="margin"> <s><margin.target id="note384"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Picartus<emph.end type="italics"></emph.end>menſurando arcum gradus unius & 22′. 55″ inter <lb></lb> <emph type="italics"></emph>Ambianum<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Malvoiſinam,<emph.end type="italics"></emph.end>invenit arcum gradus unius eſſe hexa<lb></lb>pedarum Pariſienſium 57060. Unde ambitus Terræ eſt pedum <lb></lb> Pariſienſium 123249600, ut ſupra. </s> <s>Sed cum error quadringente<lb></lb>ſimæ partis digiti, tam in fabrica inſtrumentorum quam in ap<lb></lb>plicatione eorum ad obſervationes capiendas, ſit inſenſibilis, & <lb></lb> in Sectore decempedali quo <emph type="italics"></emph>Galli<emph.end type="italics"></emph.end>obſervarunt Latitudines loco<lb></lb>rum reſpondeat minutis quatuor ſecundis, & in ſingulis obſerva<lb></lb>tionibus incidere poſſit tam ad centrum Sectoris quam ad ejus <lb></lb> circumferentiam, & errores in minoribus ar<lb></lb>cubus ſint majoris momenti:<arrow.to.target n="note385"></arrow.to.target> ideo <emph type="italics"></emph>Caſſinus<emph.end type="italics"></emph.end><lb></lb>juſſu Regio menſuram Terræ per majora loco<lb></lb>rum intervalla aggreſſus eſt, & ſubinde per <lb></lb> diſtantiam inter Obſervatorium Regium <emph type="italics"></emph>Pariſienſe<emph.end type="italics"></emph.end>& villam <emph type="italics"></emph>Coli<lb></lb>oure<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Rouſſillon<emph.end type="italics"></emph.end>& Latitudinum differentiam 6<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 18′, ſuppo<lb></lb>nendo quod figura Terræ ſit Sphærica, invenit gradum unum eſſe <lb></lb> hexapedarum 57292, prope ut <emph type="italics"></emph>Norwoodus<emph.end type="italics"></emph.end>noſter antea invenerat. <lb></lb> Hic enim circa annum 1635, menſurando diſtantiam pedum Lon<lb></lb>dinenſium 905751 inter <emph type="italics"></emph>Londinum<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Eboracum,<emph.end type="italics"></emph.end>& obſervando <lb></lb> differentiam Latitudinum 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 28′, collegit menſuram gradus unius <lb></lb> eſſe pedum Londinenſium 367196, id eſt, hexapedarum Pariſien<lb></lb>ſium 57300. Ob magnitudinem intervalli a <emph type="italics"></emph>Caſſino<emph.end type="italics"></emph.end>monſurati, pro <lb></lb> menſura gradus unius in medio intervalli illius, id eſt, inter La<lb></lb>titudines 45<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> & 46<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> uſurpabo hexapedas 57292. Unde, ſi <lb></lb> Terra ſit Sphærica, ſemidiameter ejus erit pedum Pariſienſium <lb></lb> 19695539. <lb></lb> <pb xlink:href="039/01/407.jpg" pagenum="379"></pb><lb></lb></s></p> <p type="foot"> <s><foot.target id="note385"></foot.target>Vide Hiſtoriam Aca<lb></lb>demiæ Regiæ ſcientiarum <lb></lb> anno 1700.</s></p> <p type="main"> <s>Penduli in Latitudine <emph type="italics"></emph>Lutetiæ Pariſiorum<emph.end type="italics"></emph.end>ad minuta ſecunda <lb></lb> <arrow.to.target n="note386"></arrow.to.target>oſcillantis longitudo eſt pedum trium Pariſienſium & linearum 8 5/9. <lb></lb> Et longitudo quod grave tempore minuti unius ſecundi cadendo <lb></lb> deſcribit, eſt ad dimidiam longitudinem penduli hujus, in duplicata <lb></lb> ratione circumferentiæ circuli ad diametrum ejus (ut indicavit <lb></lb> <emph type="italics"></emph>Hugenius<emph.end type="italics"></emph.end>) ideoque eſt pedum Pariſienſium 15, dig. 1, lin. (2 1/189), ſeu <lb></lb> linearum (2174 1/18). <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note386"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s>Corpus in circulo, ad diſtantiam pedum 19695539 a centro, <lb></lb> ſingulis diebus ſidereis horarum 23. 56′. 4″ uniformiter revolvens, <lb></lb> tempore minuti unius ſecundi deſcribit arcum pedum 1436,223, <lb></lb> cujus ſinus verſus eſt pedum 0,05236558, ſeu linearum 7,54064. <lb></lb> Ideoque vis qua gravia deſcendunt in Latitudine <emph type="italics"></emph>Lutetiæ,<emph.end type="italics"></emph.end>eſt ad <lb></lb> vim centrifugam corporum &c. in Æquatore, a Terræ motu diurno <lb></lb> oriundam, ut (2174 1/18) ad 7,54064. <lb></lb> </s></p> <p type="main"> <s>Vis centrifuga corporum in Æquatore, eſt ad vim centrifugam <lb></lb> qua corpora directe tendunt a Terra in Latitudine <emph type="italics"></emph>Lutetiæ<emph.end type="italics"></emph.end>gra<lb></lb>duum 48. 50′, in duplicata ratione Radii ad ſinum complementi <lb></lb> Latitudinis illius, id eſt, ut 7,54064 ad 3,267. Addatur hæc vis <lb></lb> ad vim qua gravia deſcendunt in Latitudine <emph type="italics"></emph>Lutetiæ,<emph.end type="italics"></emph.end>& corpus <lb></lb> in Latitudine <emph type="italics"></emph>Lutetiæ<emph.end type="italics"></emph.end>vi tota gravitatis cadendo, tempore minuti <lb></lb> unius ſecundi deſcriberet lineas 2177,32, ſeu pedes Pariſienſes 15, <lb></lb> dig. 1, & lin. 5,32. Et vis tota gravitatis in Latitudine illa, erit <lb></lb> ad vim centriſugam corporum &c. in Æquatore Terræ, ut 2177,32 <lb></lb> ad 7,54064, ſeu 289 ad 1. <lb></lb> </s></p> <p type="main"> <s>Unde ſi <emph type="italics"></emph>APBQ<emph.end type="italics"></emph.end>figuram Terræ deſignet jam non amplius <lb></lb> Sphæricam ſed revolutione Ellipſeos circum axem minorem <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end><lb></lb>genitam, ſitque <emph type="italics"></emph>ACQqca<emph.end type="italics"></emph.end>canalis aquæ ple<lb></lb><figure id="id.039.01.407.1.jpg" xlink:href="039/01/407/1.jpg"></figure><lb></lb>na, a polo <emph type="italics"></emph>Qq<emph.end type="italics"></emph.end>ad centrum <emph type="italics"></emph>Cc,<emph.end type="italics"></emph.end>& inde ad <lb></lb> Æquatorem <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>pergens: debebit pondus <lb></lb> aquæ in canalis crure <emph type="italics"></emph>ACca,<emph.end type="italics"></emph.end>eſſe ad pondus <lb></lb> aquæ in crure altero <emph type="italics"></emph>QCcq<emph.end type="italics"></emph.end>ut 289 ad 288, <lb></lb> eo quod vis centrifuga ex circulari motu <lb></lb> orta partem unam e ponderis partibus 289 <lb></lb> ſuſtinebit ac detrahet, & pondus 288 in al<lb></lb>tero crure ſuſtinebit reliquas. </s> <s>Porro (ex <lb></lb> Propoſitionis XCI. Corollario ſecundo, Lib.I.) <lb></lb> computationem ineundo, invenio quod ſi Terra conſtaret ex uni<lb></lb>formi materia, motuque omni privaretur, & eſſet ejus axis <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end><lb></lb><pb xlink:href="039/01/408.jpg" pagenum="380"></pb><lb></lb><arrow.to.target n="note387"></arrow.to.target>ad diametrum <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ut 100 ad 101: gravitas in loco <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>in Terram, <lb></lb> foret ad gravitatem in eodem loco <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>in Sphæram centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>radio <lb></lb> <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>QC<emph.end type="italics"></emph.end>deſcriptam, ut 126 ad 125. Et eodem argumento <lb></lb> gravitas in loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in Sphæroidem, convolutione Ellipſeos <emph type="italics"></emph>APBQ<emph.end type="italics"></emph.end><lb></lb>circa axem <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>deſcriptam, eſt ad gravitatem in eodem loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in <lb></lb> Sphæram centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>deſcriptam, ut 125 ad 126. Eſt au<lb></lb>tem gravitas in loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in Terram, media proportionalis inter <lb></lb> gravitates in dictam Sphæroidem & Sphæram: propterea quod <lb></lb> Sphæra, diminuendo diametrum <emph type="italics"></emph>PQ<emph.end type="italics"></emph.end>in ratione 101 ad 100, <lb></lb> vertitur in figuram Terræ; & hæc figura diminuendo in eadem <lb></lb> ratione diametrum tertiam, quæ diametris duabus <emph type="italics"></emph>AB, PQ<emph.end type="italics"></emph.end>per<lb></lb>pendicularis eſt, vertitur in dictam Sphæroidem; & gravitas in <lb></lb> <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>in caſu utroque, diminuitur in eadem ratione quam proxime. <lb></lb> Eſt igitur gravitas in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in Sphæram centro <lb></lb> <figure id="id.039.01.408.1.jpg" xlink:href="039/01/408/1.jpg"></figure><lb></lb><emph type="italics"></emph>C<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>deſcriptam, ad gravitatem in <lb></lb> <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in Terram ut 126 ad 125 1/2, & gravitas <lb></lb> in loco <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>in Sphæram centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>QC<emph.end type="italics"></emph.end><lb></lb>deſcriptam, eſt ad gravitatem in loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in <lb></lb> Sphæram centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>radio <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>deſcriptam, <lb></lb> in ratione diametrorum (per Prop. LXXII. <lb></lb> Lib. I.) id eſt, ut 100 ad 101. Conjungan<lb></lb>tur jam hæ tres rationes, 126 ad 125, 126 <lb></lb> ad 125 1/2, & 100 ad 101: & fiet gravitas <lb></lb> in loco <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>in Terram, ad gravitatem in loco <emph type="italics"></emph>A<emph.end type="italics"></emph.end>in Terram, ut <lb></lb> 126X126X100 ad 125X125 1/2X101, ſeu ut 501 ad 500. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note387"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s>Jam cum (per Corol. 3. Prop. XCI. Lib. I.) gravitas in canalis <lb></lb> crure utrovis <emph type="italics"></emph>ACca<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>QCcq<emph.end type="italics"></emph.end>ſit ut diſtantia loeorum a centro <lb></lb> Terræ; ſi crura illa ſuperficiebus tranſverſis & æquidiſtantibus di<lb></lb>ſtinguantur in partes totis proportionales, erunt pondera partium <lb></lb> ſingularum in crure <emph type="italics"></emph>ACca<emph.end type="italics"></emph.end>ad pondera partium totidem in crure <lb></lb> altero, ut magnitudines & gravitates acceleratrices conjunctim; id <lb></lb> eſt, ut 101 ad 100 & 500 ad 501, hoc eſt, ut 505 ad 501. Ac <lb></lb> proinde ſi vis centrifuga partis cujuſQ.E.I. crure <emph type="italics"></emph>ACca<emph.end type="italics"></emph.end>ex motu <lb></lb> diurno oriunda, fuiſſet ad pondus partis ejuſdem ut 4 ad 505, eo <lb></lb> ut de pondere partis cujuſque, in partes 505 diviſo, partes qua<lb></lb>tuor detraheret; manerent pondera in utroque crure æqualia, & <lb></lb> propterea fluidum conſiſteret in æquilibrio. </s> <s>Verum vis centrifuga <lb></lb> partis cujuſque eſt ad pondus ejuſdem ut 1 ad 289, hoc eſt, vis <lb></lb> centrifuga quæ deberet eſſe ponderis pars (4/505) eſt tantum pars (1/289). <lb></lb> <pb xlink:href="039/01/409.jpg" pagenum="381"></pb><lb></lb>Et propterea dico, ſecundum Regulam auream, quod ſi vis cen<lb></lb><arrow.to.target n="note388"></arrow.to.target>trifuga (4/505) faciat ut altitudo aquæ in crure <emph type="italics"></emph>ACca<emph.end type="italics"></emph.end>ſuperet altitu<lb></lb>dinem aquæ in crure <emph type="italics"></emph>QCcq<emph.end type="italics"></emph.end>parte centeſima totius altitudinis: <lb></lb> vis centrifuga (1/289) faciet ut exceſſus altitudinis in crure <emph type="italics"></emph>ACca<emph.end type="italics"></emph.end>ſit <lb></lb> altitudinis in crure altero <emph type="italics"></emph>QCcq<emph.end type="italics"></emph.end>pars tantum (1/229). Eſt igitur dia<lb></lb>meter Terræ ſecundum æquatorem ad ipſius diametrum per polos <lb></lb> ut 230 ad 229. Ideoque cum Terræ ſemidiameter mediocris, juxta <lb></lb> menſuram <emph type="italics"></emph>Caſſini,<emph.end type="italics"></emph.end>ſit. pedum Pariſienſium 19695539, ſeu milliarium <lb></lb> 3939 (poſito quod milliare ſit menſura pedum 5000) Terra altior <lb></lb> erit ad Æquatorem quam ad Polos exceſſu pedum 85820, ſeu <lb></lb> milliarum 17 1/6. <lb></lb> </s></p> <p type="margin"> <s><margin.target id="note388"></margin.target>LIBER <lb></lb> TERTIUS.</s></p> <p type="main"> <s>Si Planeta major ſit vel minor quam Terra manente ejus den<lb></lb>ſitate ac tempore periodico revolutionis diurnæ, manebit pro<lb></lb>portio vis centrifugæ ad gravitatem, & propterea manebit etiam <lb></lb> proportio diametri inter polos ad diametrum ſecundum æquato<lb></lb>rem. </s> <s>At ſi motus diurnus in ratione quacunque acceleretur vel <lb></lb> retardetur, augebitur vel minuetur vis centrifuga in duplicata illa <lb></lb> ratione, & propterea differentia diametrorum augebitur vel mi<lb></lb>nuetur in eadem duplicata ratione quamproxime. </s> <s>Et ſi denſitas <lb></lb> Planetæ augeatur vel minuatur in ratione quavis, gravitas etiam <lb></lb> in ipſum tendens augebitur vel minuetur in eadem ratione, & <lb></lb> differentia diametrorum viciſſim minuetur in ratione gravitatis <lb></lb> auctæ vel augebitur in ratione gravitatis diminutæ. Unde cum <lb></lb> Terra reſpectu fixarum revolvatur horis 23. 56′, Jupiter autem <lb></lb> horis 9. 56′, ſintque temporum quadrata ut 29 ad 5, & denſitates <lb></lb> ut 5 ad 1: differentia diametrorum Jovis erit ad ipſius diame<lb></lb>trum minorem ut (29/5)X(5/1)X(1/229) ad 1, ſeu 1 ad 8 quamproxime. </s> <s>Eſt <lb></lb> igitur diameter Jovis ab oriente in occidentem ducta, ad ejus dia<lb></lb>metrum inter polos ut 9 ad 8 quamproxime, & propterea diame<lb></lb>ter inter polos eſt 35 1/2″. Hæc ita ſe habent ex hypotheſi quod <lb></lb> uniformis ſit Planetarum materia. </s> <s>Nam ſi materia denſior ſit ad <lb></lb> centrum quam ad circumferentiam; diameter quæ ab oriente in <lb></lb> occidentem ducitur, erit adhuc major. <lb></lb> </s></p> <p type="main"> <s>Jovis vero diametrum quæ polis ejus interjacet minorem eſſe <lb></lb> diametro altera <emph type="italics"></emph>Caſſinus<emph.end type="italics"></emph.end>dudum obſervavit, & Terræ diametrum <lb></lb> inter polos minorem eſſe diametro altera patebit per ea quæ <lb></lb> dicentur in Propoſitione ſequente. <lb></lb> <pb xlink:href="039/01/410.jpg" pagenum="382"></pb><lb></lb><arrow.to.target n="note389"></arrow.to.target>PROPOSITIO XX. PROBLEMA IV.<lb></lb></s></p> <p type="margin"> <s><margin.target id="note389"></margin.target>DE MUNDI <lb></lb> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Invenire & inter ſe comparare Pondera corporum in Terræ hujus <lb></lb> regionibus diverſis.<emph.end type="italics"></emph.end><lb></lb></s></p> <p type="main"> <s>Quoniam pondera inæqualium crurum canalis aqueæ <emph type="italics"></emph>ACQqca<emph.end type="italics"></emph.end><lb></lb>æqualia ſunt; & pondera partium, cruribus totis proportionalium <lb></lb> & ſimiliter in totis ſitarum, ſunt ad invicem ut pondera totorum, <lb></lb> adeoque etiam æquantur inter ſe; erunt pondera æqualium & in <lb></lb> cruribus ſimiliter ſitarum partium reciproce ut crura, id eſt, reci<lb></lb>proce ut 230 ad 229. Et par eſt ratio homogeneorum & æqua<lb></lb>lium quorumvis & in canalis cruribus ſimiliter ſitorum corporum. <lb></lb> Horum pondera ſunt reciproce ut crura, id eſt, reciproce ut di<lb></lb>ſtantiæ corporum a centro Terræ. Proinde ſi corpora in ſupre<lb></lb>mis canalium partibus, ſive in ſuperficie Terræ conſiſtant; erunt <lb></lb> pondera eorum ad invicem reciproce ut diſtantiæ eorum a centro. <lb></lb> Et eodem argumento pondera, in aliis quibuſcunque per totam <lb></lb> Terræ ſuperficiem regionibus, ſunt reciproce ut diſtantiæ loeorum <lb></lb> a centro; & propterea, ex Hypotheſi quod Terra Sphærois ſit, <lb></lb> dantur proportione. <lb></lb> </s></p> <p type="main"> <s>Unde tale confit Theorema, quod incrementum ponderis per<lb></lb>gendo ab Æquatore ad Polos, ſit quam proxime ut ſinus verſus <lb></lb> Latitudinis duplicatæ, vel, quod perinde eſt, ut quadratum ſinus <lb></lb> recti Latitudinis. </s> <s>Et in eadem circiter ratione augentur arcus <lb></lb> graduum Latitudinis in Meridiano. </s> <s>Ideoque cum Latitudo <emph type="italics"></emph>Lu<lb></lb>tetiæ Pariſiorum<emph.end type="italics"></emph.end>ſit 48<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 50′, ea loeorum ſub Æquatore 00<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 00′, <lb></lb> & ea loeorum ad Polos 90<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> & duplorum ſinus verſi ſint 11334, <lb></lb> 00000 & 20000, exiſtente Radio 10000, & gravitas ad Polum ſit <lb></lb> ad gravitatem ſub Æquatore ut 230 ad 229, & exceſſus gravi<lb></lb>tatis ad Polum ad gravitatem ſub Æquatore ut 1 ad 229: erit ex<lb></lb>ceſſus gravitatis in Latitudine <emph type="italics"></emph>Lutetiæ<emph.end type="italics"></emph.end>ad gravitatem ſub Æquatore, <lb></lb> ut 1X(11334/20000) ad 229, ſeu 5667 ad 2290000. Et propterea gravitates <lb></lb> totæ in his locis erunt ad invicem ut 2295667 ad 2290000. Quare <lb></lb> cum longitudines pendulorum æqualibus temporibus oſcillantium <lb></lb> ſint ut gravitates, & in Latitudine <emph type="italics"></emph>Lutetiæ Pariſiorum<emph.end type="italics"></emph.end>longitudo <lb></lb> penduli ſingulis minutis ſecundis oſcillantis ſit pedum trium Pa<lb></lb>riſienſium & linearum 8 1/9: longitudo penduli ſub Æquatore ſu<lb></lb>perabitur a longitudine ſynchroni penduli <emph type="italics"></emph>Pariſienſis,<emph.end type="italics"></emph.end>exceſſu li<lb></lb>neæ unius & 87 partium milleſimarum lineæ. Et ſimili computo <lb></lb> confit Tabula ſequens. <lb></lb> <pb xlink:href="039/01/411.jpg" pagenum="383"></pb><lb></lb><arrow.to.target n="note390"></arrow.to.target></s> </p><table><row><cell><emph type="italics"></emph>Latitudo <lb></lb> Loci<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Longitudo <lb></lb> Penduli<emph.end type="italics"></emph.end></cell><cell><emph type="italics"></emph>Menſura <lb></lb> Gradus unius <lb></lb> in Meridiano<emph.end type="italics"></emph.end></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>Conſtat autem per hanc Tabulam, quod graduum inæqualitas <lb></lb>tam parva ſit, ut in rebus Geographicis figura Terræ pro Sphæ<lb></lb>rica haberi poſſit, quodQ.E.I.æqualitas diametrorum Terræ faci<lb></lb>lius & certius per experimenta pendulorum deprehendi poſſit vel <lb></lb>etiam per Eclipſes Lunæ, quam per arcus Geographice menſuratos <lb></lb>in Meridiano. <pb xlink:href="039/01/412.jpg" pagenum="384"></pb><arrow.to.target n="note415"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note415"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Hæc ita ſe habent ex hypotheſi quod Terra ex uniformi ma<lb></lb>teria conſtat. </s> <s>Nam ſi materia ad centrum paulo denſior ſit quam <lb></lb>ad ſuperficiem, differentiæ pendulorum & graduum Meridiani <lb></lb>paulo majores erunt quam pro Tabula præcedente, propterea <lb></lb>quod ſi materia ad centrum redundans qua denſitas ibi major <lb></lb>redditur, ſubducatur & ſeorſim ſpectetur, gravitas in Terram re<lb></lb>liquam uniformiter denſam, erit reciproce ut diſtantia ponderis <lb></lb>a centro; in materiam vero redundantem reciproce ut quadratum <lb></lb>diſtantiæ a materia illa quamproxime. </s> <s>Gravitas igitur ſub æqua<lb></lb>tore minor eſt in materiam illam redundantem quam pro com<lb></lb>puto ſuperiore: & propterea Terra ibi, propter defectum gravita<lb></lb>tis, paulo altius aſcendet, & exceſſus longitudinum Pendulorum & <lb></lb>graduum ad polos paulo majores erunt quam in præcedentibus <lb></lb>definitum eſt. </s></p> <p type="main"> <s>Jam vero Aſtronomi aliqui in longinquas regiones ad obſerva<lb></lb>tiones Aſtronomicas faciendas miſſi, invenerunt quod horologia <lb></lb>oſcillatoria tardius moverentur prope Æquatorem quam in regi<lb></lb>onibus noſtris. </s> <s>Et primo quidem <emph type="italics"></emph>D. Richer<emph.end type="italics"></emph.end>hoc obſervavit anno <lb></lb>1672 in inſula <emph type="italics"></emph>Cayennæ.<emph.end type="italics"></emph.end>Nam dum obſervaret tranſitum Fixarum <lb></lb>per meridianum menſe <emph type="italics"></emph>Auguſto,<emph.end type="italics"></emph.end>reperit horologium ſuum tardius <lb></lb>moveri quam pro medio motu Solis, exiſtente differentia 2′. </s> <s>28″ <lb></lb>ſingulis diebus. </s> <s>Deinde faciendo ut Pendulum ſimplex ad minuta <lb></lb>ſingula ſecunda per horologium optimum menſurata oſcillaret, <lb></lb>notavit longitudinem Penduli ſimplicis, & hoc fecit ſæpius ſingu<lb></lb>lis ſeptimanis per menſes decem. </s> <s>Tum in <emph type="italics"></emph>Galliam<emph.end type="italics"></emph.end>redux contulit <lb></lb>longitudinem hujus Penduli cum longitudine Penduli <emph type="italics"></emph>Pariſienſis<emph.end type="italics"></emph.end><lb></lb>(quæ erat trium pedum Pariſienſium, & octo linearum cum tribus <lb></lb>quintis partibus lineæ) & reperit breviorem eſſe, exiſtente diffe<lb></lb>rentia lineæ unius cum quadrante. </s> <s>At ex tarditate horologii <lb></lb>oſcillatorii in <emph type="italics"></emph>Cayenna,<emph.end type="italics"></emph.end>differentia Pendulorum colligitur eſſe lineæ <lb></lb>unius cum ſemiſſe. </s></p> <p type="main"> <s>Poſtea <emph type="italics"></emph>Halleius<emph.end type="italics"></emph.end>noſter circa annum 1677 ad inſulam <emph type="italics"></emph>S<emph type="sup"></emph>sa<emph.end type="sup"></emph.end> Hel<lb></lb>lenæ<emph.end type="italics"></emph.end>navigans, reperit horologium ſuum oſcillatorium ibi tardius <lb></lb>moveri quam <emph type="italics"></emph>Londini,<emph.end type="italics"></emph.end>ſed differentiam non notavit. </s> <s>Pendulum <lb></lb>vero brevius reddidit pluſquam octava parte digiti, ſeu linea una <lb></lb>cum ſemiſſe. </s> <s>Et ad hoc efficiendum, cum longitudo cochleæ in <lb></lb>ima parte penduli non ſufficeret, annulum ligneum thecæ cochleæ <lb></lb>& ponderi pendulo interpoſuit. </s></p> <p type="main"> <s>Deinde anno 1682 <emph type="italics"></emph>D. Varin<emph.end type="italics"></emph.end>& <emph type="italics"></emph>D. </s> <s>Des Hayes<emph.end type="italics"></emph.end>invenerunt lon<lb></lb>gitudinem Penduli ſingulis minutis ſecundis oſcillantis in Obſer-<pb xlink:href="039/01/413.jpg" pagenum="385"></pb>vatorio Regio <emph type="italics"></emph>Pariſienſi<emph.end type="italics"></emph.end>eſſe ped. </s> <s>3. lin. </s> <s>8 1/9. Et in inſula <emph type="italics"></emph>Gorea<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><arrow.to.target n="note416"></arrow.to.target>eadem methodo longitudinem Penduli ſynchroni invenerunt eſſe <lb></lb>ped. </s> <s>3. lin. </s> <s>6 5/9, exiſtente longitudinum differentia lin. </s> <s>2. Et eodem <lb></lb>anno ad inſulas <emph type="italics"></emph>Guadaloupam<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Martinicam<emph.end type="italics"></emph.end>navigantes, invenerunt <lb></lb>longitudinem Penduli ſynchroni in his inſulis eſſe ped. </s> <s>3. lin. </s> <s>6 1/3. </s></p> <p type="margin"> <s><margin.target id="note416"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Poſthac <emph type="italics"></emph>D. Couplet<emph.end type="italics"></emph.end>filius anno 1697 menſe <emph type="italics"></emph>Julio,<emph.end type="italics"></emph.end>horologium <lb></lb>ſuum oſcillatorium ad motum Solis medium in Obſervatorio Regio <lb></lb><emph type="italics"></emph>Pariſienſi<emph.end type="italics"></emph.end>ſic aptavit, ut tempore ſatis longo horologium cum motu <lb></lb>Solis congrueret. </s> <s>Deinde <emph type="italics"></emph>Ulyſſipponem<emph.end type="italics"></emph.end>navigans invenit quod <lb></lb>menſe <emph type="italics"></emph>Novembri<emph.end type="italics"></emph.end>proximo horologium tardius iret quam prius, <lb></lb>exiſtente differentia 2′. </s> <s>13″ in horis 24. Et menſe <emph type="italics"></emph>Martio<emph.end type="italics"></emph.end>ſe<lb></lb>quente <emph type="italics"></emph>Paraibam<emph.end type="italics"></emph.end>navigans invenit ibi horologium ſuum tardius <lb></lb>ire quam <emph type="italics"></emph>Pariſiis,<emph.end type="italics"></emph.end>exiſtente differentia 4′. </s> <s>12″ in horis 24. Et <lb></lb>affirmat Pendulum ad minuta ſecunda oſcillans brevius fuiſſe <emph type="italics"></emph>Ulyſ<lb></lb>ſipponi<emph.end type="italics"></emph.end>lineis 2 1/2 & <emph type="italics"></emph>Paraibæ<emph.end type="italics"></emph.end>lineis 3 2/3 quam <emph type="italics"></emph>Pariſiis.<emph.end type="italics"></emph.end>Rectius po<lb></lb>ſuiſſet differentias eſſe 1 1/3 & 2 5/9. Nam hæ differentiæ differen<lb></lb>tiis temporum 2′. </s> <s>13″, & 4′. </s> <s>12″ reſpondent. </s> <s>Craſſioribus hujus <lb></lb>Obſervationibus minus fidendum eſt. </s></p> <p type="main"> <s>Annis proximis (1699 & 1700) <emph type="italics"></emph>D. </s> <s>Des Hayes<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Americam<emph.end type="italics"></emph.end><lb></lb>denuo navigans, determinavit quod in inſulis <emph type="italics"></emph>Cayennæ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Granadæ<emph.end type="italics"></emph.end><lb></lb>longitudo Penduli ad minuta ſecunda oſcillantis, eſſet paulo minor <lb></lb>quam ped. </s> <s>3. lin. </s> <s>6 1/2, quodQ.E.I. inſula <emph type="italics"></emph>S. Chriſtophori<emph.end type="italics"></emph.end>longitudo <lb></lb>illa eſſet ped. </s> <s>3. lin. </s> <s>6 1/4, & quod in inſula <emph type="italics"></emph>S. Dominici<emph.end type="italics"></emph.end>eadem eſſet <lb></lb>ped. </s> <s>3. lin. </s> <s>7. </s></p> <p type="main"> <s>Annoque 1704. <emph type="italics"></emph>P. Feuelleus<emph.end type="italics"></emph.end>invenit in <emph type="italics"></emph>Porto-belo<emph.end type="italics"></emph.end>in <emph type="italics"></emph>America<emph.end type="italics"></emph.end><lb></lb>longitudinem Penduli ad minuta ſecunda oſcillantis, eſſe pedum <lb></lb>trium Pariſienſium & linearum tantum (5 7/12), id eſt, tribus fere li<lb></lb>neis breviorem quam <emph type="italics"></emph>Lutetiæ Pariſiorum,<emph.end type="italics"></emph.end>ſed errante Obſerva<lb></lb>tione. </s> <s>Nam deinde ad inſulam <emph type="italics"></emph>Martinicam<emph.end type="italics"></emph.end>navigans, invenit lon<lb></lb>gitudinem Penduli iſochroni eſſe pedum tantum trium Pariſien<lb></lb>ſium & linearum (5 10/12). </s></p> <p type="main"> <s>Latitudo autem <emph type="italics"></emph>Paraibæ<emph.end type="italics"></emph.end>eſt 6<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 38′ ad auſtrum, & ea <emph type="italics"></emph>Porto<lb></lb>beli<emph.end type="italics"></emph.end>9<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 33′ ad boream, & Latitudines inſularum <emph type="italics"></emph>Cayennæ, Goreæ, <lb></lb>Guadaloupæ, Martinicæ, Granadæ, S<emph type="sup"></emph>ti.<emph.end type="sup"></emph.end> Chriſtophori,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>S<emph type="sup"></emph>ti.<emph.end type="sup"></emph.end> Domi<lb></lb>nici<emph.end type="italics"></emph.end>ſunt reſpective 4<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 55′, 14<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 40′, 14<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 00′, 14<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 44′, 12<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 6′, <lb></lb>17<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 19′, & 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 48′ ad boream. </s> <s>Et exceſſus longitudinis Pen<lb></lb>duli <emph type="italics"></emph>Pariſienſis<emph.end type="italics"></emph.end>ſupra longitudines Pendulorum iſochronorum in <lb></lb>his latitudinibus obſervatas, ſunt paulo majores quam pro Ta<lb></lb>bula longitudinum Penduli ſuperius computata. </s> <s>Et propterea <lb></lb>Terra aliquanto altior eſt ſub Æquatore quam pro ſuperiore cal-<pb xlink:href="039/01/414.jpg" pagenum="386"></pb><arrow.to.target n="note417"></arrow.to.target>culo, & denſior ad centrum quam in fodinis prope ſuperficiem, <lb></lb>niſi forte calores in Zona torrida longitudinem Pendulorum ali<lb></lb>quantulum auxerint. </s></p> <p type="margin"> <s><margin.target id="note417"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Obſervavit utique <emph type="italics"></emph>D. Picartus<emph.end type="italics"></emph.end>quod virga ferrea, quæ tempore <lb></lb>hyberno ubi gelabant frigora erat pedis unius longitudine, ad <lb></lb>ignem calefacta evaſit pedis unius cum quarta parte lineæ. </s> <s>De<lb></lb>inde <emph type="italics"></emph>D. de la Hire<emph.end type="italics"></emph.end>obſervavit quod virga ferrea quæ tempore <lb></lb>conſimili hyberno ſex erat pedum longitudinis, ubi Soli æſtivo <lb></lb>exponebatur evaſit ſex pedum longitudinis cum duabus tertiis <lb></lb>partibus lineæ. </s> <s>In priore caſu calor major fuit quam in poſte<lb></lb>riore, in hoc vero major fuit quam calor externarum partium <lb></lb>corporis humani. </s> <s>Nam metalla ad Solem æſtivum valde incale<lb></lb>ſcunt. </s> <s>At virga penduli in horologio oſcillatorio nunquam ex<lb></lb>poni ſolet calori Solis æſtivi, nunquam calorem concipit calori <lb></lb>externæ ſuperficiei corporis humani æqualem. </s> <s>Et propterea virga <lb></lb>Penduli in horologio tres pedes longa, paulo quidem longior <lb></lb>erit tempore æſtivo quam hyberno, ſed exceſſu quartam partem <lb></lb>lineæ unius vix ſuperante. </s> <s>Proinde differentia tota longitudinis <lb></lb>pendulorum quæ in diverſis regionibus iſochrona ſunt, diverſo <lb></lb>calori attribui non poteſt. </s> <s>Sed neque erroribus Aſtronomorum è <lb></lb><emph type="italics"></emph>Gallia<emph.end type="italics"></emph.end>miſſorum tribuenda eſt hæc differentia. </s> <s>Nam quamvis <lb></lb>eorum obſervationes non perfecte congruant inter ſe, tamen erro<lb></lb>res ſunt adeo parvi ut contemni poſſint. </s> <s>Et in hoc concordant <lb></lb>omnes, quod iſochrona pendula ſunt breviora ſub Æquatore quam <lb></lb>in Obſervatorio Regio <emph type="italics"></emph>Pariſienſi,<emph.end type="italics"></emph.end>exiſtente differentia duarum cir<lb></lb>citer linearum ſeu ſextæ partis digiti. </s> <s>Per obſervationes <emph type="italics"></emph>D. Ri<lb></lb>cher<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Cayenna<emph.end type="italics"></emph.end>factas, differentia fuit lineæ unius cum ſemiſſe. </s> <s><lb></lb>Error ſemiſſis lineæ facile committitur. </s> <s>Et <emph type="italics"></emph>D. des Hayes<emph.end type="italics"></emph.end>poſtea <lb></lb>per obſervationes ſuas in eadem inſula factas errorem correxit, <lb></lb>inventa differentia linearum (2 1/18). Sed & per obſervationes in in<lb></lb>ſulis <emph type="italics"></emph>Gorea, Guadaloupa, Martinica, Granada, S. Chriſtophori,<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>S. Dominici<emph.end type="italics"></emph.end>factas & ad Æquatorem reductas, differentia illa pro<lb></lb>diit haud minor quam (1 19/20) lineæ, haud major quam 2 1/2 linearum. </s> <s><lb></lb>Et inter hos limites quantitas mediocris eſt (2 9/40) linearum. </s> <s>Prop<lb></lb>ter calores loeorum in Zona torrida negligamus (9/40) partes lineæ, <lb></lb>& manebit differentia duarum linearum. </s></p> <p type="main"> <s>Quare cum differentia illa per Tabulam præcedentem, ex hy<lb></lb>potheſi quod Terra ex materia uniformiter denſa conſtat, ſit tan<lb></lb>tum (1 87/1000) lineæ: exceſſus altitudinis Terræ ad æquatorem ſupra <lb></lb>altitudinem ejus ad polos, qui erat milliarium 17 1/6, jam auctus in <pb xlink:href="039/01/415.jpg" pagenum="387"></pb>ratione differentiarum, fiet milliarium (31 7/18). Nam tarditas Pen<lb></lb><arrow.to.target n="note418"></arrow.to.target>duli ſub Æquatore defectum gravitatis arguit; & quo levior eſt <lb></lb>materia eo major eſſe debet altitudo ejus, ut pondere ſuo mate<lb></lb>riam ſub Polis in æquilibrio ſuſtineat. </s></p> <p type="margin"> <s><margin.target id="note418"></margin.target>LIBFR <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Hinc figura umbræ Terræ per Eclipſes Lunæ determinanda, non <lb></lb>erit omnino circularis, ſed diameter ejus ab oriente in occidentem <lb></lb>ducta major erit quam diameter ejus ab auſtro in boream ducta, <lb></lb>exceſſu 55″ circiter. </s> <s>Et parallaxis maxima Lunæ in Longitudi<lb></lb>nem paulo major erit quam ejus parallaxis maxima in Latitudi<lb></lb>nem. </s> <s>Ac Terræ ſemidiameter maxima erit podum Pariſienſium <lb></lb>19767630, minima pedum 19609820 & mediocris pedum 19688725<emph type="sup"></emph>1<emph.end type="sup"></emph.end><lb></lb>quamproxime. </s></p> <p type="main"> <s>Cum gradus unus menſurante <emph type="italics"></emph>Picarto<emph.end type="italics"></emph.end>ſit hexapedarum 57060, <lb></lb>menſurante vero <emph type="italics"></emph>Caſſino<emph.end type="italics"></emph.end>ſit hexapedarum 57292: ſuſpicantur ali<lb></lb>qui gradum unumquemque, pergenda per <emph type="italics"></emph>Gallies<emph.end type="italics"></emph.end>auſtrum verſus <lb></lb>majorem eſſe gradu præcedente hexapedia plus minus: 72, ſeu <lb></lb>parte octingenteſima gradus unius; exiſtente Perra Sphæroide ob<lb></lb>longa cujus partes ad polos ſunt altiſſimæ. </s> <s>Quo poſito, corpora <lb></lb>omnia ad polos Terræ leviora forent quam ad Æquatorem, & <lb></lb>altitudo Terræ ad polos ſuperaret altitudinem ejus ad æquatorem <lb></lb>milliaribus fere 95, & pendula iſochrona longiora forent ad Æ<lb></lb>quatorem quem in Obſervatorio Regio <emph type="italics"></emph>Pariſieuſi<emph.end type="italics"></emph.end>exceſſu ſemiſſis <lb></lb>digiti circiter; ut conſerenti proportiones hic poſitas cum pro<lb></lb>portionibus in Tabula præcedente poſitis, facile conſtabit. </s> <s>Sed <lb></lb>& diameter umbræ Terræ quæ ab auſtro in boream ducitur, ma<lb></lb>jor foret quam diameter ejus quæ ab oriente in occidentem duci<lb></lb>tur, exceſſu 2′. </s> <s>46″, ſeu parte duodecima diametri Lunæ. </s> <s>Qui<lb></lb>bus omnibus Experientia contrariatur. </s> <s>Certe <emph type="italics"></emph>Caſſinus,<emph.end type="italics"></emph.end>definiendo <lb></lb>gradum unum eſſe hexapedarum 57292, medium inter menſuras <lb></lb>ſuas omnes, ex hypotheſi de æqualitate graduum aſſumpſit. </s> <s>Et <lb></lb>quamvis <emph type="italics"></emph>Picartus<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Galliæ<emph.end type="italics"></emph.end>limite boreali invenit gradum paulo <lb></lb>minorem eſſe, tamen <emph type="italics"></emph>Norwoodus<emph.end type="italics"></emph.end>noſter in regionibus magis bore<lb></lb>alibus, menſurando majus intervallum, invenit gradum paulo majo<lb></lb>rem eſſe quam <emph type="italics"></emph>Caſſinus<emph.end type="italics"></emph.end>invenerat. </s> <s>Et <emph type="italics"></emph>Caſſinus<emph.end type="italics"></emph.end>ipſe menſuram <emph type="italics"></emph>Picarti,<emph.end type="italics"></emph.end><lb></lb>ob parvitatem intervalli menſurati, non ſatis certam & exactam eſſe <lb></lb>judicavit ubi menſuram gradus unius per intervallum longe majus <lb></lb>definire aggreſſus eſt. </s> <s>Differentiæ vero inter menſuras <emph type="italics"></emph>Caſſini, Pi<lb></lb>carti,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Norwoodi<emph.end type="italics"></emph.end>ſunt prope inſenſibiles, & ab inſenſibilibus ob<lb></lb>ſervationum erroribus facilo oriri potuere, ut Nutationem axis <lb></lb>Terræ præteream. <pb xlink:href="039/01/416.jpg" pagenum="388"></pb><arrow.to.target n="note419"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note419"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXI. THEOREMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Puncta Æquinoctialia regredi, & axem Terræ ſingulis revoluti<lb></lb>onibus annuis nutando bis inclinari in Eclipticam & bis re<lb></lb>dire ad poſitionem priorem.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Patet per Corol. </s> <s>20. Prop. </s> <s>LXVI. Lib. </s> <s>I. </s> <s>Motus tamen iſte <lb></lb>nutandi perexiguus eſſet debet, & vix aut ne vix quidem ſen<lb></lb>ſibilis. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXII. THEOREMA XVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Motus omnes Lunares, omneſque motuum inæqualitates ex alla<lb></lb>tis Principiis conſequi.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Planetas majores, interea dum circa Solem feruntur, poſſe alios <lb></lb>minores circum ſe revolventes Planetas deferre, & minores illos in <lb></lb>Ellipſibus, umbilicos in centris majorum habentibus, revolvi de<lb></lb>bere patet per Prop. </s> <s>LXV. Lib. </s> <s>I. </s> <s>Actione autem Solis perturba<lb></lb>buntur eorum motus multimode, iiſque adficientur inæqualitati<lb></lb>bus quæ in Luna noſtra notantur. </s> <s>Hæc utique (per Corol. </s> <s>2, <lb></lb>3, 4, & 5. Prop. </s> <s>LXVI.) velocius movetur, ac radio ad Terram <lb></lb>ducto deſcribit aream pro tempore majorem, Orbemque habet <lb></lb>minus curvum, atque adeo propius accedit ad Terram, in Syzygiis <lb></lb>quam in Quadraturis, niſi quatenus impedit motus Eccentricitatis. </s> <s><lb></lb>Eccentricitas enim maxima eſt (per Corol. </s> <s>9. Prop. </s> <s>LXVI.) ubi <lb></lb>Apogæum Lunæ in Syzygiis verſatur, & minima ubi idem in Qua<lb></lb>draturis conſiſtit; & inde Luna in Perigæo velocior eſt & nobis <lb></lb>propior, in Apogæo autem tardior & remotior in Syzygiis quam <lb></lb>in Quadraturis. </s> <s>Progreditur inſuper Apogæum, & regrediuntur <lb></lb>Nodi, ſed motu inæquabili. </s> <s>Et Apogæum quidem (per Corol. </s> <s>7. <lb></lb>& 8. Prop. </s> <s>LXVI.) velocius progreditur in Syzygiis ſuis, tardius <lb></lb>regreditur in Quadraturis, & exceſſu progreſſus ſupra regreſſum <lb></lb>annuatim fertur in conſequentia. </s> <s>Nodi autem (per Corol. </s> <s>11. <lb></lb>Prop. </s> <s>LXVI.) quieſcunt in Syzygiis ſuis, & velociſſime regrediun<lb></lb>tur in Quadraturis. </s> <s>Sed & major eſt Lunæ latitudo maxima in <lb></lb>ipſius Quadraturis (per Corol. </s> <s>10. Prop. </s> <s>LXVI.) quam in Syzy<lb></lb>giis: & motus medius tardior in Perihelio Terræ (per Corol. </s> <s>6. <pb xlink:href="039/01/417.jpg" pagenum="389"></pb>Prop. </s> <s>LXVI,) quam in ipſius Aphelio. </s> <s>Atque hæ ſunt inæquali<lb></lb><arrow.to.target n="note420"></arrow.to.target>tates inſigniores ab Aſtronomis notatæ. </s></p> <p type="margin"> <s><margin.target id="note420"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Sunt etiam aliæ quædam nondum obſervatæ inæqualitates, qui<lb></lb>bus motus Lunares adeo perturbantur, ut nulla hactenus lege ad <lb></lb>Regulam aliquam certam reduci potuerint. </s> <s>Velocitates enim ſeu <lb></lb>motus horarii Apogæi & Nodorum Lunæ, & eorundem æquati<lb></lb>ones, ut & differentia inter Eccentricitatem maximam in Syzygiis <lb></lb>& minimam in Quadraturis, & inæqualitas quæ Variatio dicitur, <lb></lb>augentur ac diminuuntur annuatim (per Corol. </s> <s>14. Prop. </s> <s>LXVI.) <lb></lb>in triplicata ratione diametri apparentis Solaris. </s> <s>Et Variatio præ<lb></lb>terea augetur vel diminuitur in duplicata ratione temporis in<lb></lb>ter quadraturas quam proxime (per Corol. </s> <s>1. & 2. Lem. </s> <s>X. & <lb></lb>Corol. </s> <s>16. Prop. </s> <s>LXVI. Lib. </s> <s>I.) Sed hæc inæqualitas in calculo <lb></lb>Aſtronomico, ad Proſthaphæreſin Lunæ referri ſolet, & cum ea <lb></lb>confundi. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIII. PROBLEMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Motus inæquales Satellitum Jovis & Saturni à motibus Luna<lb></lb>ribus derivare.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Ex motibus Lunæ noſtræ motus analogi Lunarum ſeu Satelli<lb></lb>tum Jovis ſic derivantur. </s> <s>Motus medius Nodorum Satellitis ex<lb></lb>timi Jovialis, eſt ad motum medium Nodorum Lunæ noſtræ, in ra<lb></lb>tione compoſita ex ratione duplicata temporis periodici Terræ <lb></lb>circa Solem ad tempus periodicum Jovis circa Solem, & ratione <lb></lb>ſimplici temporis periodici Satellitis circa Jovem ad tempus perio<lb></lb>dicum Lunæ circa Terram: (per Corol. </s> <s>16. Prop. </s> <s>LXVI.) adeoque <lb></lb>annis centum conficit Nodus iſte 8<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 24′. </s> <s>in antecedentia. </s> <s>Motus <lb></lb>medii Nodorum Satellitum interiorum ſunt ad motum hujus, ut <lb></lb>illorum tempora periodica ad tempus periodicum hujus, per idem <lb></lb>Corollarium, & inde dantur. </s> <s>Motus autem Augis Satellitis cu<lb></lb>juſQ.E.I. conſequentia, eſt ad motum Nodorum ipſius in antece<lb></lb>dentia, ut motus Apogæi Lunæ noſtræ ad hujus motum Nodo<lb></lb>rum, (per idem Corol.) & inde datur. </s> <s>Diminui tamen debet <lb></lb>motus Augis ſic inventus in ratione 5 ad 9 vel 1 ad 2 circiter, ob <lb></lb>cauſam quam hic exponere non vacat. </s> <s>Æquationes maximæ No<lb></lb>dorum & Augis Satellitis cujuſque fere ſunt ad æquationes maxi<lb></lb>mas Nodorum & Augis Lunæ reſpective, ut motus Nodorum & <lb></lb>Augis Satellitum tempore unius revolutionis æquationum prio-<pb xlink:href="039/01/418.jpg" pagenum="390"></pb><arrow.to.target n="note421"></arrow.to.target>rum, ad motus Nodorum & Apogæi Lunæ tempore unius revo<lb></lb>lutionis æquationum poſteriorum. </s> <s>Variatio Satellitis è Jove ſpe<lb></lb>ctati, eſt ad Variationem Lunæ, ut ſunt ad invicem toti motus No<lb></lb>dorum temporibus quibus Satelles & Luna ad Solem revolvuntur, <lb></lb>per idem Corollarium; adeoQ.E.I. Satellite extimo non ſuperat <lb></lb>5″. </s> <s>12′. </s></p> <p type="margin"> <s><margin.target id="note421"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIV. THEOREMA XIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Fluxum & refluxum Maris ab actionibus Solis ac <lb></lb>Lunæ oriri.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Mare ſingulis diebus tam Lunaribus quam Solaribus bis intu<lb></lb>meſcere debere ac bis defluere, patet per Corol. </s> <s>19. Prop. </s> <s>LXVI. <lb></lb>Lib.I. ut & aquæ maximam altitudinem, in maribus profundis <lb></lb>& liberis, appulſum Luminarium ad Meridianum loci, minori <lb></lb>quam ſex horarum ſpatio ſequi, uti fit in Maris <emph type="italics"></emph>Atlantici<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>Æthiopici<emph.end type="italics"></emph.end>tractu toto orientali inter <emph type="italics"></emph>Galliam<emph.end type="italics"></emph.end>& Promontorium <lb></lb><emph type="italics"></emph>Bonæ Spei,<emph.end type="italics"></emph.end>ut & in Maris <emph type="italics"></emph>Pacifici<emph.end type="italics"></emph.end>littore <emph type="italics"></emph>Chilenſt<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Peruviano<emph.end type="italics"></emph.end>: <lb></lb>in quibus omnibus littoribus æſtus in horam circiter tertiam in<lb></lb>cidit, niſi ubi motus per loca vadoſa propagatus aliquantulum re<lb></lb>tardatur. </s> <s>Horas numero ab appulſu Luminaris utriuſque ad Me<lb></lb>ridianum loci, tam infra Horizontem quam ſupra, & per horas <lb></lb>diei Lunaris intelligo vigeſimas quartas partes temporis quo Luna <lb></lb>motu apparente diurno ad Meridianum loci revolvitur. </s></p> <p type="main"> <s>Motus autem bini, quos Luminaria duo excitant, non cernen<lb></lb>tur diſtincte, ſed motum Q.E.D.m mixtum efficient. </s> <s>In Lumina<lb></lb>rium Conjunctione vel Oppoſitione conjungentur eorum effectus, <lb></lb>& componetur fluxus & refluxus maximus. </s> <s>In Quadraturis Sol <lb></lb>attollet aquam ubi Luna deprimit, deprimetque ubi Sol attollit; <lb></lb>& ex effectuum differentia æſtus omnium minimus orietur. </s> <s>Et <lb></lb>quoniam, experientia teſte, major eſt effectus Lunæ quam Solis, <lb></lb>incidet aquæ maxima altitudo in horam tertiam Lunarem. </s> <s>Ex<lb></lb>tra Syzygias & Quadraturas, æſtus maximus qui ſola vi Lunari <lb></lb>incidere ſemper deberet in horam tertiam Lunarem, & ſola Solari <lb></lb>in tertiam Solarem, compoſitis viribus incidet in tempus aliquod <lb></lb>intermedium quod tertiæ Lunari propinquius eſt; adeoQ.E.I. <lb></lb>tranſitu Lunæ a Syzygiis ad Quadraturas, ubi hora tertia Solaris <lb></lb>præcedit tertiam Lunarem, maxima aquæ altitudo præcedet etiam <pb xlink:href="039/01/419.jpg" pagenum="391"></pb>tertiam Lunarem, ideque maximo intervallo paulo poſt Octantes <lb></lb><arrow.to.target n="note422"></arrow.to.target>Lunæ; & paribus intervallis æſtus maximus ſequetur horam ter<lb></lb>tiam Lunatem in tranſitu Lunæ a Quadraturis ad Syzygias. </s> <s>Hæc <lb></lb>ita ſunt in Mari aperto. </s> <s>Nam in oſtiis Fluviorum fluxus majo<lb></lb>res cæteris paribus tardius ad <foreign lang="grc">ἀκμλὼ</foreign> venient. </s></p> <p type="margin"> <s><margin.target id="note422"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Pendent autem effectus Luminarium ex eorum diſtantiis a Terra. </s> <s><lb></lb>In minoribus enim diſtantiis majores ſunt eorum effectus, in ma<lb></lb>joribus minores, idQ.E.I. triplicata ratione diametrorum appa<lb></lb>rentium. </s> <s>Igitur Sol tempore hyberno, in Perigæo exiſtens, ma<lb></lb>jores edit effectus, efficitque ut æſtus in Syzygiis paulo majores <lb></lb>ſint, & in Quadraturis paulo minores (cæteris paribus) quam <lb></lb>tempore æſtivo; & Luna in Perigæo ſingulis menſibus majores <lb></lb>ciet æſtus quam ante vel poſt dies quindecim, ubi in Apogæo ver<lb></lb>ſatur. </s> <s>Vnde fit ut æſtus duo omnino maximi in Syzygiis con<lb></lb>tinuis ſe mutuo non ſequantur. </s></p> <p type="main"> <s>Pendet etiam effectus utriuſque Luminaris ex ipſius Declina<lb></lb>tione ſeu diſtautia ab Æquatore. </s> <s>Nam ſi Luminare in polo con<lb></lb>ſtitueretur, traheret illud ſingulas aquæ partes conſtanter, abſque <lb></lb>actionis intenſione & remiſſione, adeoque nullam motus recipro<lb></lb>cationem cieret. </s> <s>Igitur Luminaria recedendo ab æquatore polum <lb></lb>verſus, effectus ſuos gradatim amittent, & propterea minores cie<lb></lb>bunt æſtus in Syzygiis Solſtitialibus quam in Æquinoctialibus. </s> <s><lb></lb>In Quadraturis autem Solſtitialibus majores ciebunt æſtus quam <lb></lb>in Quadraturis Æquinoctialibus; eo quod Lunæ jam in æquatore <lb></lb>conſtitutæ effectus maxime ſuperat effectum Solis Incidunt igi<lb></lb>tur æſtus maximi in Syzygias & minimi in Quadraturas Lumina<lb></lb>rium, circa tempora Æquinoctii utriuſque. </s> <s>Et æſtum maximum <lb></lb>in Syzygiis comitatur ſemper minimus in Quadraturis, ut experi<lb></lb>entia compertum eſt. </s> <s>Per minorem autem diſtantiam Solis a <lb></lb>Terra, tempore hyberno quam tempore æſtivo, fit ut æſtus ma<lb></lb>ximi & minimi ſæpius præcedant Æquinoctium vernum quam <lb></lb>ſequantur, & ſæpius ſequantur autumnale quam præcedant. </s></p> <p type="main"> <s>Pendent etiam effectus Luminarium ex loeorum latitudine. </s> <s>De<lb></lb>ſignet <emph type="italics"></emph>ApEP<emph.end type="italics"></emph.end>Tellurem aquis profundis undique coopertam; <emph type="italics"></emph>C<emph.end type="italics"></emph.end><lb></lb>centrum ejus; <emph type="italics"></emph>P, p<emph.end type="italics"></emph.end>polos, <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>Æquatorem; <emph type="italics"></emph>F<emph.end type="italics"></emph.end>locum quemvis <lb></lb>extra Æquatorem; <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end>parallelum loci; <emph type="italics"></emph>Dd<emph.end type="italics"></emph.end>parallelum ei re<lb></lb>ſpondentem ex altera parte æquatoris; <emph type="italics"></emph>L<emph.end type="italics"></emph.end>locum quem Luna tri<lb></lb>bus ante horis occupabat; <emph type="italics"></emph>H<emph.end type="italics"></emph.end>locum Telluris ei perpendiculariter <pb xlink:href="039/01/420.jpg" pagenum="392"></pb><arrow.to.target n="note423"></arrow.to.target>ſubjectum; <emph type="italics"></emph>h<emph.end type="italics"></emph.end>locum huic oppoſitum; <emph type="italics"></emph>K, k<emph.end type="italics"></emph.end>loca inde gradibus 90 <lb></lb>diſtantia, <emph type="italics"></emph>CH, Ch<emph.end type="italics"></emph.end>Maris altitudines maximas menſuratas a cen<lb></lb>tro Telluris; & <emph type="italics"></emph>CK, Ck<emph.end type="italics"></emph.end>altitudines minimas: & ſi axibus <emph type="italics"></emph>Hh, <lb></lb>Kk<emph.end type="italics"></emph.end>deſcribatur Ellipſis, deinde Ellipſeos hujus revolutione circa <lb></lb>axem majorem <emph type="italics"></emph>Hh<emph.end type="italics"></emph.end>deſcribatur Sphærois <emph type="italics"></emph>HPKhpk<emph.end type="italics"></emph.end>; deſignabit <lb></lb>hæc figuram Maris quam <lb></lb><figure id="id.039.01.420.1.jpg" xlink:href="039/01/420/1.jpg"></figure><lb></lb>proxime, & erunt <emph type="italics"></emph>CF, Cf, <lb></lb>CD, Cd<emph.end type="italics"></emph.end>altitudines Maris <lb></lb>in locis <emph type="italics"></emph>F, f, D, d.<emph.end type="italics"></emph.end>Quin<lb></lb>etiam ſi in præfata Ellipſeos <lb></lb>revolutione punctum quod<lb></lb>vis <emph type="italics"></emph>N<emph.end type="italics"></emph.end>deſcribat circulum <lb></lb><emph type="italics"></emph>NM,<emph.end type="italics"></emph.end>ſecantem parallelos <lb></lb><emph type="italics"></emph>Ff, Dd<emph.end type="italics"></emph.end>in locis quibuſvis <lb></lb><emph type="italics"></emph>R, T,<emph.end type="italics"></emph.end>& æquatorem <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>in <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end>; erit <emph type="italics"></emph>CN<emph.end type="italics"></emph.end>altitudo Maris <lb></lb>in locis omnibus <emph type="italics"></emph>R, S, T,<emph.end type="italics"></emph.end>ſitis in hoc circulo. </s> <s>Hinc in revolu<lb></lb>tione diurna loci cujuſvis <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>affluxus erit maximus in <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>hora <lb></lb>tertia poſt appulſum Lunæ ad Meridianum ſupra Horizontem; <lb></lb>poſtea defluxus maximus in <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>hora tertia poſt occaſum Lunæ; <lb></lb>dein affluxus maximus in <emph type="italics"></emph>f<emph.end type="italics"></emph.end>hora tertia poſt appulſum Lunæ ad <lb></lb>Meridianum infra Horizontem; ultimo defluxus maximus in <emph type="italics"></emph>Q<emph.end type="italics"></emph.end><lb></lb>hora tertia poſt ortum Lunæ; & affluxus poſterior in <emph type="italics"></emph>f<emph.end type="italics"></emph.end>erit mi<lb></lb>nor quam affluxus prior in <emph type="italics"></emph>F.<emph.end type="italics"></emph.end>Diſtinguitur enim Mare totum in <lb></lb>duos omnino fluctus Hemiſphæricos, unum in Hemiſphærio <lb></lb><emph type="italics"></emph>KHkC<emph.end type="italics"></emph.end>ad Boream vergentem, alterum in Hemiſphærio oppo<lb></lb>ſito <emph type="italics"></emph>KhkC<emph.end type="italics"></emph.end>; quos igitur fluctum Borealem & fluctum Auſtralem <lb></lb>nominare licet. </s> <s>Hi fluctus ſemper ſibi mutuo oppoſiti, veniunt <lb></lb>per vices ad Meridianos loeorum ſingulorum, interpoſito inter<lb></lb>vallo horarum Lunarium duodecim. </s> <s>Cumque regiones Boreales <lb></lb>magis participant fluctum Borealem, & Auſtrales magis Auſtra<lb></lb>lem, inde oriuntur æſtus alternis vicibus majores & minores, in <lb></lb>locis ſingulis extra æquatorem, in quibus luminaria oriuntur & <lb></lb>occidunt. </s> <s>Æſtus autem major, Luna in verticem loci declinante, <lb></lb>incidet in horam circiter tertiam poſt appulſum Lunæ ad Meri<lb></lb>dianum ſupra Horizontem, & Luna declinationem mutante verte<lb></lb>tur in minorem. </s> <s>Et fluxuum differentia maxima incidet in tem<lb></lb>pora Solſtitiorum; præſertim ſi Lunæ Nodus aſcendens verſatur <lb></lb>in principio Arietis. </s> <s>Sic experientia compertum eſt, quod æſtus <lb></lb>matutini tempore hyberno ſuperent veſpertinos & veſpertini tem-<pb xlink:href="039/01/421.jpg" pagenum="393"></pb>pore æſtivo matutinos, ad <emph type="italics"></emph>Plymuthum<emph.end type="italics"></emph.end>quidem altitudine quaſi<lb></lb>pedis unius, ad <emph type="italics"></emph>Briſtoliam<emph.end type="italics"></emph.end>vero altitudine quindecim digitorum:<lb></lb>obſervantibus <emph type="italics"></emph>Colepreſſio<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Sturmio<emph.end type="italics"></emph.end>.</s> <s>Motus autem hactenus deſcripti mutantur aliquantulum per vim<lb></lb>illam reciprocationis aquarum, qua Maris aſtus, etiam ceſſantibus Luminarium actionibus, poſſet aliquam diu perſeverare.</s> <s>Conſer<lb></lb>vatio hæcce motus impreſſi minuit differentiam æſtuum alterno<lb></lb>rum; & aſtus proxime poſt Syzygias majores reddit, eoſque pro<lb></lb>xime poſt Quadraturas minuit.</s> <s>Unde ſit ut æſtus alterni ad<emph type="italics"></emph>Ply<lb></lb>muthum & Briſtoliam<emph.end type="italics"></emph.end>non multo mafis differant ab invicem quam<lb></lb>altitudine pedis unius vel digitorum quindecim; utque æſtus om<lb></lb>nium maximi in iiſdem portubus, non ſint primi a Syzygiis, ſed<lb></lb>tertii.</s> <s>Retardantur etiam motus omnes in tranſitu per vada, adeo<lb></lb>ut æſtus omnium maximi, in fretis quibusdam & Fluviorum oſtiis,<lb></lb>ſsint quarti vel etiam quinti a Syzygiis.</s> <s><lb></lb>Porro fieri poteſt ut æſtus propagetur ab Oceano per freta di<lb></lb>verſa ad eundem portum, & citius tranſeat per aliqua freta quam<lb></lb>per alia: quo in caſu æſtus idem, in duos vel plures ſucceſſive ad<lb></lb>venientis diviſus, componere poſſit motus novos diverſorum ge<lb></lb>nerum.</s> <s>Fingamus æſtus duos æquales a diverſis locis in eundem<lb></lb>portum venire, quorum prior præcedat alterum ſpatio horarum<lb></lb>fex, incidatQ.E.I. horam tertiam ab appulſu Lunæ ad Meridia<lb></lb>num portus.</s> <s>Si Luna in hocce ſuo ad Meridianum appulſu ver<lb></lb>fabatur in æquatore, venient ſingulis horis fenis æquales affluxus,<lb></lb>qui in motuos refluxus incidendo eoſdem affluxibus æquabunt,<lb></lb> & ſic ſpatio diei illius efficient ut aqua tranquille ſtagnet.</s> <s>Si<lb></lb>Luna tunc declinabat ab Æquatore, fient æſtus in Oceano vici<lb></lb>bus alternis majores & minores, uti dictum eſt; &inde propaga<lb></lb>buntur in hunc portum affluxus bini majores & bini minores, vi<lb></lb>cibus alternis.</s> <s>Affluxus autem bini majores component aquam<lb></lb>altiſſimam in medio inter utrumque, affluxus major & minor fa<lb></lb>ciet ut aqua aſcendat ad mediocrem altitudinem in Medio ipſo<lb></lb>rum, & inter affluxus binos minores aqua aſcendet ad altitudi<lb></lb>dinem minimam.</s> <s>Sic ſpatio viginti quatuor horarum, aqua non<lb></lb>bis ut fieri ſolet, sed ſemel tantum perveniet ad maximam altitu<lb></lb>dinem & ſemel ad minimam; & altitudo maxima, ſi Luna decli<lb></lb>nat in polum ſupra Horizontem loci, incidet in horam vel ſextam<lb></lb>vel triceſimam ab appulſu Lunæ ad Meridianum, atque Luna de<lb></lb>clinationem mutante mutabitur in defluxum.</s> <s>Quorum omnium<lb></lb>exemplum, in portu regni <emph type="italics"></emph>Tunquini<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Batſham<emph.end type="italics"></emph.end>, ſub latitudine<pb xlink:href="039/01/422.jpg" pagenum="394"></pb>Boreali 20<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 50′.</s> <s><emph type="italics"></emph>Halleius<emph.end type="italics"></emph.end>ex Nautarum Obſervationibus pate<lb></lb>fecit.</s> <s>Ibi aqua di transitum Lunæ per Æquatorem ſequente<lb></lb>ſtagnat, dein Luna ad Boream declinante incipit fluere & refluere,<lb></lb> non bis, ut in aliis portubus, ſed ſemel ſingulis diebus; & æſtus<lb></lb>incidit in occasum Lunæ, defluxus maximus in ortum.</s> <s>Cum<lb></lb> Lunæ declinatione augetur hic æſtus uſque ad diem ſeptimum<lb></lb>vel octavum, dein per alios ſeptem dies iisdem gradibus decreſcit,<lb></lb>quibus antea creverat; & Luna declinationem mutante ceſſat, ac<lb></lb>mox mutator in defluxum.</s> <s>Incidit enim ſubinde defluxus in oc<lb></lb>caſum Lunæ & affluxus in ortum, donec Luna iterum mutet de<lb></lb>clinationem.</s> <s>Aditus ad hunc portum fretaque vicina duplex pa<lb></lb>ter, alter ab Oceano <emph type="italics"></emph>Sinenſi<emph.end type="italics"></emph.end>inter Continentem & Inſulam <emph type="italics"></emph>Luco<lb></lb>niam<emph.end type="italics"></emph.end>, alter a Mari <emph type="italics"></emph>Indico<emph.end type="italics"></emph.end>inter Continentem & Inſulam <emph type="italics"></emph>Borneo<emph.end type="italics"></emph.end>.<lb></lb></s> <s>An æſtus ſpatio horarum duodecim a Mari <emph type="italics"></emph>Indico<emph.end type="italics"></emph.end>& ſpatio hora<lb></lb>rum fex a Mari <emph type="italics"></emph>Sinenſi<emph.end type="italics"></emph.end>per freta illa venientes, & ſic in horam ter<lb></lb>tiam & nonam Lunarem incidentes, componant huiuſmodi motus;<lb></lb>ſitne alia Marium illorum conditio, obſervationibus vicinorum<lb></lb>littorum determinandum reliquo.</s></p> <p type="main"> <s>Hactenus cauſas motuum Lunæ & Marium reddidi.</s> <s>De quan<lb></lb>titate motuum jam convenit aliqua ſubjungere.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXV. PROBLEMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Invenire vires Solis ad perturbandos motus Lunæ.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Deſignet <emph type="italics"></emph>S<emph.end type="italics"></emph.end>Solem, <emph type="italics"></emph>T<emph.end type="italics"></emph.end>Terram, <emph type="italics"></emph>P<emph.end type="italics"></emph.end>Lunam, <emph type="italics"></emph>P A D B<emph.end type="italics"></emph.end>orbem<lb></lb>Lunæ.</s> <s>In <emph type="italics"></emph>S P<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>S K<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>S T<emph.end type="italics"></emph.end>, ſitque <emph type="italics"></emph>S L<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>S K<emph.end type="italics"></emph.end><figure id="id.039.01.422.1.jpg" xlink:href="039/01/422/1.jpg"></figure>in duplicata ratione <emph type="italics"></emph>S K<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>S P<emph.end type="italics"></emph.end>, & ipsi <emph type="italics"></emph>P T<emph.end type="italics"></emph.end>agatur parallela<lb></lb><emph type="italics"></emph>L M<emph.end type="italics"></emph.end>; & ſi gravitas acceleratrix Terræ in Solem exponatur per<lb></lb>diſtantiam <emph type="italics"></emph>S T<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>S K<emph.end type="italics"></emph.end>, erit <emph type="italics"></emph>S L<emph.end type="italics"></emph.end>gravitas acceleratrix Lunæ in<pb xlink:href="039/01/423.jpg" pagenum="395"></pb>Solem. </s> <s>Ea componitur ex partibus <emph type="italics"></emph>SM, LM,<emph.end type="italics"></emph.end>quarum <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>& <lb></lb><arrow.to.target n="note424"></arrow.to.target>ipſius <emph type="italics"></emph>SM<emph.end type="italics"></emph.end>pars <emph type="italics"></emph>TM<emph.end type="italics"></emph.end>perturbat motum Lunæ, ut in Libri primi <lb></lb>Prop. </s> <s>LXVI. & ejus Corollariis expoſitum eſt. </s> <s>Quatenus Terra <lb></lb>& Luna circum commune gravitatis centrum revolvuntur, pertur<lb></lb>babitur etiam motus Terræ circa centrum illud a viribus conſimi<lb></lb>libus; ſed ſummas tam virium quam motuum referre licet ad Lu<lb></lb>nam, & ſummas virium per lineas ipſis analogas <emph type="italics"></emph>TM<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ML<emph.end type="italics"></emph.end><lb></lb>deſignare. </s> <s>Vis <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>(in mediocri ſua quantitate) eſt ad vim <lb></lb>centripetam, qua Luna in Orbe ſuo circa Terram quieſcentem ad <lb></lb>diſtantiam <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>revolvi poſſet, in duplicata ratione temporum <lb></lb>periodieorum Lunæ circa Terram & Terræ circa Solem, (per <lb></lb>Corol. </s> <s>17. Prop. </s> <s>LXVI. Lib.I.) hoc eſt, in duplicata ratione die<lb></lb>rum 27. <emph type="italics"></emph>hor.<emph.end type="italics"></emph.end>7. <emph type="italics"></emph>min.<emph.end type="italics"></emph.end>43. ad dies 365. <emph type="italics"></emph>hor.<emph.end type="italics"></emph.end>6. <emph type="italics"></emph>min.<emph.end type="italics"></emph.end>9. id eſt, ut 1000 <lb></lb>ad 178725, ſeu 1 ad (178 39/40). Invenimus autem in Propoſitione <lb></lb>quarta quod, ſi Terra & Luna circa commune gravitatis centrum <lb></lb>revolvantur, earum diſtantia mediocris ab invicem erit 60 1/2 ſemi<lb></lb>diametrorum mediocrium Terræ quamproxime. </s> <s>Et vis qua Luna <lb></lb>in Orbe circa Terram quieſcentem, ad diſtantiam <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ſemidiame<lb></lb>trorum terreſtrium 60 1/2 revolvi poſſet, eſt ad vim, qua eodem <lb></lb>tempore ad diſtantiam ſemidiametrorum 60 revolvi poſſet, ut <lb></lb>60 1/2 ad 60; & hæc vis ad vim gravitatis apud nos ut 1 ad <lb></lb>60X60 quamproxime. </s> <s>Ideoque vis mediocris <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>eſt ad vim <lb></lb>gravitatis in ſuperficie Terræ, ut 1X60 1/2 ad 60X60X60X(178 29/40), <lb></lb>ſeu 1 ad 638092, 6. Vnde ex proportione linearum <emph type="italics"></emph>TM, ML,<emph.end type="italics"></emph.end><lb></lb>datur etiam vis <emph type="italics"></emph>TM:<emph.end type="italics"></emph.end>& hæ ſunt vires Solis quibus Lunæ motus <lb></lb>perturbantur. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note423"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="margin"> <s><margin.target id="note424"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVI. PROBLEMA VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Invenire incrementum borarium areæ quam Luna, radio ad Ter<lb></lb>ram ducto, in Orbe circulari deſcribit.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Diximus aream, quam Luna radio ad Terram ducto deſcribit, <lb></lb>eſſe tempori proportionalem, niſi quatenus motus Lunaris ab <lb></lb>actione Solis turbatur. </s> <s>Inæqualitatem momenti (vel incrementi <lb></lb>horarii) hic inveſtigandam proponimus. </s> <s>Ut computatio facilior <lb></lb>reddatur, fingamus orbem Lunæ circularem eſſe, & inæqualitates <lb></lb>omnes negligamus, ea ſola excepta, de qua hic agitur. </s> <s>Ob in<lb></lb>gentem vero Solis diſtantiam, ponamus etiam lineas <emph type="italics"></emph>SP, ST<emph.end type="italics"></emph.end>ſibi <lb></lb>invicem parallelas eſſe. </s> <s>Hoc pacto vis <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>reducetur ſemper <pb xlink:href="039/01/424.jpg" pagenum="396"></pb><arrow.to.target n="note425"></arrow.to.target>ad mediocrem ſuam quantitatem <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>ut & vis <emph type="italics"></emph>TM<emph.end type="italics"></emph.end>ad medio<lb></lb>crem ſuam quantitatem 3 <emph type="italics"></emph>PK.<emph.end type="italics"></emph.end>Hæ vires, per Legum Corol. </s> <s>2. <lb></lb>componunt vim <emph type="italics"></emph>TL<emph.end type="italics"></emph.end>; & hæc vis, ſi in radium <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>demittatur <lb></lb>perpendiculum <emph type="italics"></emph>LE,<emph.end type="italics"></emph.end>reſolvitur in vires <emph type="italics"></emph>TE, EL,<emph.end type="italics"></emph.end>quarum <emph type="italics"></emph>TE,<emph.end type="italics"></emph.end><lb></lb>agendo ſemper ſecundum radium <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>nec accelerat nec retardat <lb></lb>deſcriptionem areæ <emph type="italics"></emph>TPC<emph.end type="italics"></emph.end>radio illo <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>factam; & <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>agendo <lb></lb>ſecundum perpendiculum, accelerat vel retardat ipſam, quan<lb></lb>tum accelerat vel retardat Lunam. </s> <s>Acceleratio illa Lunæ, in <lb></lb>tranſitu ipſius a Quadratura <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ad Conjunctionem <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ſingulis <lb></lb>temporis momentis facta, eſt ut ipſa vis accelerans <emph type="italics"></emph>EL,<emph.end type="italics"></emph.end>hoc eſt, <lb></lb>ut (<emph type="italics"></emph>3PKXTK/TP<emph.end type="italics"></emph.end>). Exponatur tempus per motum medium Luna<lb></lb>rem, vel (quod eodem fere recidit) per angulum <emph type="italics"></emph>CTP,<emph.end type="italics"></emph.end>vel <lb></lb><figure id="id.039.01.424.1.jpg" xlink:href="039/01/424/1.jpg"></figure>etiam per arcum <emph type="italics"></emph>CP.<emph.end type="italics"></emph.end>Ad <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>erigatur normalis <emph type="italics"></emph>CG<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>CT<emph.end type="italics"></emph.end><lb></lb>æqualis. </s> <s>Et diviſo arcu quadrantali <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>in particulas innumeras <lb></lb>æquales <emph type="italics"></emph>Pp,<emph.end type="italics"></emph.end>&c. </s> <s>per quas æquales totidem particulæ temporis <lb></lb>exponi poſſint, ductaque <emph type="italics"></emph>pk<emph.end type="italics"></emph.end>perpendiculari ad <emph type="italics"></emph>CT,<emph.end type="italics"></emph.end>jungatur <lb></lb><emph type="italics"></emph>TG<emph.end type="italics"></emph.end>ipſis <emph type="italics"></emph>KP, kp<emph.end type="italics"></emph.end>productis occurrens in <emph type="italics"></emph>F<emph.end type="italics"></emph.end>& <emph type="italics"></emph>f<emph.end type="italics"></emph.end>; & erit <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>PK<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Pp<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Tp,<emph.end type="italics"></emph.end>hoc eſt in data ratione, adeoque <emph type="italics"></emph>FKXKk<emph.end type="italics"></emph.end><lb></lb>ſeu area <emph type="italics"></emph>FKkf,<emph.end type="italics"></emph.end>ut (<emph type="italics"></emph>3PKXTK/TP<emph.end type="italics"></emph.end>), id eſt, ut <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>; & compoſite, <lb></lb>area tota <emph type="italics"></emph>GCKF<emph.end type="italics"></emph.end>ut ſumma omnium virium <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>tempore toto <lb></lb><emph type="italics"></emph>CP<emph.end type="italics"></emph.end>impreſſarum in Lunam, atque adeo etiam ut velocitas hac <pb xlink:href="039/01/425.jpg" pagenum="397"></pb>ſumma genita, id eſt, ut acceleratio deſcriptionis areæ <emph type="italics"></emph>CTP,<emph.end type="italics"></emph.end>ſeu <lb></lb><arrow.to.target n="note426"></arrow.to.target>incrementum momenti. </s> <s>Vis qua Luna circa Terram quieſcentem <lb></lb>ad diſtantiam <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>tempore ſuo periodico <emph type="italics"></emph>CADBC<emph.end type="italics"></emph.end>dierum 27. <lb></lb><emph type="italics"></emph>hor.<emph.end type="italics"></emph.end>7. <emph type="italics"></emph>min.<emph.end type="italics"></emph.end>43. revolvi poſſet, efficeret ut corpus, tempore <emph type="italics"></emph>CT<emph.end type="italics"></emph.end><lb></lb>cadendo, deſcriberet longitudinem 1/2 <emph type="italics"></emph>CT,<emph.end type="italics"></emph.end>& velocitatem ſimul <lb></lb>acquireret æqualem velocitati, qua Luna in Orbe ſuo movetur. </s> <s><lb></lb>Patet hoc per Corol. </s> <s>9. Prop. </s> <s>IV. Lib. </s> <s>I. </s> <s>Cum autem perpen<lb></lb>diculum <emph type="italics"></emph>Kd<emph.end type="italics"></emph.end>in <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>demiſſum ſit ipſius <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>pars tertia, & ip<lb></lb>ſius <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>in Octantibus pars dimidia, vis <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>in Octan<lb></lb>tibus, ubi maxima eſt, ſuperabit vim <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>in ratione 3 ad 2, <lb></lb>adeoque erit ad vim illam, qua Luna tempore ſuo periodico circa <lb></lb>Terram quieſcentem revolvi poſſet, ut 100 ad 2/3X17872 1/2 ſeu <lb></lb>11915, & tempore <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>velocitatem generare deberet quæ eſſet <lb></lb>pars (100/11915) velocitatis Lunaris, tempore autem <emph type="italics"></emph>CPA<emph.end type="italics"></emph.end>velocitatem <lb></lb>majorem generaret in ratione <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>TP.<emph.end type="italics"></emph.end>Exponatur <lb></lb>vis maxima <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>in Octantibus per aream <emph type="italics"></emph>FKXKk<emph.end type="italics"></emph.end>rectangulo <lb></lb>1/2 <emph type="italics"></emph>TPXPp<emph.end type="italics"></emph.end>æqualem. </s> <s>Et velocitas, quam vis maxima tempore <lb></lb>quovis <emph type="italics"></emph>CP<emph.end type="italics"></emph.end>generare poſſet, erit ad velocitatem quam vis omnis <lb></lb>minor <emph type="italics"></emph>EL<emph.end type="italics"></emph.end>eodem tempore generat, ut rectangulum 1/2 <emph type="italics"></emph>TPXCP<emph.end type="italics"></emph.end><lb></lb>ad aream <emph type="italics"></emph>KCGF<emph.end type="italics"></emph.end>: tempore autem toto <emph type="italics"></emph>CPA,<emph.end type="italics"></emph.end>velocitates ge<lb></lb>nitæ erunt ad invicem ut rectangulum 1/2<emph type="italics"></emph>TPXCA<emph.end type="italics"></emph.end>& triangulum <lb></lb><emph type="italics"></emph>TCG,<emph.end type="italics"></emph.end>ſive ut arcus quadrantalis <emph type="italics"></emph>CA<emph.end type="italics"></emph.end>& radius <emph type="italics"></emph>TP.<emph.end type="italics"></emph.end>Ideoque <lb></lb>(per Prop. </s> <s>IX. Lib. </s> <s>V. Elem.) velocitas poſterior, toto tempore <lb></lb>genita, erit pars (100/11915) velocitatis Lunæ. </s> <s>Huic Lunæ velocitati, <lb></lb>quæ areæ momento mediocri analoga eſt, addatur & auferatur <lb></lb>dimidium velocitatis alterius; & ſi momentum mediocre expona<lb></lb>tur per numerum 11915, ſumma 11915+50 ſeu 11965 exhi<lb></lb>bebit momentum maximum areæ in Syzygia <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ac differentia <lb></lb>11915-50 ſeu 11865 ejuſdem momentum minimum in Quadra<lb></lb>turis. </s> <s>Igitur areæ temporibus æqualibus in Syzygiis & Quadra<lb></lb>turis deſcriptæ, ſunt ad invicem ut 11965 ad 11865. Ad mo<lb></lb>mentum minimum 11865 addatur momentum, quod ſit ad mo<lb></lb>mentorum differentiam 100 ut trapezium <emph type="italics"></emph>FKCG<emph.end type="italics"></emph.end>ad triangu<lb></lb>lum <emph type="italics"></emph>TCG<emph.end type="italics"></emph.end>(vel quod perinde eſt, ut quadratum Sinus <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>ad <lb></lb>quadratum Radii <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>Pd<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>) & ſumma exhi<lb></lb>bebit momentum areæ, ubi Luna eſt in loco quovis interme<lb></lb>dio <emph type="italics"></emph>P.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note425"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="margin"> <s><margin.target id="note426"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Hæc omnia ita ſe habent, ex Hypotheſi quod Sol & Terra qui<lb></lb>eſcunt, & Luna tempore Synodico dierum 27. <emph type="italics"></emph>hor.<emph.end type="italics"></emph.end>7. <emph type="italics"></emph>min.<emph.end type="italics"></emph.end>43. re<lb></lb>volvitur. </s> <s>Cum autem periodus Synodica Lunaris vere ſit die-<pb xlink:href="039/01/426.jpg" pagenum="398"></pb><arrow.to.target n="note427"></arrow.to.target>rum 29. <emph type="italics"></emph>hor.<emph.end type="italics"></emph.end>12. & <emph type="italics"></emph>min.<emph.end type="italics"></emph.end>44. augeri debent momentorum incre<lb></lb>menta in ratione temporis, id eſt, in ratione 1080853 ad 1000000. <lb></lb>Hoc pacto incrementum totum, quod erat pars (100/11915) momenti <lb></lb>mediocris, jam fiet ejuſdem pars (100/11023). Ideoque momentum <lb></lb>areæ in Quadratura Lunæ erit ad ejus momentum in Syzygia <lb></lb>ut 11023-50 ad 11023+50, ſeu 10973 ad 11073, & ad ejus <lb></lb>momentum, ubi Luna in alio quovis loco intermedio <emph type="italics"></emph>P<emph.end type="italics"></emph.end>verſatur, <lb></lb>ut 10973 ad 10973+<emph type="italics"></emph>Pd,<emph.end type="italics"></emph.end>exiſtente videlicet <emph type="italics"></emph>TP<emph.end type="italics"></emph.end>æquali 100. </s></p> <p type="margin"> <s><margin.target id="note427"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Area igitur, quam Luna radio ad Terram ducto ſingulis tem<lb></lb>poris particulis æqualibus deſcribit, eſt quam proxime ut ſumma <lb></lb>numeri 219,46 & Sinus verſi duplicatæ diſtantiæ Lunæ a Quadra<lb></lb>tura proxima, in circulo cujus radius eſt unitas. </s> <s>Hæc ita ſe ha<lb></lb>bent ubi Variatio in Octantibus eſt magnitudinis mediocris. </s> <s>Sin <lb></lb>Variatio ibi major ſit vel minor, augeri debet vel minui Sinus ille <lb></lb>verſus in eadem ratione. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVII. PROBLEMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Ex motu horario Lunæ invenire ipſius diſtantiam a Terra.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Area, quam Luna radio ad Terram ducto, ſingulis temporis <lb></lb>momentis, deſcribit, eſt ut motus horarius Lunæ & quadratum <lb></lb>diſtantiæ Lunæ a Terra conjunctim; & propterea diſtantia Lunæ <lb></lb>a Terra eſt in ratione compoſita ex ſubduplicata ratione Areæ di<lb></lb>recte & ſubduplicata ratione motus horarii inverſe. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc datur Lunæ diameter apparens: quippe quæ ſit <lb></lb>reciproce ut ipſius diſtantia a Terra. </s> <s>Tentent Aſtronomi quam <lb></lb>probe hæc Regula cum Phænomenis congruat. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Hinc etiam Orbis Lunaris accuratius ex Phænomenis <lb></lb>quam antehac definiri poteſt. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXVIII. PROBLEMA IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire diametros Orbis in quo Luna, abſque eccentricitate, <lb></lb>moveri deberet.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Curvatura Trajectoriæ, quam mobile, ſi ſecundum Trajectoriæ <lb></lb>illius perpendiculum trahatur, deſcribit, eſt ut attractio directe & <lb></lb>quadratum velocitatis inverſe, Curvaturas linearum pono eſſe in-<pb xlink:href="039/01/427.jpg" pagenum="399"></pb>ter ſe in ultima proportione Sinuum vel Tangentium angulorum <lb></lb><arrow.to.target n="note428"></arrow.to.target>contactuum ad radios æquales pertinentium, ubi radii illi in infi<lb></lb>nitum diminuuntur. </s> <s>Attractio autem Lunæ in Terram in Syzy<lb></lb>giis eſt exceſſus gravitatis ipſius in Terram ſupra vim Solarem <lb></lb>2 <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>(Vide <emph type="italics"></emph>Figur. </s> <s>pag.<emph.end type="italics"></emph.end>394) qua gravitas acceleratrix Lunæ in <lb></lb>Solem ſuperat gravitatem acceleratricem Terræ in Solem. </s> <s>In Qua<lb></lb>draturis autem attractio illa eſt ſumma gravitatis Lunæ in Terram <lb></lb>& vis Solaris <emph type="italics"></emph>KT,<emph.end type="italics"></emph.end>qua Luna in Terram trahitur. </s> <s>Et hæ attra<lb></lb>ctiones, ſi (<emph type="italics"></emph>AT+CT<emph.end type="italics"></emph.end>/2) dicatur N, ſunt ut (178725/<emph type="italics"></emph>ATq<emph.end type="italics"></emph.end>)-(2000/<emph type="italics"></emph>CTXN<emph.end type="italics"></emph.end>) & <lb></lb>(178725/<emph type="italics"></emph>CIq<emph.end type="italics"></emph.end>)+(1000/<emph type="italics"></emph>ATXN<emph.end type="italics"></emph.end>) quam proxime; ſeu ut 178725NX<emph type="italics"></emph>CTq<emph.end type="italics"></emph.end><lb></lb>-2000 <emph type="italics"></emph>ATqXCT<emph.end type="italics"></emph.end>& 178725 NX<emph type="italics"></emph>ATq<emph.end type="italics"></emph.end>+1000 <emph type="italics"></emph>CTqXAT.<emph.end type="italics"></emph.end>Nam <lb></lb>ſi gravitas acceleratrix Lunæ in Terram exponatur per numerum <lb></lb>178725, vis mediocris <emph type="italics"></emph>ML,<emph.end type="italics"></emph.end>quæ in Quadraturis eſt <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>vel <lb></lb><emph type="italics"></emph>TK<emph.end type="italics"></emph.end>& Lunam trahit in Ter<lb></lb><figure id="id.039.01.427.1.jpg" xlink:href="039/01/427/1.jpg"></figure><lb></lb>ram, erit 1000, & vis me<lb></lb>diocris <emph type="italics"></emph>TM<emph.end type="italics"></emph.end>in Syzygiis erit <lb></lb>3000; de qua, ſi vis medio<lb></lb>cris <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>ſubducatur, mane<lb></lb>bit vis 2000 qua Luna in <lb></lb>Syzygiis diſtrahitur a Terra, <lb></lb>quamque jam ante nominavi <lb></lb>2 <emph type="italics"></emph>PK.<emph.end type="italics"></emph.end>Velocitas autem Lu<lb></lb>næ in Syzygiis <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B<emph.end type="italics"></emph.end>eſt ad <lb></lb>ipſius velocitatem in Qua<lb></lb>draturis <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& <emph type="italics"></emph>D,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>AT<emph.end type="italics"></emph.end>& momentum areæ quam <lb></lb>Luna radio ad Terram du<lb></lb>cto deſcribit in Syzygiis ad <lb></lb>momentum ejuſdem areæ in <lb></lb>Quadraturis conjunctim; i.e. </s> <s><lb></lb>ut 11073 <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>ad 10973 <emph type="italics"></emph>AT.<emph.end type="italics"></emph.end><lb></lb>Sumatur hæc ratio bis in<lb></lb>verſe & ratio prior ſemel directe, & fiet curvatura Orbis Lu<lb></lb>naris in Syzygiis ad ejuſdem curvaturam in Quadraturis ut <lb></lb>120406729X178725 <emph type="italics"></emph>ATqXCTq<emph.end type="italics"></emph.end>XN-120406729X2000 <emph type="italics"></emph>ATqq <lb></lb>XCT<emph.end type="italics"></emph.end>ad 122611329X178725 <emph type="italics"></emph>ATqXCTq<emph.end type="italics"></emph.end>XN+122611329X <lb></lb>1000 <emph type="italics"></emph>CTqqXAT, i.e.<emph.end type="italics"></emph.end>ut 2151969 <emph type="italics"></emph>ATXCT<emph.end type="italics"></emph.end>XN-24081 <emph type="italics"></emph>AT cub.<emph.end type="italics"></emph.end><lb></lb>ad 2191371 <emph type="italics"></emph>ATXCT<emph.end type="italics"></emph.end>XN+12261 <emph type="italics"></emph>CT cub.<emph.end type="italics"></emph.end><pb xlink:href="039/01/428.jpg" pagenum="400"></pb><arrow.to.target n="note429"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note428"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note429"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Quoniam Figura orbis Lunaris ignoratur, hujus vice aſſuma<lb></lb>mus Ellipſin <emph type="italics"></emph>DBCA,<emph.end type="italics"></emph.end>in cujus centro <emph type="italics"></emph>T<emph.end type="italics"></emph.end>Terra collocetur, & cu<lb></lb>jus axis major <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>Quadraturis, minor <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>Syzygiis interja<lb></lb>ceat. </s> <s>Cum autem planum Ellipſeos hujus motu angulari circa <lb></lb>Terram revolvatur, & Trajectoria cujus curvaturam conſideramus, <lb></lb>deſcribi debet in plano quod omni motu angulari omnino deſti<lb></lb>tuitur: conſideranda erit Figura, quam Luna in Ellipſi illa revol<lb></lb>vendo deſcribit in hoc plano, hoc eſt Figura <emph type="italics"></emph>Cpa,<emph.end type="italics"></emph.end>cujus puncta <lb></lb>ſingula <emph type="italics"></emph>p<emph.end type="italics"></emph.end>inveniuntur capiendo punctum quodvis <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Ellipſi, <lb></lb>quod locum Lunæ repreſentet, & ducendo <emph type="italics"></emph>Tp<emph.end type="italics"></emph.end>æqualem <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>ea <lb></lb>lege ut angulus <emph type="italics"></emph>PTp<emph.end type="italics"></emph.end>æqualis ſit motui apparenti Solis a tem<lb></lb>pore Quadraturæ <emph type="italics"></emph>C<emph.end type="italics"></emph.end>confecto; vel (quod eodem fere recidit) ut <lb></lb>angulus <emph type="italics"></emph>CTp<emph.end type="italics"></emph.end>ſit ad angulum <lb></lb><figure id="id.039.01.428.1.jpg" xlink:href="039/01/428/1.jpg"></figure><lb></lb><emph type="italics"></emph>CTP<emph.end type="italics"></emph.end>ut tempus revolutio<lb></lb>nis Synodicæ Lunaris ad tem<lb></lb>pus revolutionis Periodicæ <lb></lb>ſeu 29<emph type="sup"></emph>d.<emph.end type="sup"></emph.end> 12<emph type="sup"></emph>h.<emph.end type="sup"></emph.end> 44′, ad 27<emph type="sup"></emph>d.<emph.end type="sup"></emph.end> 7<emph type="sup"></emph>h.<emph.end type="sup"></emph.end> 43′. </s> <s><lb></lb>Capiatur igitur angulus <emph type="italics"></emph>CTa<emph.end type="italics"></emph.end><lb></lb>in eadem ratione ad angu<lb></lb>lum rectum <emph type="italics"></emph>CTA,<emph.end type="italics"></emph.end>& ſit <lb></lb>longitudo <emph type="italics"></emph>Ta<emph.end type="italics"></emph.end>æqualis lon<lb></lb>gitudini <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>; & erit <emph type="italics"></emph>a<emph.end type="italics"></emph.end><lb></lb>Apſis ima & <emph type="italics"></emph>C<emph.end type="italics"></emph.end>Apſis ſum<lb></lb>ma Orbis hujus <emph type="italics"></emph>Cpa.<emph.end type="italics"></emph.end>Ra<lb></lb>tiones autem ineundo inve<lb></lb>nio quod differentia inter <lb></lb>curvaturam Orbis <emph type="italics"></emph>Cpa<emph.end type="italics"></emph.end>in <lb></lb>vertice <emph type="italics"></emph>a,<emph.end type="italics"></emph.end>& curvaturam Cir<lb></lb>culi centro <emph type="italics"></emph>T<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>TA<emph.end type="italics"></emph.end><lb></lb>deſcripti, ſit ad differentiam <lb></lb>inter curvaturam Ellipſeos in <lb></lb>vertice <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& curvaturam ejuſdem Circuli, in duplicata ratione an<lb></lb>guli <emph type="italics"></emph>CTP<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>CTp<emph.end type="italics"></emph.end>; & quod curvatura Ellipſeos in <emph type="italics"></emph>A<emph.end type="italics"></emph.end><lb></lb>ſit ad curvaturam Circuli illius, in duplicata ratione <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>; <lb></lb>& curvatura Circuli illius ad curvaturam Circuli centro <emph type="italics"></emph>T<emph.end type="italics"></emph.end>in<lb></lb>tervallo <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>deſcripti, ut <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>; hujus autem curvatura ad <lb></lb>curvaturam Ellipſeos in <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>in duplicata ratione <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>; & <lb></lb>differentia inter curvaturam Ellipſeos in vertice <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& curvaturam <lb></lb>Circuli noviſſimi, ad differentiam inter curvaturam Figuræ <emph type="italics"></emph>Tpa<emph.end type="italics"></emph.end><lb></lb>in vertice <emph type="italics"></emph>C<emph.end type="italics"></emph.end>& curvaturam ejuſdem Circuli, in duplicata ratione <pb xlink:href="039/01/429.jpg" pagenum="401"></pb>anguli <emph type="italics"></emph>CTp<emph.end type="italics"></emph.end>ad angulum <emph type="italics"></emph>CTP.<emph.end type="italics"></emph.end>Quæ quidem rationes ex ſinu</s></p> <p type="main"> <s><arrow.to.target n="note430"></arrow.to.target>bus angulorum contactus ac differentiarum angulorum facile colli<lb></lb>guntur. </s> <s>His autem inter ſe collatis, prodit curvatura Figuræ <emph type="italics"></emph>Cpa<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>a<emph.end type="italics"></emph.end>ad ipſius curvaturam in <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AT cub<emph.end type="italics"></emph.end>+(16824/100000)<emph type="italics"></emph>CTqXAT<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>CT cub<emph.end type="italics"></emph.end>+(16824/100000) <emph type="italics"></emph>ATqXCT.<emph.end type="italics"></emph.end>Ubi numerus (16824/100000) deſignat <lb></lb>differentiam quadratorum angulorum <emph type="italics"></emph>CTP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CTp<emph.end type="italics"></emph.end>appli<lb></lb>catam ad quadratum anguli minoris <emph type="italics"></emph>CTP,<emph.end type="italics"></emph.end>ſeu (quod per<lb></lb>inde eſt) differentiam quadratorum temporum 27<emph type="sup"></emph>d.<emph.end type="sup"></emph.end> 7<emph type="sup"></emph>h.<emph.end type="sup"></emph.end> 43′, & <lb></lb>29<emph type="sup"></emph>d.<emph.end type="sup"></emph.end> 12<emph type="sup"></emph>h.<emph.end type="sup"></emph.end> 44′, applicatam ad quadratum temporis 27<emph type="sup"></emph>d.<emph.end type="sup"></emph.end> 7<emph type="sup"></emph>h.<emph.end type="sup"></emph.end> 43′, </s></p> <p type="margin"> <s><margin.target id="note430"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Igitur cum <emph type="italics"></emph>a<emph.end type="italics"></emph.end>deſignet Syzygiam Lunæ, & <emph type="italics"></emph>C<emph.end type="italics"></emph.end>ipſius Quadratu<lb></lb>ram, proportio jam inventa eadem eſſe debet cum proportione <lb></lb>curvaturæ Orbis Lunæ in Syzygiis ad ejuſdem curvaturam in <lb></lb>Quadraturis, quam ſupra invenimus. </s> <s>Proinde ut inveniatur pro<lb></lb>portio <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AT,<emph.end type="italics"></emph.end>duco extrema & media in ſe invicem. </s> <s>Et <lb></lb>termini prodeuntes ad <emph type="italics"></emph>ATXCT<emph.end type="italics"></emph.end>applicati, fiunt 2062, 79 <emph type="italics"></emph>CTqq<emph.end type="italics"></emph.end><lb></lb>-2151969 NX<emph type="italics"></emph>CTcub<emph.end type="italics"></emph.end>+368676 NX<emph type="italics"></emph>ATXCTq<emph.end type="italics"></emph.end>+36342 <emph type="italics"></emph>ATq <lb></lb>XCTq<emph.end type="italics"></emph.end>-362047 NX<emph type="italics"></emph>ATqXCT<emph.end type="italics"></emph.end>+2191371 NX<emph type="italics"></emph>AT cub<emph.end type="italics"></emph.end>+ <lb></lb>4051, 4 <emph type="italics"></emph>ATqq<emph.end type="italics"></emph.end>=0. Hic pro terminorum <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>ſemiſum<lb></lb>ma N ſcribo 1, & pro eorundem ſemidifferentia ponendo <emph type="italics"></emph>x,<emph.end type="italics"></emph.end>fit <lb></lb><emph type="italics"></emph>CT<emph.end type="italics"></emph.end>=1+<emph type="italics"></emph>x,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>=1-<emph type="italics"></emph>x<emph.end type="italics"></emph.end>: quibus in æquatione ſcriptis, & <lb></lb>æquatione prodeunte reſoluta, obtinetur <emph type="italics"></emph>x<emph.end type="italics"></emph.end>æqualis 0,00719, & <lb></lb>inde ſemidiameter <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>fit 1,00719, & ſemidiameter <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>0,99281, <lb></lb>qui numeri ſunt ut (70 1/24) & (69 1/24) quam proxime. </s> <s>Eſt igitur di<lb></lb>ſtantia Lunæ a Terra in Syzygiis ad ipſius diſtantiam in Quadra<lb></lb>turis (ſepoſita ſcilicet Eccentricitatis conſideratione) ut (69 1/24) ad <lb></lb>(70 1/24), vel numeris rotundis ut 69 ad 70. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXIX. PROBLEMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire Variationem Lunæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Oritur hæc inæqualitas partim ex forma Elliptica orbis Luna<lb></lb>ris, partim ex inæqualitate momentorum areæ, quam Luna radio <lb></lb>ad Terram ducto deſcribit. </s> <s>Si Luna <emph type="italics"></emph>P<emph.end type="italics"></emph.end>in Ellipſi <emph type="italics"></emph>DBCA<emph.end type="italics"></emph.end>circa <lb></lb>Terram in centro Ellipſeos quieſcentem moveretur, & radio <emph type="italics"></emph>TP<emph.end type="italics"></emph.end><lb></lb>ad Terram ducto deſcriberet aream <emph type="italics"></emph>CTP<emph.end type="italics"></emph.end>tempori proportiona<lb></lb>lem; eſſet autem Ellipſeos ſemidiameter maxima <emph type="italics"></emph>CT<emph.end type="italics"></emph.end>ad ſemi<lb></lb>diametrum minimam <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>ut 70 ad 69: foret tangens anguli <lb></lb><emph type="italics"></emph>CTP<emph.end type="italics"></emph.end>ad tangentem anguli motus medii a Quadratura <emph type="italics"></emph>C<emph.end type="italics"></emph.end>compu<lb></lb>tati, ut Ellipſeos ſemidiameter <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>ad ejuſdem ſemidiametrum <pb xlink:href="039/01/430.jpg" pagenum="402"></pb><arrow.to.target n="note431"></arrow.to.target><emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ſeu 69 ad 70. Debet autem deſcriptio areæ <emph type="italics"></emph>CTP,<emph.end type="italics"></emph.end>in pro<lb></lb>greſſu Lunæ a Quadratura ad Syzygiam, ea ratione accelerari, ut <lb></lb>ejus momentum in Syzygia Lunæ ſit ad ejus momentum in Qua<lb></lb>dratura ut 11073 ad 10973, utque exceſſus momenti in loco <lb></lb>quovis intermedio <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ſupra momentum in Quadratura ſit ut qua<lb></lb>dratum ſinus anguli <emph type="italics"></emph>CTP.<emph.end type="italics"></emph.end>Id quod ſatis accurate fiet, ſi tan<lb></lb>gens anguli <emph type="italics"></emph>CTP<emph.end type="italics"></emph.end>diminuatur in ſubduplicata ratione numeri <lb></lb>10973 ad numerum 11073, id eſt, in ratione numeri 68,6877 ad <lb></lb>numerum 69. Quo pacto <lb></lb><figure id="id.039.01.430.1.jpg" xlink:href="039/01/430/1.jpg"></figure><lb></lb>tangens anguli <emph type="italics"></emph>CTP<emph.end type="italics"></emph.end>jam e<lb></lb>rit ad tangentem motus me<lb></lb>dii ut 68,6877 ad 70, & an<lb></lb>gulus <emph type="italics"></emph>CTP<emph.end type="italics"></emph.end>in Octantibus, <lb></lb>ubi motus medius eſt 45<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end><lb></lb>invenietur 44<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 27′. </s> <s>28″. </s> <s>qui <lb></lb>ſubductus de angulo motus <lb></lb>medii 45<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> relinquit Varia<lb></lb>tionem maximam 32′. </s> <s>32″. </s> <s><lb></lb>Hæc ita ſe haberent ſi Luna, <lb></lb>pergendo a Quadratura ad <lb></lb>Syzygiam, deſcriberet angu<lb></lb>lum <emph type="italics"></emph>CTA<emph.end type="italics"></emph.end>graduum tantum <lb></lb>nonaginta. </s> <s>Verum ob mo<lb></lb>tum Terræ, quo Sol in con<lb></lb>ſequentia motu apparente <lb></lb>transfertur, Luna, priuſquam <lb></lb>Solem aſſequitur, deſcribit <lb></lb>angulum <emph type="italics"></emph>CTa<emph.end type="italics"></emph.end>angulo recto majorem in ratione temporis revo<lb></lb>lutionis Lunaris Synodicæ ad tempus revolutionis Periodicæ, id <lb></lb>eſt, in ratione 29<emph type="sup"></emph>d.<emph.end type="sup"></emph.end> 12<emph type="sup"></emph>h.<emph.end type="sup"></emph.end> 44′. </s> <s>ad 27<emph type="sup"></emph>d.<emph.end type="sup"></emph.end> 7<emph type="sup"></emph>h.<emph.end type="sup"></emph.end> 43′. </s> <s>Et hoc pacto an<lb></lb>guli omnes circa centrum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>dilatantur in eadem ratione, & Va<lb></lb>riatio maxima quæ ſecus eſſet 32′. </s> <s>32″, jam aucta in eadem ratione <lb></lb>fit 35′. </s> <s>10″. </s></p> <p type="margin"> <s><margin.target id="note431"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Hæc eſt ejus magnitudo in mediocri diſtantia Solis a Terra, <lb></lb>neglectis differentiis quæ a curvatura Orbis magni majorique So<lb></lb>lis actione in Lunam falcatam & novam quam in gibboſam & <lb></lb>plenam, oriri poſſint. </s> <s>In aliis diſtantiis Solis a Terra, Variatio <lb></lb>maxima eſt in ratione quæ componitur ex duplicata ratione tem<lb></lb>poris revolutionis Synodicæ Lunaris (dato anni tempore) directe, <lb></lb>& triplicata ratione diſtantiæ Solis a Terra inverſe. </s> <s>IdeoQ.E.I. <pb xlink:href="039/01/431.jpg" pagenum="403"></pb>Apogæo Solis, Variatio maxima eſt 33′. </s> <s>14″, & in ejus Perigæo <lb></lb><arrow.to.target n="note432"></arrow.to.target>37′. </s> <s>11″, ſi modo Eccentricitas Solis ſit ad Orbis magni ſemidia<lb></lb>metrum tranſverſam ut (16 15/16) ad 1000. </s></p> <p type="margin"> <s><margin.target id="note432"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Hactenus Variationem inveſtigavimus in Orbe non eccentrico, <lb></lb>in quo utique Luna in Octantibus ſuis ſemper eſt in mediocri ſua <lb></lb>diſtantia a Terra. </s> <s>Si Luna propter eccentricitatem ſuam, magis <lb></lb>vel minus diſtat a Terra quam ſi locaretur in hoc Orbe, Variatio <lb></lb>paulo major eſſe poteſt vel paulo minor quam pro Regula hic <lb></lb>allata: ſed exceſſum vel defectum ab Aſtronomis per Phænomena <lb></lb>determinandum relinquo. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXX. PROBLEMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire motum borarium Nodorum Lunæ in Orbe circulari.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Deſignet <emph type="italics"></emph>S<emph.end type="italics"></emph.end>Solem, <emph type="italics"></emph>T<emph.end type="italics"></emph.end>Terram, <emph type="italics"></emph>P<emph.end type="italics"></emph.end>Lunam, <emph type="italics"></emph>NPn<emph.end type="italics"></emph.end>Orbem Lunæ, <lb></lb><emph type="italics"></emph>Npn<emph.end type="italics"></emph.end>veſtigium Orbis in plano Eclipticæ; <emph type="italics"></emph>N, n<emph.end type="italics"></emph.end>Nodos, <emph type="italics"></emph>nTNm<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.431.1.jpg" xlink:href="039/01/431/1.jpg"></figure><lb></lb>lineam Nodorum infinite productam; <emph type="italics"></emph>PI, PK<emph.end type="italics"></emph.end>perpendicula de<lb></lb>miſſa in lineas <emph type="italics"></emph>ST, <expan abbr="Qq;">Qque</expan> Pp<emph.end type="italics"></emph.end>perpendiculum demiſſum in planum <pb xlink:href="039/01/432.jpg" pagenum="404"></pb>Eclipticæ; <emph type="italics"></emph>Q, q<emph.end type="italics"></emph.end>Quadraturas Lunæ in plano Eclipticæ, & <emph type="italics"></emph>p K<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note433"></arrow.to.target>perpendiculum in lineam <emph type="italics"></emph>Qq<emph.end type="italics"></emph.end>Quadraturis interjacentem. </s> <s>Vis <lb></lb>Solis ad perturbandum motum Lunæ (per Prop.xxv.) duplex eſt, <lb></lb>altera lineæ <emph type="italics"></emph>LM,<emph.end type="italics"></emph.end>altera lineæ <emph type="italics"></emph>MT<emph.end type="italics"></emph.end>proportionalis. </s> <s>Et Luna vi <lb></lb>priore in Terram, poſteriore in Solem ſecundum lineam rectæ <emph type="italics"></emph>ST<emph.end type="italics"></emph.end><lb></lb>a Terra ad Solem ductæ parallelam trahitur. </s> <s>Vis prior <emph type="italics"></emph>LM<emph.end type="italics"></emph.end><lb></lb>agit ſecundum planum orbis Lunaris, & propterea ſitum plani nil <lb></lb>mutat. </s> <s>Hæc igitur negligenda eſt. </s> <s>Vis poſterior <emph type="italics"></emph>MT<emph.end type="italics"></emph.end>qua planum <lb></lb>Orbis Lunaris perturbatur eadem eſt cum vi 3<emph type="italics"></emph>PK<emph.end type="italics"></emph.end>vel 3<emph type="italics"></emph>IT.<emph.end type="italics"></emph.end><lb></lb>Et hæc vis (per Prop.xxv.) eſt ad vim qua Luna in circulo circa <lb></lb><figure id="id.039.01.432.1.jpg" xlink:href="039/01/432/1.jpg"></figure><lb></lb>Terram quicſcentem tempore ſuo periodico uniformiter revolvi <lb></lb>poſſet, ut 3<emph type="italics"></emph>IT<emph.end type="italics"></emph.end>ad Radium circuli multiplicatum per numerum <lb></lb>178,725, ſive ut <emph type="italics"></emph>IT<emph.end type="italics"></emph.end>ad Radium multiplicatum per 59,575. Cæte<lb></lb>rum in hoc calculo & eo omni qui ſequitur, conſidero lineas om<lb></lb>nes a Luna ad Solem ductas tanquam parallelas lineæ quæ a Terra <lb></lb>ad Solem ducitur, propterea quod inclinatio tantum fere minuit <lb></lb>effectus omnes in aliquibus caſibus, quantum auget in aliis; & <lb></lb>Nodorum motus mediocres quærimus, neglectis iſtiuſmodi minu<lb></lb>tiis, quæ calculum nimis impeditum redderent. </s></p><pb xlink:href="039/01/433.jpg" pagenum="405"></pb> <p type="margin"> <s><margin.target id="note433"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Deſignet jam <emph type="italics"></emph>PM<emph.end type="italics"></emph.end>arcum, quem Luna dato tempore quam <lb></lb><arrow.to.target n="note434"></arrow.to.target>minimo deſcribit, & <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>lineolam quam Luna, impellente vi <lb></lb>præfata 3<emph type="italics"></emph>IT,<emph.end type="italics"></emph.end>eodem tempore deſcribere poſſet. </s> <s>Jungantur <lb></lb><emph type="italics"></emph>PL, MP,<emph.end type="italics"></emph.end>& producantur eæ ad <emph type="italics"></emph>m<emph.end type="italics"></emph.end>& <emph type="italics"></emph>l,<emph.end type="italics"></emph.end>ubi ſecent planum E<lb></lb>clipticæ; inque <emph type="italics"></emph>Tm<emph.end type="italics"></emph.end>demittatur perpendiculum <emph type="italics"></emph>PH.<emph.end type="italics"></emph.end>Et quo<lb></lb>niam recta <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>parallela eſt plano Eclipticæ, ideoque cum recta <lb></lb><emph type="italics"></emph>ml<emph.end type="italics"></emph.end>quæ in plano illo jacet concurrere non poteſt, & tamen ja<lb></lb>cent hæ rectæ in plano communi <emph type="italics"></emph>LMP ml<emph.end type="italics"></emph.end>; parallelæ erunt hæ<lb></lb>rectæ, & propterea ſimilia erunt triangula <emph type="italics"></emph>LMP, Lmp.<emph.end type="italics"></emph.end>Jam <lb></lb>cum <emph type="italics"></emph>MPm<emph.end type="italics"></emph.end>ſit in plano Orbis, in quo Luna in loco <emph type="italics"></emph>P<emph.end type="italics"></emph.end>moveba<lb></lb>tur, incidet punctum <emph type="italics"></emph>m<emph.end type="italics"></emph.end>in lineam <emph type="italics"></emph>Nn<emph.end type="italics"></emph.end>per Orbis illius Nodos. <lb></lb><emph type="italics"></emph>N, n<emph.end type="italics"></emph.end>dictam. </s> <s>Et quoniam vis qua lineola <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>generatur, ſi <lb></lb>tota ſimul & ſemel in loco <emph type="italics"></emph>P<emph.end type="italics"></emph.end>impreſſa eſſet, efficeret ut Luna <lb></lb>moveretur in arcu, cujus chorda eſſet <emph type="italics"></emph>LP,<emph.end type="italics"></emph.end>atque adeo trans<lb></lb>ferret Lunam de plano <emph type="italics"></emph>MPmT<emph.end type="italics"></emph.end>in planum <emph type="italics"></emph>LPIT<emph.end type="italics"></emph.end>; motus an<lb></lb>gularis Nodorum a vi illa genitus, æqualis erit angulo <emph type="italics"></emph>mTl.<emph.end type="italics"></emph.end>Eſt <lb></lb>autem <emph type="italics"></emph>ml<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>mP<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MP,<emph.end type="italics"></emph.end>adeoque cum <emph type="italics"></emph>MP<emph.end type="italics"></emph.end>ob da<lb></lb>tum tempus data ſit, eſt <emph type="italics"></emph>ml<emph.end type="italics"></emph.end>ut rectangulum <emph type="italics"></emph>MLXmP,<emph.end type="italics"></emph.end>id eſt, <lb></lb>ut rectangulum <emph type="italics"></emph>ITXmP.<emph.end type="italics"></emph.end>Et angulus <emph type="italics"></emph>mTl,<emph.end type="italics"></emph.end>ſi modo angulus <lb></lb><emph type="italics"></emph>Tml<emph.end type="italics"></emph.end>rectus ſit, eſt ut (<emph type="italics"></emph>ml/Tm<emph.end type="italics"></emph.end>), & propterea ut (<emph type="italics"></emph>ITXPm/Tm<emph.end type="italics"></emph.end>), id eſt, <lb></lb>(ob proportionales <emph type="italics"></emph>Tm<emph.end type="italics"></emph.end>& <emph type="italics"></emph>mP, TP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>) ut (<emph type="italics"></emph>ITXPH/TP<emph.end type="italics"></emph.end>), <lb></lb>adeoque ob datam <emph type="italics"></emph>TP,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>ITXPH.<emph.end type="italics"></emph.end>Quod ſi angulus <emph type="italics"></emph>Tml,<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>STN<emph.end type="italics"></emph.end>obliquus fit, erit angulus <emph type="italics"></emph>mTl<emph.end type="italics"></emph.end>adhuc minor, in rati<lb></lb>one ſinus anguli <emph type="italics"></emph>STN<emph.end type="italics"></emph.end>ad Radium. </s> <s>Eſt igitur velocitas No<lb></lb>dorum ut <emph type="italics"></emph>ITXPHXAZ,<emph.end type="italics"></emph.end>ſive ut contentum ſub ſinubus trium <lb></lb>angulorum <emph type="italics"></emph>TPI, PTN<emph.end type="italics"></emph.end>& <emph type="italics"></emph>STN.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note434"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Si anguli illi, Nodis in Quadraturis & Luna in Syzygia exiſten<lb></lb>tibus, recti ſint, lineola <emph type="italics"></emph>ml<emph.end type="italics"></emph.end>abibit in infinitum, & angulus <emph type="italics"></emph>mTl<emph.end type="italics"></emph.end><lb></lb>evadet angulo <emph type="italics"></emph>mPl<emph.end type="italics"></emph.end>æqualis. </s> <s>Hoc autem in caſu, angulus <emph type="italics"></emph>mPl<emph.end type="italics"></emph.end><lb></lb>eſt ad angulum <emph type="italics"></emph>PTM,<emph.end type="italics"></emph.end>quem Luna eodem tempore motu ſuo <lb></lb>apparente circa Terram deſcribit ut 1 ad 59,575. Nam angulus <lb></lb><emph type="italics"></emph>mPl<emph.end type="italics"></emph.end>æqualis eſt angulo <emph type="italics"></emph>LPM,<emph.end type="italics"></emph.end>id eſt, angulo deflexionis Lunæ <lb></lb>a recto tramite, quem ſola vis præfata Solaris 3<emph type="italics"></emph>IT<emph.end type="italics"></emph.end>ſi tum ceſſa<lb></lb>ret Lunæ gravitas dato illo tempore generare poſſet; & angulus <lb></lb><emph type="italics"></emph>PTM<emph.end type="italics"></emph.end>æqualis eſt angulo deflexionis Lunæ a recto tramite, quem <lb></lb>vis illa, qua Luna in Orbe ſuo retinetur, ſi tum ceſſaret vis Sola<lb></lb>ris 3<emph type="italics"></emph>IT<emph.end type="italics"></emph.end>eodem tempore generaret. </s> <s>Et hæ vires, ut ſupra dixi-<pb xlink:href="039/01/434.jpg" pagenum="406"></pb>mus, ſunt ad invicem ut 1 ad 59,575. Ergo cum motus medius <lb></lb><arrow.to.target n="note435"></arrow.to.target>horarius Lunæ (reſpectu fixarum) ſit 32′. </s> <s>56″. </s> <s>27′. </s> <s>12<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>1/2, motus <lb></lb>horarius Nodi in hoc caſu erit 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>12<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>Aliis autem in <lb></lb>caſibus motus iſte horarius erit ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>12<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ut conten<lb></lb>tum ſub ſinubus angulorum trium <emph type="italics"></emph>TPI, PTN,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>STN<emph.end type="italics"></emph.end>(ſeu <lb></lb>diſtantiarum Lunæ a Quadratura, Lunæ a Nodo, & Nodi a Sole) <lb></lb>ad cubum Radii. </s> <s>Et quoties ſignum anguli alicujus de affirmativo <lb></lb>in negativum, deque negativo in affirmativum mutatur, debebit <lb></lb>motus regreſſivus in progreſſivum & progreſſivus in regreſſivum <lb></lb>mutari. </s> <s>Unde fit ut Nodi progrediantur quoties Luna inter Qua<lb></lb>draturam alterutram & Nodum Quadraturæ proximum verſatur. </s> <s><lb></lb>Aliis in caſibus regrediuntur, & per exceſſum regreſſus ſupra pro<lb></lb>greſſum, ſingulis menſibus ſeruntur in antecedentia. </s></p> <p type="margin"> <s><margin.target id="note435"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi a dati arcus quam minimi <emph type="italics"></emph>PM<emph.end type="italics"></emph.end>terminis <emph type="italics"></emph>P<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>M<emph.end type="italics"></emph.end>ad lineam Quadraturas jungentem <emph type="italics"></emph>Qq<emph.end type="italics"></emph.end>demittantur perpen<lb></lb>dicula <emph type="italics"></emph>PK, Mk,<emph.end type="italics"></emph.end>eademque producantur donec ſecent lineam <lb></lb>Nodorum <emph type="italics"></emph>Nn<emph.end type="italics"></emph.end>in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>d<emph.end type="italics"></emph.end>; erit motus horarius Nodorum ut area <lb></lb><emph type="italics"></emph>MPDd<emph.end type="italics"></emph.end>& quadratum lineæ <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>conjunctim. </s> <s>Sunto enim <lb></lb><figure id="id.039.01.434.1.jpg" xlink:href="039/01/434/1.jpg"></figure><lb></lb><emph type="italics"></emph>PK, PH<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>prædicti tres ſinus. </s> <s>Nempe <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>ſinus di<lb></lb>ſtantiæ Lunæ a Quadratura, <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>ſinus diſtantiæ Lunæ a Nodo, & <lb></lb><emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>ſinus diſtantiæ Nodi a Sole: & erit velocitas Nodi ut conten<lb></lb>tum <emph type="italics"></emph>PKXPHXAZ.<emph.end type="italics"></emph.end>Eſt autem <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>PM<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Kk,<emph.end type="italics"></emph.end><lb></lb>adeoque ob datas <emph type="italics"></emph>PT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PM<emph.end type="italics"></emph.end>eſt <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>proportionalis. </s> <s><lb></lb>Eſt & <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PH,<emph.end type="italics"></emph.end>& propterea <emph type="italics"></emph>PH<emph.end type="italics"></emph.end>rectangulo <pb xlink:href="039/01/435.jpg" pagenum="407"></pb><emph type="italics"></emph>PDXAZ<emph.end type="italics"></emph.end>proportionalis, & conjunctis rationibus, <emph type="italics"></emph>PKXPH<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note436"></arrow.to.target>eſt ut contentum <emph type="italics"></emph>KkXPDXAZ,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>PKXPHXAZ<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>KkXPDXAZ qu.<emph.end type="italics"></emph.end>id eſt, ut area <emph type="italics"></emph>PDdM<emph.end type="italics"></emph.end>& <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>con<lb></lb>junctim. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note436"></margin.target>LIBER <lb></lb>TIRTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol<emph.end type="italics"></emph.end>2. In data quavis Nodorum poſitione, motus horarius <lb></lb>mediocris eſt ſemiſſis motus horarii in Syzygiis Lunæ, ideoque eſt <lb></lb>ad 16″. </s> <s>35′. </s> <s>16<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>36<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ut quadratum ſinus diſtantiæ Nodorum a <lb></lb>Syzygiis ad quadratum Radii, five ut <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>AD <emph type="italics"></emph>AT.qu.<emph.end type="italics"></emph.end>Nam <lb></lb>ſi Luna uniformi cum motu perambulet ſemicirculum <emph type="italics"></emph>QAq,<emph.end type="italics"></emph.end>ſum<lb></lb>ma omnium arearum <emph type="italics"></emph>PDdM,<emph.end type="italics"></emph.end>quo tempore Luna pergit a <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>M,<emph.end type="italics"></emph.end>erit area <emph type="italics"></emph>QMdE<emph.end type="italics"></emph.end>quæ ad circuli tangentem <emph type="italics"></emph>QE<emph.end type="italics"></emph.end>termina<lb></lb>tur; & quo tempore Luna attingit punctum <emph type="italics"></emph>n,<emph.end type="italics"></emph.end>ſumma illa erit <lb></lb>area tota <emph type="italics"></emph>EQAn<emph.end type="italics"></emph.end>quam linea <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>deſcribit, dein Luna pergente <lb></lb>ab <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>q,<emph.end type="italics"></emph.end>linea <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>cadet extra circulum, & aream <emph type="italics"></emph>nqe<emph.end type="italics"></emph.end>ad <lb></lb>circuli tangentem <emph type="italics"></emph>qe<emph.end type="italics"></emph.end>terminatam deſcribet; quæ, quoniam Nodi <lb></lb>prius regrediebantur, jam vero progrediuntur, ſubduci debet de <lb></lb>area priore, & cum æqualis ſit areæ <emph type="italics"></emph>QEN,<emph.end type="italics"></emph.end>relinquet ſemicircu<lb></lb>lum <emph type="italics"></emph>NQAn.<emph.end type="italics"></emph.end>Igitur ſumma omnium arearum <emph type="italics"></emph>PDdM,<emph.end type="italics"></emph.end>quo <lb></lb>tempore Luna ſemicirculum deſcribit, eſt area ſemicirculi; & <lb></lb>ſumma omnium quo tempore Luna circulum deſcribit eſt area cir<lb></lb>culi totius. </s> <s>At area <emph type="italics"></emph>PDdM,<emph.end type="italics"></emph.end>ubi Luna verſatur in Syzygiis, eſt <lb></lb>rectangulum ſub arcu <emph type="italics"></emph>PM<emph.end type="italics"></emph.end>& radic <emph type="italics"></emph>MT<emph.end type="italics"></emph.end>; & ſumma omnium huic <lb></lb>æqualium arearum, quo tempore Luna circulum deſcribit, eſt <lb></lb>rectangulum ſub circumferentia tota & radio circuli; & hoc <lb></lb>rectangulum, cum ſit æquale duobus circulis, duplo majus eſt <lb></lb>quam rectangulum prius. </s> <s>Proinde Nodi, ea cum velocitate uNI<lb></lb>formiter continuata quam habent in Syzygiis Lunaribus, ſpatium <lb></lb>duplo majus deſcriberent quam revera deſcribunt; & propterea <lb></lb>motus mediocris quocum, ſi uniformiter continuaretur, ſpatium <lb></lb>a ſe inæquabili cum motu revera confectum deſcribere poſſent, eſt <lb></lb>ſemiſſis motus quem habent in Syzygiis Lunæ. </s> <s>Unde cum mo<lb></lb>tus horarius maximus, ſi Nodi in Quadraturis verſantur, ſit <lb></lb>33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>12<emph type="sup"></emph>v<emph.end type="sup"></emph.end>, motus mediocris horarius in hoc caſu erit <lb></lb>16″. </s> <s>35′. </s> <s>16<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>36<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>Et cum motus horarius Nodorum ſemper ſit <lb></lb>ut <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>& area <emph type="italics"></emph>PDdM<emph.end type="italics"></emph.end>conjunctim, & propterea motus ho<lb></lb>rarius Nodorum in Syzygiis Lunæ ut <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>& area <emph type="italics"></emph>PDdM<emph.end type="italics"></emph.end><lb></lb>conjunctim, id eſt (ob datam aream <emph type="italics"></emph>PDdM<emph.end type="italics"></emph.end>in Syzygiis de<lb></lb>ſcriptam) ut <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>erit etiam motus mediocris ut <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>atque <lb></lb>adeo hic motus, ubi Nodi extra Quadraturas verſantur, erit ad <lb></lb>16″. </s> <s>35′. </s> <s>16<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>36<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ut <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATqu. </s> <s>Q.E.D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/436.jpg" pagenum="408"></pb><arrow.to.target n="note437"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note437"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXI. PROBLEMA XII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire motum horarium Nodorum Lunæ in Orbe Elliptico.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Deſignet <emph type="italics"></emph>Qpmaq<emph.end type="italics"></emph.end>Ellipſin, axe majore <emph type="italics"></emph>Qq,<emph.end type="italics"></emph.end>minore <emph type="italics"></emph>ab<emph.end type="italics"></emph.end>de<lb></lb>ſcriptam, <emph type="italics"></emph>QAq<emph.end type="italics"></emph.end>Circulum circumſcriptum, <emph type="italics"></emph>T<emph.end type="italics"></emph.end>Terram in utriuſque <lb></lb>centro communi, <emph type="italics"></emph>S<emph.end type="italics"></emph.end>Solem, <emph type="italics"></emph>p<emph.end type="italics"></emph.end>Lunam in Ellipſi motam, & <emph type="italics"></emph>pm<emph.end type="italics"></emph.end>ar<lb></lb>cum quem data temporis particula quam minima deſcribit, <emph type="italics"></emph>N<emph.end type="italics"></emph.end>& <emph type="italics"></emph>n<emph.end type="italics"></emph.end><lb></lb>Nodos linea <emph type="italics"></emph>Nn<emph.end type="italics"></emph.end>junctos, <emph type="italics"></emph>pK<emph.end type="italics"></emph.end>& <emph type="italics"></emph>mk<emph.end type="italics"></emph.end>perpendicula in axem <emph type="italics"></emph>Qq<emph.end type="italics"></emph.end><lb></lb>demiſſa & hinc inde producta, donec occurrant Circulo in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>& <emph type="italics"></emph>M,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.436.1.jpg" xlink:href="039/01/436/1.jpg"></figure><lb></lb>& lineæ Nodorum in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>& <emph type="italics"></emph>d.<emph.end type="italics"></emph.end>Et ſi Luna, radio ad Terram du<lb></lb>cto, aream deſcribat tempori proportionalem, erit motus Nodi in <lb></lb>Ellipſi ut area <emph type="italics"></emph>pDdm.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam ſi <emph type="italics"></emph>PF<emph.end type="italics"></emph.end>tangat Circulum in <emph type="italics"></emph>P,<emph.end type="italics"></emph.end>& producta occurrat <emph type="italics"></emph>TN<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>pf<emph.end type="italics"></emph.end>tangat Ellipſin in <emph type="italics"></emph>p<emph.end type="italics"></emph.end>& producta occurrat eidem <emph type="italics"></emph>TN<emph.end type="italics"></emph.end><pb xlink:href="039/01/437.jpg" pagenum="409"></pb>in <emph type="italics"></emph>f,<emph.end type="italics"></emph.end>conveniant autem hæ tangentes in axe <emph type="italics"></emph>TQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Y<emph.end type="italics"></emph.end>; & ſi <lb></lb><arrow.to.target n="note438"></arrow.to.target><emph type="italics"></emph>ML<emph.end type="italics"></emph.end>deſignet ſpatium quod Luna in Circulo revolvens, interea <lb></lb>dum deſcribit arcum <emph type="italics"></emph>PM,<emph.end type="italics"></emph.end>urgente & impellente vi prædicta <lb></lb>3<emph type="italics"></emph>IT,<emph.end type="italics"></emph.end>motu tranſverſo deſcribere poſſet, & <emph type="italics"></emph>ml<emph.end type="italics"></emph.end>deſignet ſpatium <lb></lb>quod Luna in Ellipſi revolvens eodem tempore, urgente etiam vi <lb></lb>3<emph type="italics"></emph>IT,<emph.end type="italics"></emph.end>deſcribere poſſet; & producantur <emph type="italics"></emph>LP<emph.end type="italics"></emph.end>& <emph type="italics"></emph>lp<emph.end type="italics"></emph.end>donec occurrant <lb></lb>plano Eclipticæ in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>g<emph.end type="italics"></emph.end>; & jungantur <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>& <emph type="italics"></emph>fg,<emph.end type="italics"></emph.end>quarum <emph type="italics"></emph>FG<emph.end type="italics"></emph.end><lb></lb>producta ſecet <emph type="italics"></emph>pf, pg<emph.end type="italics"></emph.end>& <emph type="italics"></emph>TQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>c, e<emph.end type="italics"></emph.end>& <emph type="italics"></emph>R<emph.end type="italics"></emph.end>reſpective, & <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>pro<lb></lb>ducta ſecet <emph type="italics"></emph>TQ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>r<emph.end type="italics"></emph.end>: Quoniam vis 3<emph type="italics"></emph>IT<emph.end type="italics"></emph.end>ſeu 3<emph type="italics"></emph>PK<emph.end type="italics"></emph.end>in Circulo <lb></lb>eſt ad vim 3<emph type="italics"></emph>IT<emph.end type="italics"></emph.end>ſeu 3<emph type="italics"></emph>pK<emph.end type="italics"></emph.end>in Ellipſi, ut <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>pK,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>aT<emph.end type="italics"></emph.end>; erit ſpatium <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>vi priore genitum, ad ſpatium <emph type="italics"></emph>ml<emph.end type="italics"></emph.end>vi po<lb></lb>ſteriore genitum, ut <emph type="italics"></emph>PK<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>pK,<emph.end type="italics"></emph.end>id eſt, ob ſimiles figuras <lb></lb><emph type="italics"></emph>PYKp<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FYRc,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>FR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>cR.<emph.end type="italics"></emph.end>Eſt autem <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>(ob <lb></lb>ſimilia triangula <emph type="italics"></emph>PLM, PGF<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>PL<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PG,<emph.end type="italics"></emph.end>hoc eſt (ob <lb></lb>parallelas <emph type="italics"></emph>Lk, PK, GR<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>pl<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>pe,<emph.end type="italics"></emph.end>id eſt, (ob ſimilia trian<lb></lb>gula <emph type="italics"></emph>plm, cpe<emph.end type="italics"></emph.end>) ut <emph type="italics"></emph>lm<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ce<emph.end type="italics"></emph.end>; & inverſe ut <emph type="italics"></emph>LM<emph.end type="italics"></emph.end>eſt ad <emph type="italics"></emph>lm,<emph.end type="italics"></emph.end>ſeu <lb></lb><emph type="italics"></emph>FR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>cR,<emph.end type="italics"></emph.end>ita eſt <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ce.<emph.end type="italics"></emph.end>Et propterea ſi <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>eſſet ad <emph type="italics"></emph>ce<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>fY<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>cY,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>fr<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>cR<emph.end type="italics"></emph.end>(hoc eſt, ut <emph type="italics"></emph>fr<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FR<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FR<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>cR<emph.end type="italics"></emph.end><lb></lb>conjunctim, id eſt, ut <emph type="italics"></emph>fT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FT<emph.end type="italics"></emph.end>& <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ce<emph.end type="italics"></emph.end>conjunctim,) quo<lb></lb>niam ratio <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ce<emph.end type="italics"></emph.end>utrinque ablata relinquit rationes <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FG<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>fT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FT,<emph.end type="italics"></emph.end>foret <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>fT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FT<emph.end type="italics"></emph.end>; atque adeo anguli, <lb></lb>quos <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>& <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>ſubtenderent ad Terram <emph type="italics"></emph>T,<emph.end type="italics"></emph.end>æquarentur inter ſe. </s> <s><lb></lb>Sed anguli illi (per ca quæ in præcedente Propoſitione expoſui<lb></lb>mus) ſunt motus Nodorum, quo tempore Luna in Circulo ar<lb></lb>cum <emph type="italics"></emph>PM,<emph.end type="italics"></emph.end>in Ellipſi arcum <emph type="italics"></emph>pm<emph.end type="italics"></emph.end>percurrit: & propterea motus <lb></lb>Nodorum in Circulo & Ellipſi æquarentur inter ſe. </s> <s>Hæc ita ſe <lb></lb>haberent, ſi modo <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>eſſet ad <emph type="italics"></emph>ce<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>fY<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>cY,<emph.end type="italics"></emph.end>id eſt, ſi <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>æqua<lb></lb>lis eſſet (<emph type="italics"></emph>ceXfY/cY<emph.end type="italics"></emph.end>). Verum ob ſimilia triangula <emph type="italics"></emph>fgp, cep,<emph.end type="italics"></emph.end>eſt <emph type="italics"></emph>fg<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>ce<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>fp<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>cp<emph.end type="italics"></emph.end>; ideoque <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>æqualis eſt (<emph type="italics"></emph>ceXfp/cp<emph.end type="italics"></emph.end>); & propterea <lb></lb>angulus, quem <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>revera ſubtendit, eſt ad angulum priorem, quem <lb></lb><emph type="italics"></emph>FG<emph.end type="italics"></emph.end>ſubtendit, hoc eſt, motus Nodorum in Ellipſi ad motum <lb></lb>Nodorum in Circulo, ut hæc <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>ſeu (<emph type="italics"></emph>ceXfp/cp<emph.end type="italics"></emph.end>) ad priorem <emph type="italics"></emph>fg<emph.end type="italics"></emph.end>ſeu <lb></lb>(<emph type="italics"></emph>ceXfY/cY<emph.end type="italics"></emph.end>), id eſt, ut <emph type="italics"></emph>fpXcY<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>fYXcp,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>fp<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>fY<emph.end type="italics"></emph.end>& <emph type="italics"></emph>cY<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>cp,<emph.end type="italics"></emph.end><lb></lb>hoc eſt, ſi <emph type="italics"></emph>ph<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>TN<emph.end type="italics"></emph.end>parallela occurrat <emph type="italics"></emph>FP<emph.end type="italics"></emph.end>in <emph type="italics"></emph>h,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Fh<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FY<emph.end type="italics"></emph.end><lb></lb>& <emph type="italics"></emph>FY<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FP<emph.end type="italics"></emph.end>; hoc eſt, ut <emph type="italics"></emph>Fh<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>FP<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>Dp<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DP,<emph.end type="italics"></emph.end>adeoque <lb></lb>ut area <emph type="italics"></emph>Dpmd<emph.end type="italics"></emph.end>ad aream <emph type="italics"></emph>DPMd.<emph.end type="italics"></emph.end>Et propterea, cum area po-<pb xlink:href="039/01/438.jpg" pagenum="410"></pb><arrow.to.target n="note439"></arrow.to.target>ſterior proportionalis ſit motui Nodorum in Circulo, erit area <lb></lb>prior proportionalis motui Nodorum in Ellipſi. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note438"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note439"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Igitur cum, in data Nodorum poſitione, ſumma omnium <lb></lb>arearum <emph type="italics"></emph>pDdm,<emph.end type="italics"></emph.end>quo tempore Luna pergit a Quadratura ad lo<lb></lb>cum quemvis <emph type="italics"></emph>m,<emph.end type="italics"></emph.end>ſit area <emph type="italics"></emph>mpQEd,<emph.end type="italics"></emph.end>quæ ad Ellipſeos tangentem <lb></lb><emph type="italics"></emph>QE<emph.end type="italics"></emph.end>terminatur; & ſumma omnium arearum illarum, in revolu<lb></lb>tione integra, ſit area Ellipſeos totius: motus mediocris Nodorum <lb></lb>in Ellipſi erit ad motum mediocrem Nodorum in Circulo, ut El<lb></lb>lipſis ad Circulum; id eſt, ut <emph type="italics"></emph>Ta<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>TA,<emph.end type="italics"></emph.end>ſeu 69 ad 70. Et <lb></lb>propterea, cum motus mediocris horarius Nodorum in Circulo <lb></lb>ſit ad 16″. </s> <s>35′. </s> <s>16<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>36<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ut <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATqu.<emph.end type="italics"></emph.end>ſi capiatur angu<lb></lb>lus 16″. </s> <s>21′. </s> <s>3<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>30<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ad angulum 16″. </s> <s>35′. </s> <s>16<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>36<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ut 69 ad 70, <lb></lb>erit motus mediocris horarius Nodorum in Ellipſi ad 16″. </s> <s>21′. </s> <s><lb></lb>3<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>30<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ut <emph type="italics"></emph>AZq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATq<emph.end type="italics"></emph.end>; hoc eſt, ut quadratum ſinus diſtantiæ <lb></lb>Nodi a Sole ad quadratum Radii. </s></p> <p type="main"> <s>Cæterum Luna, radio ad Terram ducto, aream velocius deſcri<lb></lb>bit in Syzygiis quam in Quadraturis, & eo nomine tempus in Sy<lb></lb>zygiis contrahitur, in Quadraturis producitur; & una cum tem<lb></lb>pore motus Nodorum augetur ac diminuitur. </s> <s>Erat autem mo<lb></lb>mentum areæ in Quadraturis Lunæ ad ejus momentum in Syzygiis <lb></lb>ut 10973 ad 11073, & propterea momentum mediocre in Octan<lb></lb>tibus eſt ad exceſſum in Syzygiis, defectumQ.E.I. Quadraturis, ut <lb></lb>numerorum ſemiſumma 11023 ad eorundem ſemidifferentiam 50. <lb></lb>Unde cum tempus Lunæ in ſingulis Orbis particulis æqualibus ſit <lb></lb>reciproce ut ipſius velocitas, erit tempus mediocre in Octantibus <lb></lb>ad exceſſum temporis in Quadraturis, ac defectum in Syzygiis, ab <lb></lb>hac cauſa oriundum, ut 11023 ad 50 quam proxime. </s> <s>Pergendo <lb></lb>autem a Quadraturis ad Syzygias, invenio quod exceſſus momen<lb></lb>torum areæ in locis ſingulis, ſupra momentum minimum in Qua<lb></lb>draturis, ſit ut quadratum ſinus diſtantiæ Lunæ a Quadraturis <lb></lb>quam proxime; & propterea differentia inter momentum in loco <lb></lb>quocunque & momentum mediocre in Octantibus, eſt ut diffe<lb></lb>rentia inter quadratum ſinus diſtantiæ Lunæ a Quadraturis & <lb></lb>quadratum ſinus graduum 45, ſeu ſemiſſem quadrati Radii; & <lb></lb>incrementum temporis in locis ſingulis inter Octantes & Quadra<lb></lb>turas, & decrementum ejus inter Octantes & Syzygias, eſt in ea<lb></lb>dem ratione. </s> <s>Motus autem Nodorum, quo tempore Luna per<lb></lb>currit ſingulas Orbis particulas æquales, acceleratur vel retardatur <lb></lb>in duplicata ratione temporis. </s> <s>Eſt enim motus iſte, dum Luna <pb xlink:href="039/01/439.jpg" pagenum="411"></pb>percurrit <emph type="italics"></emph>PM,<emph.end type="italics"></emph.end>(cæteris paribus) ut <emph type="italics"></emph>ML,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>eſt in dupli<lb></lb><arrow.to.target n="note440"></arrow.to.target>cata ratione temporis. </s> <s>Quare motus Nodorum in Syzygiis, eo <lb></lb>tempore confectus quo Luna datas Orbis particulas percurrit, di<lb></lb>minuitur in duplicata ratione numeri 11073 ad numerum 11023; <lb></lb>eſtQ.E.D.crementum ad motum reliquum ut 100 ad 10973, ad <lb></lb>motum vero totum ut 100 ad 11073 quam proxime. </s> <s>Decre<lb></lb>mentum autem in locis inter Octantes & Syzygias, & incremen<lb></lb>tum in locis inter Octantes & Quadraturas, eſt quam proxime ad <lb></lb>hoc decrementum, ut motus totus in locis illis ad motum totum <lb></lb>in Syzygiis & differentia inter quadratum ſinus diſtantiæ Lunæ a <lb></lb>Quadratura & ſemiſſem quadrati Radii ad ſemiſſem quadrati Ra<lb></lb>dii, conjunctim. </s> <s>Unde ſi Nodi in Quadraturis verſentur, & ca<lb></lb>piantur loca duo æqualiter ab Octante hinc inde diſtantia, & alia <lb></lb>duo a Syzygia & Quadratura iiſdem intervallis diſtantia, deque <lb></lb>decrementis motuum in locis duobus inter Syzygiam & Octantem, <lb></lb>ſubducantur incrementa motuum in locis reliquis duobus, quæ <lb></lb>ſunt inter Octantem & Quadraturam; decrementum reliquum <lb></lb>æquale erit decremento in Syzygia: uti rationem ineunti facile <lb></lb>conſtabit. </s> <s>ProindeQ.E.D.crementum mediocre, quod de Nodo<lb></lb>rum motu mediocri ſubduci debet, eſt pars quarta decrementi in <lb></lb>Syzygia. </s> <s>Motus totus horarius Nodorum in Syzygiis (ubi Luna <lb></lb>radio ad Terram ducto aream tempori proportionalem deſcribere <lb></lb>ſupponebatur) erat 32″. </s> <s>42′. </s> <s>7<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>Et decrementum motus Nodo<lb></lb>rum, quo tempore Luna jam velocior deſcribit idem ſpatium, <lb></lb>diximus eſſe ad hunc motum ut 100 ad 11073; adeoQ.E.D.cre<lb></lb>mentum illud eſt 17′. </s> <s>43<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>11<emph type="sup"></emph>v<emph.end type="sup"></emph.end>, cujus pars quarta 4′. </s> <s>25<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>48<emph type="sup"></emph>v<emph.end type="sup"></emph.end>, <lb></lb>motui horario mediocri ſuperius invento 16″. </s> <s>21′. </s> <s>3<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>30<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ſub<lb></lb>ducta, relinquit 16″. </s> <s>16′. </s> <s>37<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>42<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>motum mediocrem horarium <lb></lb>correctum. </s></p> <p type="margin"> <s><margin.target id="note440"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Si Nodi verſantur extra Quadraturas, & ſpectentur loca bina a <lb></lb>Syzygiis hinc inde æqualiter diſtantia; ſumma motuum Nodo<lb></lb>rum, ubi Luna verſatur in his locis, erit ad ſummam motuum, <lb></lb>ubi Luna in iiſdem locis & Nodi in Quadraturis verſantur, ut <lb></lb><emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATqu.<emph.end type="italics"></emph.end>Et decrementa motuum, a cauſis jam expo<lb></lb>ſitis oriunda, erunt ad invicem ut ipſi motus, adeoque motus reli<lb></lb>qui erunt ad invicem ut <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATqu.<emph.end type="italics"></emph.end>& motus mediocres <lb></lb>ut motus reliqui. </s> <s>Eſt itaque motus mediocris horarius correctus, <lb></lb>in dato quocunque Nodorum ſitu, ad 16″. </s> <s>16′. </s> <s>37<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>42<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>ut. <emph type="italics"></emph>AZqu.<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>ATqu.<emph.end type="italics"></emph.end>; id eſt, ut quadratum ſinus diſtantiæ Nodorum a Sy<lb></lb>zygiis ad quadratum Radii. <pb xlink:href="039/01/440.jpg" pagenum="412"></pb><arrow.to.target n="note441"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note441"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXII. PROBLEMA XIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire motum medium Nodorum Lunæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Motus medius annuus eſt ſumma motuum omnium horariorum <lb></lb>mediocrium in anno. </s> <s>Concipe Nodum verſari in <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>& ſingulis <lb></lb>horis completis retrahi in locum ſuum priorem, ut non obſtante <lb></lb>motu ſuo proprio, datum ſemper ſervet ſitum ad Stellas Fixas. </s> <s><lb></lb>Interea vero Solem <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>per motum Terræ, progredi a Nodo, & <lb></lb>curſum annuum apparentem uniformiter complere. </s> <s>Sit autem <lb></lb><emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>arcus datus quam minimus, quem recta <emph type="italics"></emph>TS<emph.end type="italics"></emph.end>ad Solem ſemper <lb></lb>ducta, interſectione ſui & circuli <emph type="italics"></emph>NAn,<emph.end type="italics"></emph.end>dato tempore quam mi<lb></lb>nimo deſcribit: & motus horarius mediocris (per jam oſtenſa) <lb></lb>erit ut <emph type="italics"></emph>AZq,<emph.end type="italics"></emph.end>id eſt (ob proportionales <emph type="italics"></emph>AZ, ZY<emph.end type="italics"></emph.end>) ut rectan<lb></lb>gulum ſub <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>& <emph type="italics"></emph>ZY,<emph.end type="italics"></emph.end>hoc eſt, ut area <emph type="italics"></emph>AZYa.<emph.end type="italics"></emph.end>Et ſumma om<lb></lb>nium horariorum motuum mediocrium ab initio, ut ſumma om<lb></lb>nium arearum <emph type="italics"></emph>aYZA,<emph.end type="italics"></emph.end>id eſt, ut area <emph type="italics"></emph>NAZ.<emph.end type="italics"></emph.end>Eſt autem maxima <lb></lb><figure id="id.039.01.440.1.jpg" xlink:href="039/01/440/1.jpg"></figure><lb></lb><emph type="italics"></emph>AZYa<emph.end type="italics"></emph.end>æqualis rectangulo ſub arcu <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>& radio circuli; & prop<lb></lb>terea ſumma omnium rectangulorum in circulo toto ad ſummam <lb></lb>totidem maximorum, ut area circuli totius ad rectangulum ſub <lb></lb>circumferentia tota & radio; id eſt, ut 1 ad 2. Motus autem ho<lb></lb>rarius, rectangulo maximo reſpondens, erat 16″. </s> <s>16′. </s> <s>37<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>42<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>Et <lb></lb>hic motus, anno toto ſidereo dierum 365. <emph type="italics"></emph>hor.<emph.end type="italics"></emph.end>6. <emph type="italics"></emph>min.<emph.end type="italics"></emph.end>9 fit <lb></lb>39<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 38′. </s> <s>7″. </s> <s>50′. </s> <s>Ideoque hujus dimidium 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 49′. </s> <s>3″. </s> <s>55′. </s> <s>eſt mo-<pb xlink:href="039/01/441.jpg" pagenum="413"></pb>tus medius Nodorum circulo toti reſpondens. </s> <s>Et motus Nodo<lb></lb><arrow.to.target n="note442"></arrow.to.target>rum, quo tempore Sol pergit ab <emph type="italics"></emph>N<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>eſt ad 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 49′. </s> <s>3″. </s> <s>55′. </s> <s><lb></lb>ut area <emph type="italics"></emph>NAZ<emph.end type="italics"></emph.end>ad circulum totum. </s></p> <p type="margin"> <s><margin.target id="note442"></margin.target>LIBER. <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Hæc ita ſe habent, ex Hypotheſi quod Nodus horis ſingulis in <lb></lb>locum priorem retrahitur, lic ut Sol anno toto completo ad No<lb></lb>dum eundem redeat a quo ſub initio digreſſus fuerat. </s> <s>Verum per <lb></lb>motum Nodi fit ut Sol citius ad Nodum revertatur, & compu<lb></lb>tanda jam eſt abbreviatio temporis. </s> <s>Cum Sol anno toto conficiat <lb></lb>360 gradus, & Nodus motu maximo eodem tempore conficeret <lb></lb>39<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 38′. </s> <s>7″. </s> <s>50′, ſeu 39,6355 gradus; & motus mediocris. </s> <s>Nodi <lb></lb>in loco quovis <emph type="italics"></emph>N<emph.end type="italics"></emph.end>ſit ad ipſius motum mediocrem in Quadraturis <lb></lb>ſuis, ut <emph type="italics"></emph>AZq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATq<emph.end type="italics"></emph.end>: erit motus Solis ad motum Nodi in <emph type="italics"></emph>N,<emph.end type="italics"></emph.end>ut <lb></lb>360 <emph type="italics"></emph>ATq<emph.end type="italics"></emph.end>ad 39,6355 <emph type="italics"></emph>AZq<emph.end type="italics"></emph.end>; id eſt, ut 9,0827646 <emph type="italics"></emph>ATq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="AZq.">AZque</expan><emph.end type="italics"></emph.end><lb></lb>Unde ſi circuli totius circumferentia <emph type="italics"></emph>NAn<emph.end type="italics"></emph.end>dividatur in particu<lb></lb>las æquales <emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end>tempus quo Sol percurrat particulam <emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end>ſi cir<lb></lb>culus quieſceret, erit ad tempus quo percurrit eandem parti<lb></lb>culam, ſi circulus una cum Nodis circa centrum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>revolvatur, <lb></lb>reciproce ut 9,0827646 <emph type="italics"></emph><expan abbr="ATq.">ATque</expan><emph.end type="italics"></emph.end>ad 9,0827646 <emph type="italics"></emph><expan abbr="ATq+AZq.">ATq+AZque</expan><emph.end type="italics"></emph.end>Nam <lb></lb>tempus eſt reciproce ut velocitas qua particula percurritur, & <lb></lb>hæc velocitas eſt ſumma velocitatum Solis & Nodi. </s> <s>Igitur ſi tem<lb></lb>pus, quo Sol abſque motu Nodi percurreret arcum <emph type="italics"></emph>NA,<emph.end type="italics"></emph.end>expo<lb></lb>natur per Sectorem <emph type="italics"></emph>NTA,<emph.end type="italics"></emph.end>& particula temporis quo percurreret. </s> <s><lb></lb>arcum quam minimum <emph type="italics"></emph>Aa,<emph.end type="italics"></emph.end>exponatur per Sectoris particulam <lb></lb><emph type="italics"></emph>ATa<emph.end type="italics"></emph.end>; & (perpendiculo <emph type="italics"></emph>aY<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Nn<emph.end type="italics"></emph.end>demiſſo) ſi in <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>capiatur <lb></lb><emph type="italics"></emph>dZ,<emph.end type="italics"></emph.end>ejus longitudinis ut ſit rectangulum <emph type="italics"></emph>dZ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>ZY<emph.end type="italics"></emph.end>ad Sectoris <lb></lb>particulam <emph type="italics"></emph>ATa<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AZq<emph.end type="italics"></emph.end>ad 9,0827646 <emph type="italics"></emph>ATq+AZq,<emph.end type="italics"></emph.end>id eſt, ut <lb></lb>ſit <emph type="italics"></emph>dZ<emph.end type="italics"></emph.end>ad 1/2 <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>ATq<emph.end type="italics"></emph.end>ad 9,0827646 <emph type="italics"></emph>ATq+AZq<emph.end type="italics"></emph.end>; rectangu<lb></lb>lum <emph type="italics"></emph>dZ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>ZY<emph.end type="italics"></emph.end>deſignabit decrementum temporis ex motu Nodi <lb></lb>oriundum, tempore toto quo arcus <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>percurritur. </s> <s>Et ſi pun<lb></lb>ctum <emph type="italics"></emph>d<emph.end type="italics"></emph.end>tangit Curvam <emph type="italics"></emph>NdGn,<emph.end type="italics"></emph.end>area curvilinea <emph type="italics"></emph>NdZ<emph.end type="italics"></emph.end>erit decre<lb></lb>mentum totum, quo tempore arcus totus <emph type="italics"></emph>NA<emph.end type="italics"></emph.end>percurritur; & <lb></lb>propterea exceſſus Sectoris <emph type="italics"></emph>NAT<emph.end type="italics"></emph.end>ſupra aream <emph type="italics"></emph>NdZ<emph.end type="italics"></emph.end>erit tempus <lb></lb>illud totum. </s> <s>Et quoniam motus Nodi tempore minore minor eſt <lb></lb>in ratione temporis, debebit etiam area <emph type="italics"></emph>AaYZ<emph.end type="italics"></emph.end>diminui in eadem <lb></lb>ratione. </s> <s>Id quod fiet ſi capiatur in <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>longitudo <emph type="italics"></emph>eZ,<emph.end type="italics"></emph.end>quæ ſit <lb></lb>ad longitudinem <emph type="italics"></emph>AZ<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AZq<emph.end type="italics"></emph.end>ad 9,0827646 <emph type="italics"></emph><expan abbr="ATq+AZq.">ATq+AZque</expan><emph.end type="italics"></emph.end>Sic <lb></lb>enim rectangulum <emph type="italics"></emph>eZ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>ZY<emph.end type="italics"></emph.end>erit ad aream <emph type="italics"></emph>AZYa<emph.end type="italics"></emph.end>ut decremen<lb></lb>tum temporis quo arcus <emph type="italics"></emph>Aa<emph.end type="italics"></emph.end>percurritur, ad tempus totum quo <lb></lb>percurreretur ſi Nodus quieſceret: Et propterea rectangulum illud <lb></lb>reſpondebit decremento motus Nodi. </s> <s>Et ſi punctum <emph type="italics"></emph>e<emph.end type="italics"></emph.end>tangat <pb xlink:href="039/01/442.jpg" pagenum="414"></pb><arrow.to.target n="note443"></arrow.to.target>Curvam <emph type="italics"></emph>NeFn,<emph.end type="italics"></emph.end>area tota <emph type="italics"></emph>NeZ,<emph.end type="italics"></emph.end>quæ ſumma eſt omnium decre<lb></lb>mentorum, reſpondebit decremento toti, quo tempore arcus <emph type="italics"></emph>AN<emph.end type="italics"></emph.end><lb></lb>percurritur; & area reliqua <emph type="italics"></emph>NAe<emph.end type="italics"></emph.end>reſpondebit motui reliquo, qui <lb></lb>verus eſt Nodi motus quo tempore arcus totus <emph type="italics"></emph>NA,<emph.end type="italics"></emph.end>per Solis & <lb></lb>Nodi conjunctos motus, percurritur. </s> <s>Jam vero area ſemicirculi <lb></lb>eſt ad aream Figuræ <emph type="italics"></emph>NeFnT,<emph.end type="italics"></emph.end>per methodum Serierum infinita<lb></lb>rum quæſitam, ut 793 ad 60 quamproxime. </s> <s>Motus autem qui <lb></lb>reſpondet Circulo toti erat 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 49′. </s> <s>3″. </s> <s>55′; & propterea motus, <lb></lb>qui Figuræ <emph type="italics"></emph>NeFnT<emph.end type="italics"></emph.end>duplicatæ reſpondet, eſt 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 29′. </s> <s>58″. </s> <s>2′. </s> <s><lb></lb>Qui de motu priore ſubductus relinquit 18<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 19′. </s> <s>5″. </s> <s>53′. </s> <s>motum <lb></lb>totum Nodi inter ſui ipſius Conjunctiones cum Sole; & hic mo<lb></lb>tus de Solis motu annuo graduum 360 ſubductus, relinquit 341<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end><lb></lb>40′. </s> <s>54″. </s> <s>7′. </s> <s>motum Solis inter eaſdem Conjunctiones. </s> <s>Iſte au<lb></lb>tem motus eſt ad motum annuum 360<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> ut Nodi motus jam in<lb></lb>ventus 18<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 19′. </s> <s>5″. </s> <s>53′. </s> <s>ad ipſius motum annuum, qui propterea <lb></lb>erit 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 18′. </s> <s>1″. </s> <s>23′. </s> <s>Hic eſt motus medius Nodorum in anno <lb></lb>Sidereo. </s> <s>Idem per Tabulas Aſtronomicas eſt 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 21′. </s> <s>21″. </s> <s>50′. </s> <s><lb></lb>Differentia minor eſt parte trecenteſima motus totius, & ab Or<lb></lb>bis Lunaris Eccentricitate & Inclinatione ad planum Eclipticæ <lb></lb>oriri videtur. </s> <s>Per Eccentricitatem Orbis motus Nodorum nimis <lb></lb>acceleratur, & per ejus Inclinationem viciſſim retardatur aliquan<lb></lb>tulum, & ad juſtam velocitatem reducitur. </s></p> <p type="margin"> <s><margin.target id="note443"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXIII. PROBLEMA XIV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire motum verum Nodorum Lunæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>In tempore quod eſt ut area <emph type="italics"></emph>NTA-NdZ, (in Fig. </s> <s>præced.)<emph.end type="italics"></emph.end><lb></lb>motus iſte eſt ut area <emph type="italics"></emph>NAeN,<emph.end type="italics"></emph.end>& inde datur. </s> <s>Verum ob nimiam <lb></lb>calculi difficultatem, præſtat ſequentem Problematis conſtructio<lb></lb>nem adhibere. </s> <s>Centro <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>intervallo quovis <emph type="italics"></emph>CD,<emph.end type="italics"></emph.end>deſcribatur <lb></lb>circulus <emph type="italics"></emph>BEFD.<emph.end type="italics"></emph.end>Producatur <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC<emph.end type="italics"></emph.end><lb></lb>ut motus medius ad ſemiſſem motus veri mediocris, ubi Nodi <lb></lb>ſunt in Quadraturis, (id eſt, ut 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 18′. </s> <s>1″. </s> <s>23′. </s> <s>ad 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 49′. </s> <s><lb></lb>3″. </s> <s>55′, atque adeo <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ut motuum differentia 0<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 31′. </s> <s><lb></lb>2″. </s> <s>32′, ad motum poſteriorem 19′<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 49. 3″. </s> <s>55′) hoc eſt, ut <lb></lb>1 ad (38 1/10) dein per punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ducatur infinita <emph type="italics"></emph>Gg,<emph.end type="italics"></emph.end>quæ tangat <lb></lb>circulum in <emph type="italics"></emph>D<emph.end type="italics"></emph.end>; & ſi capiatur angulus <emph type="italics"></emph>BCE<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>BCF<emph.end type="italics"></emph.end>æqualis <lb></lb>duplæ diſtantiæ Solis a loco Nodi, per motum medium invento; <pb xlink:href="039/01/443.jpg" pagenum="415"></pb>& agatur <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>AF<emph.end type="italics"></emph.end>ſecans perpendiculum <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>in <emph type="italics"></emph>G<emph.end type="italics"></emph.end>; & ca<lb></lb><arrow.to.target n="note444"></arrow.to.target>piatur angulus qui ſit ad motum totum Nodi inter ipſius Syzy<lb></lb>gias (id eſt, ad 9<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 11′. </s> <s>3″.) ut tangens <emph type="italics"></emph>DG<emph.end type="italics"></emph.end>ad circuli <emph type="italics"></emph>BED<emph.end type="italics"></emph.end><lb></lb>circumferentiam totam; atque angulus iſte (pro quo angulus <emph type="italics"></emph>DAG<emph.end type="italics"></emph.end><lb></lb>uſurpari poteſt) ad motum medium Nodorum addatur ubi Nodi <lb></lb><figure id="id.039.01.443.1.jpg" xlink:href="039/01/443/1.jpg"></figure><lb></lb>tranſeunt a Quadraturis ad Syzygias, & ab eodem motu medio <lb></lb>ſubducatur ubi tranſeunt a Syzygiis ad Quadraturas; habebitur <lb></lb>eorum motus verus. </s> <s>Nam motus verus ſic inventus congruet <lb></lb>quam proxime cum motu vero qui prodit exponendo tempus per <lb></lb>aream <emph type="italics"></emph>NTA-NdZ,<emph.end type="italics"></emph.end>& motum Nodi per aream <emph type="italics"></emph>NAeN<emph.end type="italics"></emph.end>; ut <lb></lb>rem perpendenti & computationes inſtituenti conſtabit. </s> <s>Hæc eſt <lb></lb>æquatio ſemeſtris motus Nodorum. </s> <s>Eſt & æquatio menſtrua, ſed <lb></lb>quæ ad inventionem Latitudinis Lunæ minime neceſſaria eſt. </s> <s>Nam <lb></lb>cum Variatio Inclinationis Orbis Lunaris ad planum Eclipticæ du<lb></lb>plici inæqualitati obnoxia ſit, alteri ſemeſtri, alteri autem men<lb></lb>ſtruæ; &c. </s> <s>hujus menſtrua inæqualitas & æquatio menſtrua Nodorum <lb></lb>ita ſe mutuo contemperant & corrigunt, ut ambæ in determinan<lb></lb>da Latitudine Lunæ negligi poſſint. </s></p> <p type="margin"> <s><margin.target id="note444"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Ex hac & præcedente Propoſitione liquet quod Nodi in <lb></lb>Syzygiis ſuis quieſcunt, in Quadraturis autem regrediuntur motu <lb></lb>horario 16″. </s> <s>19′. </s> <s>26<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>Et quod æquatio motus Nodorum in <lb></lb>Octantibus ſit 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′. </s> <s>Quæ omnia cum Phænomenis cœleſtibus <lb></lb>probe quadrant. <pb xlink:href="039/01/444.jpg" pagenum="416"></pb><arrow.to.target n="note445"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note445"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXIV. PROBLEMA XV.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire Variationem horariam Inclinationis Orbis Lunaris ad <lb></lb>planum Eclipticæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Deſignent <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>a<emph.end type="italics"></emph.end>Syzygias; <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>& <emph type="italics"></emph>q<emph.end type="italics"></emph.end>Quadraturas; <emph type="italics"></emph>N<emph.end type="italics"></emph.end>& <emph type="italics"></emph>n<emph.end type="italics"></emph.end>No<lb></lb>dos; <emph type="italics"></emph>P<emph.end type="italics"></emph.end>locum Lunæ in Orbe ſuo; <emph type="italics"></emph>p<emph.end type="italics"></emph.end>veſtigium loci illius in plano <lb></lb>Eclipticæ, & <emph type="italics"></emph>mTl<emph.end type="italics"></emph.end>motum momentaneum Nodorum ut ſupra. </s> <s><lb></lb>Et ſi ad lineam <emph type="italics"></emph>Tm<emph.end type="italics"></emph.end>demittatur perpendiculum <emph type="italics"></emph>PG,<emph.end type="italics"></emph.end>jungatur <emph type="italics"></emph>pG,<emph.end type="italics"></emph.end><lb></lb>& producatur ea donec occurrat <emph type="italics"></emph>Tl<emph.end type="italics"></emph.end>in <emph type="italics"></emph>g,<emph.end type="italics"></emph.end>& jungatur etiam <emph type="italics"></emph>Pg<emph.end type="italics"></emph.end>: <lb></lb>erit angulus <emph type="italics"></emph>PGp<emph.end type="italics"></emph.end>Inclinatio orbis Lunaris ad planum Eclipticæ, <lb></lb><figure id="id.039.01.444.1.jpg" xlink:href="039/01/444/1.jpg"></figure><lb></lb>ubi Luna verſatur in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>; & angulus <emph type="italics"></emph>Pgp<emph.end type="italics"></emph.end>Inclinatio ejuſdem poſt <lb></lb>momentum temporis completum; adeoque angulus <emph type="italics"></emph>GPg<emph.end type="italics"></emph.end>Variatio <lb></lb>momentanea Inclinationis. </s> <s>Eſt autem hic angulus <emph type="italics"></emph>GPg<emph.end type="italics"></emph.end>ad an<lb></lb>gulum <emph type="italics"></emph>GTg,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>TG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PG<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Pp<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PG<emph.end type="italics"></emph.end>conjunctim. </s> <s>Et prop<lb></lb>terea ſi pro momento temporis ſubſtituatur hora; cum angulus <lb></lb><emph type="italics"></emph>GTg<emph.end type="italics"></emph.end>(per Propoſit. </s> <s>xxx.) ſit ad angulum 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>ut <pb xlink:href="039/01/445.jpg" pagenum="417"></pb><emph type="italics"></emph>ITXPGXAZ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATcub,<emph.end type="italics"></emph.end>erit angulus <emph type="italics"></emph>GPg<emph.end type="italics"></emph.end>(ſeu Inclinationis <lb></lb><arrow.to.target n="note446"></arrow.to.target>horaria Variatio) ad angulum 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>, ut <emph type="italics"></emph>ITXAZXTG <lb></lb>X(Pp/PG)<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATcub. </s> <s>q.EI.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note446"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Hæc ita ſe habent ex Hypotheſi quod Luna in Orbe Circulari <lb></lb>uniformiter gyratur. </s> <s>Quod ſi Orbis ille Ellipticus ſit, motus me<lb></lb>diocris Nodorum minuetur in ratione axis minoris ad axem majo<lb></lb>rem; uti ſupra expoſitum eſt. </s> <s>Et in eadem ratione minuetur <lb></lb>etiam Inclinationis Variatio. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Si ad <emph type="italics"></emph>Nn<emph.end type="italics"></emph.end>erigatur perpendiculum <emph type="italics"></emph>TF,<emph.end type="italics"></emph.end>ſitque <emph type="italics"></emph>pM<emph.end type="italics"></emph.end><lb></lb>motus horarius Lunæ in plano Eclipticæ; & perpendicula <emph type="italics"></emph>pK, Mk<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>QT<emph.end type="italics"></emph.end>demiſſa & utrinque producta occurrant <emph type="italics"></emph>TF<emph.end type="italics"></emph.end>in <emph type="italics"></emph>H<emph.end type="italics"></emph.end>& <emph type="italics"></emph>h<emph.end type="italics"></emph.end>: <lb></lb>erit <emph type="italics"></emph>IT<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AT<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>Kk<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Mp,<emph.end type="italics"></emph.end>& <emph type="italics"></emph>TG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Hp<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>TZ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AT;<emph.end type="italics"></emph.end><lb></lb>ideoque <emph type="italics"></emph>ITXTG<emph.end type="italics"></emph.end>æquale (<emph type="italics"></emph>KkXHpXTZ/Mp<emph.end type="italics"></emph.end>), hoc eſt, æquale areæ <lb></lb><emph type="italics"></emph>HpMh<emph.end type="italics"></emph.end>ductæ in rationem (<emph type="italics"></emph>TZ/Mp<emph.end type="italics"></emph.end>): & propterea Inclinationis Varia<lb></lb>tio horaria ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>, ut <emph type="italics"></emph>HpMh<emph.end type="italics"></emph.end>ducta in <emph type="italics"></emph>AZX(TZ/Mp)X(Pp/PG)<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>AT cub.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Ideoque ſi Terra & Nodi ſingulis horis completis re<lb></lb>traherentur à locis ſuis novis, & in loca priora in inſtanti ſemper <lb></lb>reducerentur, ut ſitus eorum, per menſem integrum periodicum, <lb></lb>datus maneret; tota Inclinationis Variatio tempore menſis illius <lb></lb>foret ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>, ut aggregatum omnium arearum <emph type="italics"></emph>Hp Mh,<emph.end type="italics"></emph.end><lb></lb>in revolutione puncti <emph type="italics"></emph>p<emph.end type="italics"></emph.end>genitarum, & ſub ſignis propriis + & <lb></lb>conjunctarum, ductum in <emph type="italics"></emph>AZXTZX(Pp/PG)<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MpXAT cub.<emph.end type="italics"></emph.end>id <lb></lb>eſt, ut circulus totus <emph type="italics"></emph>QAqa<emph.end type="italics"></emph.end>ductus in <emph type="italics"></emph>AZXTZX(Pp/PG)<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MpX <lb></lb>ATcub.<emph.end type="italics"></emph.end>hoc eſt, ut circumferentia <emph type="italics"></emph>QAqa<emph.end type="italics"></emph.end>ducta in <emph type="italics"></emph>AZXTZX(Pp/PG)<emph.end type="italics"></emph.end><lb></lb>ad 2 <emph type="italics"></emph>MpXAT quad.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Proinde in dato Nodorum ſitu, Variatio mediocris <lb></lb>horaria, ex qua per menſem uniformiter continuata Variatio illa <lb></lb>menſtrua generari poſſet, eſt ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>, ut <emph type="italics"></emph>AZXTZ <lb></lb>X(Pp/PG)<emph.end type="italics"></emph.end>ad 2 <emph type="italics"></emph>ATq,<emph.end type="italics"></emph.end>ſive ut <emph type="italics"></emph>PpX(AZXTZ/1/2AT)<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PGX4AT,<emph.end type="italics"></emph.end>id <pb xlink:href="039/01/446.jpg" pagenum="418"></pb><arrow.to.target n="note447"></arrow.to.target>eſt (cum <emph type="italics"></emph>Pp<emph.end type="italics"></emph.end>ſit ad <emph type="italics"></emph>PG<emph.end type="italics"></emph.end>ut ſinus Inclinationis prædictæ ad ra<lb></lb>dium, & (<emph type="italics"></emph>AZXTZ/1/2AT<emph.end type="italics"></emph.end>) ſit ad 4<emph type="italics"></emph>AT<emph.end type="italics"></emph.end>ut ſinus duplicati anguli <emph type="italics"></emph>ATn<emph.end type="italics"></emph.end><lb></lb>ad radium quadruplicatum) ut Inclinationis ejuſdem ſinus ductus <lb></lb>in ſinum duplicatæ diſtantiæ Nodorum a Sole, ad quadruplum <lb></lb>quadratum radii. </s></p> <p type="margin"> <s><margin.target id="note447"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Quoniam Inclinationis horaria Variatio, ubi Nodi in <lb></lb>Quadraturis verſantur, eſt (per hanc Propoſitionem) ad angu<lb></lb>lum 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end> ut <emph type="italics"></emph>ITXAZXTGX(Pp/PG)<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ATcub.<emph.end type="italics"></emph.end>id eſt, <lb></lb>ut <emph type="italics"></emph>(ITXTG/1/2AT)X(Pp/PG)<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>AT<emph.end type="italics"></emph.end>; hoc eſt, ut ſinus duplicatæ di<lb></lb>ſtantiæ Lunæ à Quadraturis ductus in (<emph type="italics"></emph>Pp/PG<emph.end type="italics"></emph.end>) ad radium duplica<lb></lb>tum: ſumma omnium Variationum horariarum, quo tempore <lb></lb>Luna in hoc ſitu Nodorum tranſit à Quadratura ad Syzygiam, <lb></lb>(id eſt, ſpatio horarum 177 1/6,) erit ad ſummam totidem angulo<lb></lb>rum 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>, ſeu 5878″, ut ſumma omnium ſinuum dupli<lb></lb>catæ diſtantiæ Lunæ à Quadraturis ducta in (<emph type="italics"></emph>Pp/PG<emph.end type="italics"></emph.end>) ad ſummam to<lb></lb>tidem diametrorum; hoc eſt, ut diameter ducta in (<emph type="italics"></emph>Pp/PG<emph.end type="italics"></emph.end>) ad cir<lb></lb>cumferentiam; id eſt, ſi Inclinatio ſit 5<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 1′, ut 7X(874/10000) ad 22, <lb></lb>ſeu 278 ad 10000. Proindeque Variatio tota, ex ſumma om<lb></lb>nium horariarum Variationum tempore prædicto conflata, eſt <lb></lb>163″, ſeu 2′. </s> <s>43″. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXV. PROBLEMA XVI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Dato tempore invenire Inclinationem Orbis Lunaris ad planum <lb></lb>Eclipticæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sit <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>ſinus Inclinationis maximæ, & <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ſinus Inclinatio<lb></lb>nis minimæ. </s> <s>Biſecetur <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>in <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>& centro <emph type="italics"></emph>C,<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>BC,<emph.end type="italics"></emph.end><lb></lb>deſcribatur Circulus <emph type="italics"></emph>BGD.<emph.end type="italics"></emph.end>In <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>in ea ratione <lb></lb>ad <emph type="italics"></emph>EB<emph.end type="italics"></emph.end>quam <emph type="italics"></emph>EB<emph.end type="italics"></emph.end>habet ad 2<emph type="italics"></emph>BA:<emph.end type="italics"></emph.end>Et ſi dato tempore conſti<lb></lb>tuatur angulus <emph type="italics"></emph>AEG<emph.end type="italics"></emph.end>æqualis duplicatæ diſtantiæ Nodorum à <pb xlink:href="039/01/447.jpg" pagenum="419"></pb>Quadraturis, & ad <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>demittatur perpendiculum <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>: erit <lb></lb><arrow.to.target n="note448"></arrow.to.target><emph type="italics"></emph>AH<emph.end type="italics"></emph.end>ſinus Inclinationis quæſitæ. </s></p> <p type="margin"> <s><margin.target id="note448"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Nam <emph type="italics"></emph>GEq<emph.end type="italics"></emph.end>æquale eſt <emph type="italics"></emph>GHq+HEq=BHD+HEq= <lb></lb>HBD+HEq-BHq=HBD+BEq<emph.end type="italics"></emph.end>-2<emph type="italics"></emph>BHXBE= <lb></lb>BEq<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>ECXBH<emph.end type="italics"></emph.end>=2<emph type="italics"></emph>ECXAB<emph.end type="italics"></emph.end>+2<emph type="italics"></emph>ECXBH<emph.end type="italics"></emph.end>=2<emph type="italics"></emph>ECXAH.<emph.end type="italics"></emph.end><lb></lb>Ideoque cum 2<emph type="italics"></emph>EC<emph.end type="italics"></emph.end>detur, eſt <emph type="italics"></emph>GEq<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AH.<emph.end type="italics"></emph.end>Deſignet jam <emph type="italics"></emph>AEg<emph.end type="italics"></emph.end><lb></lb>duplicatam diſtantiam Nodorum à Quadraturis poſt datum ali<lb></lb>quod momentum temporis completum, & arcus <emph type="italics"></emph>Gg.,<emph.end type="italics"></emph.end>ob datum <lb></lb><figure id="id.039.01.447.1.jpg" xlink:href="039/01/447/1.jpg"></figure><lb></lb>angulum <emph type="italics"></emph>GEg,<emph.end type="italics"></emph.end>erit ut diſtantia <emph type="italics"></emph>GE.<emph.end type="italics"></emph.end>Eſt autem <emph type="italics"></emph>Hh<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Gg<emph.end type="italics"></emph.end><lb></lb>ut <emph type="italics"></emph>GH<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>GC,<emph.end type="italics"></emph.end>& propterea <emph type="italics"></emph>Hh<emph.end type="italics"></emph.end>eſt ut contentum <emph type="italics"></emph>GHXGg,<emph.end type="italics"></emph.end><lb></lb>ſeu <emph type="italics"></emph>GHXGE<emph.end type="italics"></emph.end>; id eſt, ut <emph type="italics"></emph>(GH/GE)XGEq<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>(GH/GE)XAH,<emph.end type="italics"></emph.end>id eſt, <lb></lb>ut <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>& ſinus anguli <emph type="italics"></emph>AEG<emph.end type="italics"></emph.end>conjunctim. </s> <s>Igitur ſi <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>in <lb></lb>caſu aliquo ſit ſinus Inclinationis, augebitur ea iiſdem incremen<lb></lb>tis cum ſinu Inclinationis, per Corol. </s> <s>3. Propoſitionis ſuperioris, <lb></lb>& propterea ſinui illi æqualis ſemper manebit. </s> <s>Sed <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>ubi <lb></lb>punctum <emph type="italics"></emph>G<emph.end type="italics"></emph.end>incidit in punctum alterutrum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>D<emph.end type="italics"></emph.end>huic ſinui <lb></lb>æqualis eſt, & propterea eidem ſemper æqualis manet. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>In hac demonſtratione ſuppoſui angulum <emph type="italics"></emph>BEG,<emph.end type="italics"></emph.end>qui eſt du<lb></lb>plicata diſtantia Nodorum à Quadraturis, uniformiter augeri. </s> <s><lb></lb>Nam omnes inæqualitatum minutias expendeve non vacat. </s> <s>Con<lb></lb>cipe jam angulum <emph type="italics"></emph>BEG<emph.end type="italics"></emph.end>rectum eſſe, & in hoc eaſe <emph type="italics"></emph>Gg<emph.end type="italics"></emph.end>eſſe <lb></lb>augmentum horarium duplæ diſtantiæ Nodorum & Solis ab invi<lb></lb>cem; & Inclinationis Variatio horaria in eodem caſu (per Corol. </s> <s><lb></lb>3. Prop. </s> <s>noviſſimæ) erit ad 33′. </s> <s>10′. </s> <s>33<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>ut contentum ſub In<lb></lb>clinationis ſinu <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>& ſinu anguli recti <emph type="italics"></emph>BEG,<emph.end type="italics"></emph.end>qui eſt dupli<lb></lb>cata diſtantia Nodorum a Sole, ad quadruplum quadratum radii; <lb></lb>id. </s> <s>eſt, ut mediocris Inclinationis ſinus <emph type="italics"></emph>AH<emph.end type="italics"></emph.end>ad radium quadru<lb></lb>plicatum; hoc eſt (cum Inclinatio illa mediocris ſit quafi 5<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 8′1/2) <lb></lb>ut ejus ſinus 896 ad radium quadruplicatum 40000, ſive ut 224 <lb></lb>ad 10000. Eſt autem Variatio tota, ſinuum differentiæ <emph type="italics"></emph>BD<emph.end type="italics"></emph.end><lb></lb>reſpondens, ad Variationem illam horariam ut diameter <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad <pb xlink:href="039/01/448.jpg" pagenum="420"></pb><arrow.to.target n="note449"></arrow.to.target>arcum <emph type="italics"></emph>Gg<emph.end type="italics"></emph.end>; id eſt, ut diameter <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad ſemicircum ferentiam <lb></lb><emph type="italics"></emph>BGD<emph.end type="italics"></emph.end>& tempus horarum (2079 1/10), quo Nodus pergit à Quadra<lb></lb>turis ad Syzygias, ad horam unam conjunctim; hoc eſt, ut 7 ad <lb></lb>11 & (2079 7/10) ad 1. Quare ſi rationes omnes conjungantur, fiet <lb></lb>Variatio tota <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"></emph>ix<emph.end type="sup"></emph.end> ut 224X7X2079 (7/10) ad <lb></lb>110000, id eſt, ut 29645 ad 1000, & inde Variatio illa <emph type="italics"></emph>BD<emph.end type="italics"></emph.end><lb></lb>prodibit 16′. </s> <s>23″ 1/2. </s></p> <p type="margin"> <s><margin.target id="note449"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Hæc eſt Inclinationis Variatio maxima quatenus locus Lunæ in <lb></lb>Orbe ſuo non conſideratur. </s> <s>Nam Inclinatio, ſi Nodi in Syzygiis <lb></lb>verſantur, nil mutatur ex vario ſitu Lunæ. </s> <s>At ſi Nodi in Qua<lb></lb>draturis conſiſtunt, Inclinatio minor eſt ubi Luna verſatur in Sy<lb></lb>zygiis, quam ubi ea verſatur in Quadraturis, exceſſu 2′. </s> <s>43″; uti <lb></lb>in Propoſitionis ſuperioris Corollario quarto indicavimus. </s> <s>Et <lb></lb>hujus exceſſus dimidio 1′. </s> <s>21″ 1/2. Variatio tota mediocris <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>in <lb></lb>Quadraturis Lunaribus diminuta fit 15′, 2″, in ipſius autem Syzy<lb></lb>giis aucta fit 17′. </s> <s>45″. </s> <s>Si Luna igitur in Syzygiis conſtituatur, <lb></lb>Variatio tota, in tranſitu Nodorum à Quadraturis ad Syzygias, <lb></lb>erit 17′. </s> <s>45″: adeoque ſi Inclinatio, ubi Nodi in Syzygiis verſan<lb></lb>tur, ſit 5<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 17′. </s> <s>20″; eadem, ubi Nodi ſunt in Quadraturis, & <lb></lb>Luna in Syzygiis, erit 4<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 59′. </s> <s>35″. </s> <s>Atque hæc ita ſe habere <lb></lb>confirmatur ex Obſervationibus. </s></p><figure id="id.039.01.448.1.jpg" xlink:href="039/01/448/1.jpg"></figure> <p type="main"> <s>Si jam deſideretur Orbis Inclinatio illa, ubi Luna in Syzygiis <lb></lb>& Nodi ubivis verſantur; fiat <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>ut ſinus graduum 4. <lb></lb>59′. </s> <s>35″ ad ſinum graduum 5. </s> <s>17′, 20″, & capiatur angulus <emph type="italics"></emph>AEG<emph.end type="italics"></emph.end><lb></lb>æqualis duplicatæ diſtantiæ Nodorum à Quadraturis; & erit <emph type="italics"></emph>AH<emph.end type="italics"></emph.end><lb></lb>ſinus Inclinationis quæſitæ. </s> <s>Huic Orbis Inclinationi æqualis eſt <lb></lb>ejuſdem Inclinatio, ubi Luna diſtat 90<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> à Nodis. </s> <s>In aliis Lunæ <lb></lb>locis inæqualitas menſtrua, quam Inclinationis variatio admittit, <lb></lb>in calculo Latitudinis Lunæ compenſatur & quodammodo tolli<lb></lb>tur per inæqualitatem menſtruam motus Nodorum, (ut ſupra dixi<lb></lb>mus) adeoQ.E.I. calculo Latitudinis illius negligi poteſt. <pb xlink:href="039/01/449.jpg" pagenum="421"></pb><arrow.to.target n="note450"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note450"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hiſce motuum Lunarium computationibus oſtendere volui, <lb></lb>quod motus Lunares, per Theoriam Gravitatis, a cauſis ſuis com<lb></lb>putari poſſint. </s> <s>Per eandem Theoriam inveni præterea quod Æ<lb></lb>quatio Annua medii motus Lunæ oriatur a varia dilatatione Or<lb></lb>bis Lunæ per vim Solis, juxta Corol. </s> <s>6. Prop. </s> <s>LXVI. Lib. </s> <s>I. </s> <s>Hæc <lb></lb>vis in Perigæo Solis major eſt, & Orbem Lunæ dilatat; in Apo<lb></lb>gæo ejus minor eſt, & Orbem illum contrahi permittit. </s> <s>In Orbe <lb></lb>dilatato Luna tardius revolvitur, in contracto citius; & Æquatio <lb></lb>Annua per quam hæc inæqualitas compenſatur, in Apogæo & <lb></lb>Perigæo Solis nulla eſt, in mediocri Solis a Terra diſtantia ad <lb></lb>11′. </s> <s>50″ circiter aſcendit, in aliis locis Æquationi centri Solis <lb></lb>proportionalis eſt; & additur medio motui Lunæ ubi Terra per<lb></lb>git ab Aphelio ſuo ad Perihelium, & in oppoſita Orbis parte, ſub<lb></lb>ducitur. </s> <s>Aſſumendo radium Orbis magni 1000 & Eccentricita<lb></lb>tem Terræ 16 7/8, hæc Æquatio ubi maxima eſt, per Theoriam Gra<lb></lb>vitatis prodiit 11′. </s> <s>49″. </s> <s>Sed Eccentricitas Terræ paulo major eſſe <lb></lb>videtur, & aucta Eccentricitate hæc Æquatio augeri debet in ea<lb></lb>dem ratione. </s> <s>Sit Eccentricitas (16 11/16), & Æquatio maxima erit <lb></lb>11′. </s> <s>52″. </s></p> <p type="main"> <s>Inveni etiam quod in Perihelio Terræ, propter majorem vim <lb></lb>Solis, Apogæum & Nodi Lunæ velocius moventur quam in Aphe<lb></lb>lio ejus, idQ.E.I. triplicata ratione diſtantiæ Terræ a Sole inverſe, <lb></lb>Et inde oriuntur Æquationes Annuæ horum motuum Æquationi <lb></lb>centri Solis proportionales. </s> <s>Motus autem Solis eſt in duplicata <lb></lb>ratione diſtantiæ Terræ a Sole inverſe, & maxima centri Æquatio <lb></lb>quam hæc inæqualitas generat, eſt 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 56′. </s> <s>26″ prædictæ Solis <lb></lb>Eccentricitati (16 15/16) congruens. </s> <s>Quod ſi motus Solis eſſet in tri<lb></lb>plicata ratione diſtantiæ inverſe, hæc inæqualitas generaret Æqua<lb></lb>tionem maximam 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 56′. </s> <s>9″. </s> <s>Et propterea Æquationes maxi<lb></lb>mæ quas inæqualitates motuum Apogæi & Nodorum Lunæ gene<lb></lb>rant, ſunt ad 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 56′. </s> <s>9″, ut motus medius diurnus Apogæi & <lb></lb>motus medius diurnus Nodorum Lunæ ſunt ad motum medium <lb></lb>diurnum Solis. </s> <s>Unde prodit Æquatio maxima medii motus <lb></lb>Apogæi 19′. </s> <s>52″: & Æquatio maxima medii motus Nodorum <lb></lb>9′. </s> <s>27″. </s> <s>Additur vero Æquatio prior & ſubducitur poſterior, ubi <lb></lb>Terra pergit a Perihelio ſuo ad Aphelium: & contrarium fit in <lb></lb>oppoſita Orbis parte. <pb xlink:href="039/01/450.jpg" pagenum="422"></pb><arrow.to.target n="note451"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note451"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Per Theoriam Gravitatis conſtitit etiam quod actio Solis in <lb></lb>Lunam paulo major ſit ubi tranſverſa diameter Orbis Lunaris <lb></lb>tranſit per Solem, quam ubi eadem ad rectos eſt angulos cum <lb></lb>linea Terram & Solem jungente: & propterea Orbis Lunaris <lb></lb>paulo major eſt in priore caſu quam in poſteriore. </s> <s>Et hinc ori<lb></lb>tur alia Æquatio motus medii Lunaris, pendens a ſitu Apogæi <lb></lb>Lunæ ad Solem, quæ quidem maxima eſt cum Apogæum Lunæ <lb></lb>verſatur in Octante cum Sole; & nulla cum illud ad Quadraturas <lb></lb>vel Syzygias pervenit: & motui medio additur in tranſitu Apo<lb></lb>gæi Lunæ a Solis Quadratura ad Syzygiam, & ſubducitur in tran<lb></lb>ſitu Apogæi a Syzygia ad Quadraturam. </s> <s>Hæc Æquatio quam <lb></lb>Semeſtrem vocabo, in Octantibus Apogæi quando maxima eſt, <lb></lb>aſcendit ad 3′. </s> <s>45″ circiter, quantum ex Phænomenis colligere <lb></lb>potui. </s> <s>Hæc eſt ejus quantitas in mediocri Solis diſtantia a Terra. </s> <s><lb></lb>Augetur vero ac diminuitur in triplicata ratione diſtantiæ Solis <lb></lb>inverſe, adeoQ.E.I. maxima Solis diſtantia eſt 3′. </s> <s>34″, & in mi<lb></lb>nima 3′. </s> <s>56″ quamproxime: ubi vero Apogæum Lunæ ſitum eſt <lb></lb>extra Octantes, evadit minor; eſtque ad Æquationem maximam, <lb></lb>ut ſinus duplæ diſtantiæ Apogæi Lunæ a proxima Syzygia vel <lb></lb>Quadratura ad radium. </s></p> <p type="main"> <s>Per eandem Gravitatis Theoriam actio Solis in Lunam paulo <lb></lb>major eſt ubi linea recta per Nodos Lunæ ducta tranſit per So<lb></lb>lem, quam ubi linea ad rectos eſt angulos cum recta Solem ac <lb></lb>Terram jungente. </s> <s>Et inde oritur alia medii motus Lunaris Æqua<lb></lb>tio, quam Semeſtrem ſecundam vocabo, quæque maxima eſt ubi <lb></lb>Nodi in Solis Octantibus verſantur, & evaneſcit ubi ſunt in Syzy<lb></lb>giis vel Quadraturis, & in aliis Nodorum poſitionibus proportio<lb></lb>nalis eſt ſinui duplæ diſtantiæ Nodi alterutrius a proxima Syzygia <lb></lb>aut Quadratura: additur vero medio motui Lunæ dum Nodi <lb></lb>tranſeunt a Solis Quadraturis ad proximas Syzygias, & ſubduci<lb></lb>tur in eorum tranſitu a Syzygiis ad Quadraturas; & in Octanti<lb></lb>bus ubi maxima eſt, aſcendit ad 47″ in mediocri Solis diſtantia a <lb></lb>Terra, uti ex Theoria Gravitatis colligo. </s> <s>In aliis Solis diſtantiis <lb></lb>hæe Æquatio, in Octantibus Nodorum, eſt reciproce ut cubus di<lb></lb>ſtantiæ Solis a Terra, ideoQ.E.I. Perigæo Solis ad 45″ in Apo<lb></lb>gæo ejus ad 49″ circiter aſcendit. </s></p> <p type="main"> <s>Per eandem Gravitatis Theoriam Apogæum Lunæ progreditur <lb></lb>quam maxime ubi vel cum Sole conjungitur vel eidem opponitur, <lb></lb>& regreditur ubi cum Sole Quadraturam facit. </s> <s>Et Eccentricitas <lb></lb>fit maxima in priore caſu & minima in poſteriore, per Corol. <pb xlink:href="039/01/451.jpg" pagenum="423"></pb>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"></arrow.to.target>Corollaria permagnæ ſunt, & Æquationem principalem Apogæi <lb></lb>generant, quam Semeſtrem vocabo. </s> <s>Et Æquatio maxima Seme<lb></lb>ſtris eſt 12<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 18′ circiter, quantum ex Obſervationibus colligere <lb></lb>potui. <emph type="italics"></emph>Horroxius<emph.end type="italics"></emph.end>noſter Lunam in Ellipſi circum Terram, in ejus <lb></lb>umbilico inferiore conſtitutam, revolvi primus ſtatuit. <emph type="italics"></emph>Halleius<emph.end type="italics"></emph.end><lb></lb>centrum Ellipſeos in Epicyclo locavit, cujus centrum uniformiter <lb></lb>revolvitur circum Terram. </s> <s>Et ex motu in Epicyclo oriuntur in<lb></lb>æqualitates jam dictæ in progreſſu & regreſſu Apogæi & quanti<lb></lb>tate Eccentricitatis. </s> <s>Dividi intelligatur diſtantia mediocris Lunæ <lb></lb>a Terra in partes 100000, & referat <emph type="italics"></emph>T<emph.end type="italics"></emph.end>Terram & <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>Eccentri<lb></lb>citatem mediocrem Lunæ partium 5505. Producatur <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>B,<emph.end type="italics"></emph.end><lb></lb>ut ſit <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>ſinus Æquationis maximæ Semeſtris 12<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 18′ ad ra<lb></lb>dium <emph type="italics"></emph>TC,<emph.end type="italics"></emph.end>& circulus <emph type="italics"></emph>BDA<emph.end type="italics"></emph.end>centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>intervallo <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>deſcriptus, <lb></lb>erit Epicyclus ille in quo centrum Orbis Lunaris locatur & ſe<lb></lb>cundum ordinem literarum <emph type="italics"></emph>BDA<emph.end type="italics"></emph.end>revolvitur. </s> <s>Capiatur angulus <lb></lb><emph type="italics"></emph>BCD<emph.end type="italics"></emph.end>æqualis duplo argumento annuo, ſeu duplæ diſtantiæ veri <lb></lb>loci Solis ab Apogæo Lunæ ſemel æquato, & erit <emph type="italics"></emph>CTD<emph.end type="italics"></emph.end>Æquatio <lb></lb><figure id="id.039.01.451.1.jpg" xlink:href="039/01/451/1.jpg"></figure><lb></lb>Semeſtris Apogæi Lunæ & <emph type="italics"></emph>TD<emph.end type="italics"></emph.end>Eccentricitas Orbis ejus in Apo<lb></lb>gæum ſecundo æquatum tendens. </s> <s>Habitis autem Lunæ motu <lb></lb>medio & Apogæo & Eccentricitate, ut & Orbis axe majore par<lb></lb>tium 200000; ex his eruetur verus Lunæ locus in Orbe & di<lb></lb>ſtantia ejus a Terra, idque per Methodos notiſſimas. </s></p> <p type="margin"> <s><margin.target id="note452"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>In Perihelio Terræ, propter majorem vim Solis, centrum Or<lb></lb>bis Lunæ velocius movetur circum centrum <emph type="italics"></emph>C<emph.end type="italics"></emph.end>quam in Aphelio, <lb></lb>idQ.E.I. triplicata ratione diſtantiæ Terræ a Sole inverſe. </s> <s>Ob <lb></lb>Æquationem centri Solis in Argumento annuo comprehenſam, cen<lb></lb>trum Orbis Lunæ velocius movetur in Epicyclo <emph type="italics"></emph>BDA<emph.end type="italics"></emph.end>in du<lb></lb>plicata ratione diſtantiæ Terræ a Sole inverſe. </s> <s>Ut idem adhuc <lb></lb>velocius moveatur in ratione ſimplici diſtantiæ inverſe; ab Orbis <lb></lb>centro <emph type="italics"></emph>D<emph.end type="italics"></emph.end>agatur recta <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>verſus Apogæum Lunæ, ſeu rectæ <lb></lb><emph type="italics"></emph>TC<emph.end type="italics"></emph.end>parallela, & capiatur angulus <emph type="italics"></emph>EDF<emph.end type="italics"></emph.end>æqualis exceſſui Argu-<pb xlink:href="039/01/452.jpg" pagenum="424"></pb><arrow.to.target n="note453"></arrow.to.target>menti annui prædicti ſupra diſtantiam Apogæi Lunæ a Perigæo <lb></lb>Solis in conſequentia; vel quod perinde eſt, capiatur angulus <lb></lb><emph type="italics"></emph>CDF<emph.end type="italics"></emph.end>æqualis complemento Anomaliæ veræ Solis ad gradus 360. <lb></lb>Et ſit <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>DC<emph.end type="italics"></emph.end>ut dupla Eccentricitas Orbis magni ad diſtan<lb></lb>tiam mediocrem Solis a Terra, & motus medius diurnus Solis ab <lb></lb>Apogæo Lunæ ad motum medium diurnum Solis ab Apogæo <lb></lb>proprio conjunctim, id eſt, ut 33 7/8 ad 1000 & 52′. </s> <s>27″. </s> <s>16′ ad <lb></lb>59′. </s> <s>8″. </s> <s>10′ conjunctim, ſive ut 3 ad 100. Et concipe centrum <lb></lb>Orbis Lunæ locari in puncto <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>& in Epicyclo cujus centrum eſt <lb></lb><emph type="italics"></emph>D<emph.end type="italics"></emph.end>& radius <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>interea revolvi dum punctum <emph type="italics"></emph>D<emph.end type="italics"></emph.end>progreditur <lb></lb>in circumferentia circuli <emph type="italics"></emph>DABD.<emph.end type="italics"></emph.end>Hac enim ratione velocitas <lb></lb>qua centrum Orbis Lunæ in linea quadam curva circum centrum <lb></lb><emph type="italics"></emph>C<emph.end type="italics"></emph.end>deſcripta movebitur, erit reciproce ut cubus diſtantiæ Solis a <lb></lb>Terra quamproxime, ut oportet. </s></p> <p type="margin"> <s><margin.target id="note453"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Computatio motus hujus difficilis eſt, ſed facilior reddetur per <lb></lb>approximationem ſequentem. </s> <s>Si diſtantia mediocris Lunæ a Terra <lb></lb>ſit partium 100000, & Eccentricitas <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ſit partium 5505 ut ſu<lb></lb>pra: recta <emph type="italics"></emph>CB<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>invenietur partium 1172 1/4, & recta <emph type="italics"></emph>DF<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.452.1.jpg" xlink:href="039/01/452/1.jpg"></figure><lb></lb>partium 35 1/3. Et hæc recta ad diſtantiam <emph type="italics"></emph>TC<emph.end type="italics"></emph.end>ſubtendit angulum <lb></lb>ad Terram quem tranſlatio centri Orbis a loco <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ad locum <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ge<lb></lb>nerat in motu centri hujus: & eadem recta duplicata in ſitu paral<lb></lb>lelo ad diſtantiam ſuperioris umbilici Orbis Lunæ a Terra, ſubten<lb></lb>dit eundem angulum, quem utique tranſlatio illa generat in motu <lb></lb>umbilici, & ad diſtantiam Lunæ a Terra ſubtendit angulum quem <lb></lb>eadem tranſlatio generat in motu Lunæ, quique propterea Æqua<lb></lb>tio centri Secunda dici poteſt. </s> <s>Et hæc Æquatio in mediocri Lunæ <lb></lb>diſtantia a Terra, eſt ut ſinus anguli quem recta illa <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>cum recta <lb></lb>a puncto <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ad Lunam ducta continet quamproxime, & ubi ma<lb></lb>xima eſt evadit 2′. </s> <s>25″. </s> <s>Angulus autem quem recta <emph type="italics"></emph>DF<emph.end type="italics"></emph.end>& recta <lb></lb>a puncto <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ad Lunam ducta comprehendunt, invenitur vel ſub<lb></lb>ducendo angulum <emph type="italics"></emph>EDF<emph.end type="italics"></emph.end>ab Anomalia media Lunæ, vel addendo <lb></lb>diſtantiam Lunæ a Sole ad diſtantiam Apogæi Lunæ ab Apogæo <pb xlink:href="039/01/453.jpg" pagenum="425"></pb>Solis. </s> <s>Et ut radius eſt ad ſinum anguli ſic inventi, ita 2′. </s> <s>25″ <lb></lb><arrow.to.target n="note454"></arrow.to.target>ſunt ad Æquationem centri Secundam, addendam ſi ſumma illa <lb></lb>ſit minor ſemicirculo, ſubducendam ſi major. </s> <s>Sic habebitur ejus <lb></lb>Longitudo in ipſis Luminarium Syzygiis. </s></p> <p type="margin"> <s><margin.target id="note454"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Si computatio accuratior deſideretur, corrigendus eſt locus <lb></lb>Lunæ in Orbe ut ſupra inventus per Variationem duplicem. </s> <s>De <lb></lb>Variatione Prima & principali diximus ſupra, hæc maxima eſt <lb></lb>in Octantibus Lunæ. </s> <s>Variatio altera maxima eſt in Quadrantibus, <lb></lb>& oritur a varia Solis actione in Orbem Lunæ pro varia poſitione <lb></lb>Apogæi Lunæ ad Solem, computatur vero in hunc modum. </s> <s><lb></lb>Ut radius ad ſinum verſum diſtantiæ Apogæi Lunæ a Perigæo <lb></lb>Solis in conſequentia, ita angulus quidam P ad quartum propor<lb></lb>tionalem. </s> <s>Et ut radius ad ſinum diſtantiæ Lunæ a Sole, ita ſum<lb></lb>ma hujus quarti proportionalis & anguli cujuſdam alterius Q ad <lb></lb>Variationem Secundam, ſubducendam ſi Lunæ lumen augetur, ad<lb></lb>dendam ſi diminuitur. </s> <s>Sic habebitur locus verus Lunæ in Orbe, <lb></lb>& per Reductionem loci hujus ad Eclipticam habebitur Longi<lb></lb>tudo Lunæ. </s> <s>Anguli vero P & Q ex Obſervationibus determi<lb></lb>nandi ſunt. </s> <s>Et interea ſi pro angulo P uſurpentur 2′, & pro <lb></lb>angulo Q 1′, non multum errabitur. </s></p> <p type="main"> <s>Cum Atmoſphæra Terræ ad uſque altitudinem milliarium 35 <lb></lb>vel 40 refringat lucem Solis, & refringendo ſpargat eandem in <lb></lb>Umbram Terræ, & ſpargendo lucem in confinio Umbræ dilatat <lb></lb>Umbram: ad diametrum Umbræ quæ per Parallaxim prodit, <lb></lb>addo minutum unum primum in Eclipſibus Lunæ, vel minutum <lb></lb>unum cum triente. </s></p> <p type="main"> <s>Theoria vero Lunæ primo in Syzygiis, deinde in Quadraturis, <lb></lb>& ultimo in Octantibus per Phænomena examinari & ſtabiliri de<lb></lb>bet. </s> <s>Et opus hocce aggreſſurus motus medios Solis & Lunæ ad <lb></lb>tempus meridianum in Obſervatorio Regio <emph type="italics"></emph>Grenovicenſi,<emph.end type="italics"></emph.end>die ul<lb></lb>timo menſis <emph type="italics"></emph>Decembris<emph.end type="italics"></emph.end>anni 1700. ſt. </s> <s>vet. </s> <s>non incommode ſe<lb></lb>quentes adhibebit: nempe motum medium Solis <gap></gap> 20<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 43′. </s> <s>40″, & <lb></lb>Apogæi ejus <gap></gap> 7<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 44′. </s> <s>30″, & motum medium Lunæ <gap></gap> 15<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end><lb></lb>20′. </s> <s>00″, & Apogæi ejus <gap></gap> 8<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 20′. </s> <s>00″, & Nodi aſcendentis <lb></lb><gap></gap> 27<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 24′. </s> <s>20″; & differentiam meridianorum Obſervatorii hu<lb></lb>jus & Obſervatorii Regii <emph type="italics"></emph>Pariſienſis<emph.end type="italics"></emph.end>0<emph type="sup"></emph>hor.<emph.end type="sup"></emph.end> 9<emph type="sup"></emph>min.<emph.end type="sup"></emph.end> 20<emph type="sup"></emph>ſec.<emph.end type="sup"></emph.end>. <pb xlink:href="039/01/454.jpg" pagenum="426"></pb><arrow.to.target n="note455"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note455"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXVI. PROBLEMA XVII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire vim Solis ad Mare movendum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Solis vis <emph type="italics"></emph>ML<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>PT,<emph.end type="italics"></emph.end>in Quadraturis Lunaribus, ad pertur<lb></lb>bandos motus Lunares, erat (per Prop. </s> <s>XXV. hujus) ad vim <lb></lb>gravitatis apud nos, ut 1 ad 638092, 6. Et vis <emph type="italics"></emph>TM-LM<emph.end type="italics"></emph.end>ſeu <lb></lb>2<emph type="italics"></emph>PK<emph.end type="italics"></emph.end>in Syzygiis Lunaribus, eſt duplo major. </s> <s>Hæ autem vires, <lb></lb>ſi deſcendatur ad ſuperficiem Terræ, diminuuntur in ratione di<lb></lb>ſtantiarum a centro Terræ, id eſt, in ratione 60 1/2 ad 1; adeo<lb></lb>que vis prior in ſuperficie Terræ, eſt ad vim gravitatis, ut 1 ad <lb></lb>38604600. Hac vi Mare deprimitur in locis quæ 90 gradibus diſtant <lb></lb><figure id="id.039.01.454.1.jpg" xlink:href="039/01/454/1.jpg"></figure><lb></lb>a Sole. </s> <s>Vi altera quæ duplo major eſt, Mare elevatur & ſub Sole <lb></lb>& in regione Soli oppoſita. </s> <s>Summa virium eſt ad vim gravitatis <lb></lb>ut 1 ad 12868200. Et quoniam vis eadem eundem ciet motum, <lb></lb>ſive ea deprimat Aquam in regionibus quæ 90 gradibus diſtant à <lb></lb>Sole, ſive elevet eandem in regionibus ſub Sole & Soli oppoſitis, <lb></lb>hæc ſumma erit tota Solis vis ad Mare agitandum; & eundem <lb></lb>habebit effectum ac ſi tota in regionibus ſub Sole & Soli oppo<lb></lb>ſitis Mare elevaret, in regionibus autem quæ 90 gradibus diſtant <lb></lb>a Sole nil ageret. </s></p> <p type="main"> <s>Hæc eſt vis Solis ad Mare ciendum in loco quo vis dato, ubi Sol <lb></lb>tam in vertice loci verſatur quam in mediocri ſua diſtantia a <lb></lb>Terra. </s> <s>In aliis Solis poſitionibus vis ad Mare accollendum, eſt <lb></lb>ut ſinus verſus duplæ altitudinis Solis ſupra horizontem loci di<lb></lb>recte & cubus diſtantiæ Solis a Terra inverſe. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Cum vis centrifuga partium Terræ à diurno Terræ motu <lb></lb>oriunda, quæ eſt ad vim gravitatis ut 1 ad 289, efficiat ut alti-<pb xlink:href="039/01/455.jpg" pagenum="427"></pb>tudo Aquæ ſub Æquatore ſuperet ejus altitudinem ſub Polis men<lb></lb><arrow.to.target n="note456"></arrow.to.target>ſura pedum Pariſienſium 85820; vis Solaris de qua egimus, cum <lb></lb>ſit ad vim gravitatis ut 1 ad 12868200, atque adeo ad vim illam <lb></lb>centrifugam ut 289 ad 12868200 ſeu 1 ad 44527, efficiet ut al<lb></lb>titudo Aquæ in regionibus ſub Sole & Soli oppoſitis, ſuperet alti<lb></lb>tudinem ejus in locis quæ 90 gradibus diſtant a Sole, menſura <lb></lb>tantum pedis unius Pariſienſis & digitorum undecim cum octava <lb></lb>parte digiti. </s> <s>Eſt enim hæc menſura ad menſuram pedum 85820 <lb></lb>ut 1 ad 44527. </s></p> <p type="margin"> <s><margin.target id="note456"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXVII. PROBLEMA XVIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire vim Lunæ ad Mare movendum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Vis Lunæ ad Mare movendum colligendà eſt ex ejus propor<lb></lb>tione ad vim Solis, & hæc proportio colligenda eſt ex propor<lb></lb>tione motuum Maris, qui ab his viribus oriuntur. </s> <s>Ante oſtium <lb></lb>fluvii <emph type="italics"></emph>Avonæ<emph.end type="italics"></emph.end>ad lapidem tertium infra <emph type="italics"></emph>Briſtoliam,<emph.end type="italics"></emph.end>tempore verno <lb></lb>& autumnali totus Aquæ aſcenſus in Conjunctione & Oppoſitione <lb></lb>Luminarium (obſervante <emph type="italics"></emph>Samuele Sturmio<emph.end type="italics"></emph.end>) eſt pedum plus mi<lb></lb>nus 45, in Quadraturis autem eſt pedum tantum 25. Altitudo <lb></lb>prior ex ſumma virium, poſterior ex earundem differentia oritur. </s> <s><lb></lb>Solis igitur & Lunæ in Æquatore verſantium & mediocriter a <lb></lb>Terra diſtantium ſunto vires S & L, & erit L+S ad L-S ut <lb></lb>45 ad 25, ſeu 9 ad 5. </s></p> <p type="main"> <s>In portu <emph type="italics"></emph>Plymuthi<emph.end type="italics"></emph.end>Æſtus maris (ex obſervatione <emph type="italics"></emph>Samuelis Cole<lb></lb>preſſi<emph.end type="italics"></emph.end>) ad pedes plus minus ſexdecim altitudine mediocri attolli<lb></lb>tur, ac tempore verno & autumnali altitudo Æſtus in Syzygiis ſu<lb></lb>perare poteſt altitudinem ejus in Quadraturis, pedibus plus ſeptem <lb></lb>vel octo. </s> <s>Si maxima harum altitudinum differentia ſit pedum no<lb></lb>vem, erit L+S ad L-S ut 20 1/2 ad 11 1/2 ſeu 41 ad 23. Quæ <lb></lb>proportio ſatis congruit cum priore. </s> <s>Ob magnitudinem Æſtus in <lb></lb>portu <emph type="italics"></emph>Biſtoliæ,<emph.end type="italics"></emph.end>obſervationibus <emph type="italics"></emph>Sturmii<emph.end type="italics"></emph.end>magis fidendum eſſe vi<lb></lb>detur, ideoQ.E.D.nec aliquid certius conſtiterit, proportionem 9 <lb></lb>ad 5 uſurpabimus. </s></p> <p type="main"> <s>Cæterum ob aquarum reciprocos motus, Æſtus maximi non in<lb></lb>cidunt in ipſas Luminarium Syzygias, ſed ſunt tertii a Syzygiis <lb></lb>ut dictum fuit, ſeu proxime ſequuntur tertium Lunæ poſt Syzy<lb></lb>gias appulſum ad meridianum loci, vel potius (ut a <emph type="italics"></emph>Sturmio<emph.end type="italics"></emph.end>no<lb></lb>tatur) ſunt tertii poſt diem novilunii vel plenilunii, ſeu poſt ho-<pb xlink:href="039/01/456.jpg" pagenum="428"></pb><arrow.to.target n="note457"></arrow.to.target>ram a novilunio vel plenilunio plus minus duodecimam, adeoque <lb></lb>incidunt in horam a novilunio vel plenilunio plus minus quadra<lb></lb>geſimam tertiam. </s> <s>Incidunt vero in hoc portu in horam ſepti<lb></lb>mam circiter ab appulſu Lunæ ad meridianum loci; ideoque pro<lb></lb>xime ſequuntur appulſum Lunæ ad meridianum, ubi Luna diſtat a <lb></lb>Sole vel ab oppoſitione Solis gradibus plus minus octodecim vel <lb></lb>novendecim in conſequentia. </s> <s>Æſtas & Hyems maxime vigent, <lb></lb>non in ipſis Solſtitiis, ſed ubi Sol diſtat a Solſtitiis decima circi<lb></lb>ter parte totius circuitus, ſeu gradibus plus minus 36 vel 37. Et <lb></lb>ſimiliter maximus Æſtus maris oritur ab appulſu Lunæ ad meri<lb></lb>dianum loci, ubi Luna diſtat a Sole decima circiter parte motus <lb></lb>totius ab Æſtu ad Æſtum. </s> <s>Sit diſtantia illa graduum plus mi<lb></lb>nus 18 1/2. Et vis Solis in hac diſtantia Lunæ a Syzygiis & Qua<lb></lb>draturis, minor erit ad augendum & ad minuendum motum ma<lb></lb>ris a vi Lunæ oriundum, quam in ipſis Syzygiis & Quadraturis, in <lb></lb>ratione radii ad ſinum complementi diſtantiæ hujus duplicatæ ſeu <lb></lb>anguli graduum 37, hoc eſt, in ratione 10000000 ad 7986355. <lb></lb>IdeoQ.E.I. analogia ſuperiore pro S ſcribi debet 0, 7986355 S. </s></p> <p type="margin"> <s><margin.target id="note457"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Sed & vis Lunæ in Quadraturis, ob declinationem Lunæ ab <lb></lb>Æquatore, diminui debet. </s> <s>Nam Luna in Quadraturis, vel potius <lb></lb>in gradu 18 1/2 poſt Quadraturas, in declinatione graduum plus <lb></lb>minus 22. 13′ verſatur. </s> <s>Et Luminaris ab Æquatore declinantis <lb></lb>vis ad Mare movendum diminuitur in duplicata ratione ſinus <lb></lb>complementi declinationis quamproxime. </s> <s>Et propterea vis <lb></lb>Lunæ in his Quadraturis eſt tantum 0,8570327 L. </s> <s>Eſt igitur <lb></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 abſque Eccentricitate mo<lb></lb>veri deberet, ſunt ad invicem ut 69 ad 70; ideoQ.E.D.ſtantia <lb></lb>Lunæ a Terra in Syzygiis eſt ad diſtantiam ejus in Quadraturis, <lb></lb>ut 69 ad 70, cæteris paribus. </s> <s>Et diſtantiæ ejus in gradu 18 1/2 a <lb></lb>Syzygiis ubi Æſtus maximus generatur, & in gradu 18 1/2 a Qua<lb></lb>draturis ubi Æſtus minimus generatur, ſunt ad mediocrem ejus <lb></lb>diſtantiam, ut 69,098747 & 69,897345 ad 69 1/2. Vires autem Lu<lb></lb>næ ad Mare movendum ſunt in triplicata ratione diſtantiarum in<lb></lb>verſe, ideoque vires in maxima & minima harum diſtantiarum ſunt <lb></lb>ad vim in mediocri diſtantia, ut 0,9830427 & 1,017522 ad 1. Unde fit <lb></lb>1,017522 L+0,7986355 S ad 0,9830427X0,8570327 L-0,7986355 S <lb></lb>ut 9 ad 5. Et S ad L ut 1 ad 4,4815. Itaque cum vis Solis fit <lb></lb>ad vim gravitatis ut 1 ad 12868200, vis Lunæ erit ad vim gravi<lb></lb>tatis ut 1 ad 2871400. </s></p><pb xlink:href="039/01/457.jpg" pagenum="429"></pb> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Cum Aqua vi Solis agitata aſcendat ad altitudinem <lb></lb><arrow.to.target n="note458"></arrow.to.target>pedis unius & undecim digitorum cum octava parte digiti, eadem <lb></lb>vi Lunæ aſcendet ad altitudinem octo pedum & digitorum octo, <lb></lb>& vi utraque ad altitudinem pedum decem cum ſemiſſe, & ubi <lb></lb>Luna eſt in Perigæo ad altitudinem pedum duodecim cum ſemiſſe <lb></lb>& ultra, præſertim ubi Æſtus ventis ſpirantibus adjuvatur. </s> <s>Tanta <lb></lb>autem vis ad omnes Maris motus excitandos abunde ſufficit, & <lb></lb>quantitati motuum probe reſpondet. </s> <s>Nam in maribus quæ ab <lb></lb>Oriente in Occidentem late patent, uti in Mari <emph type="italics"></emph>Pacifico,<emph.end type="italics"></emph.end>& Maris <lb></lb><emph type="italics"></emph>Atlantici<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Æthiopici<emph.end type="italics"></emph.end>partibus extra Tropicos, aqua attolli ſo<lb></lb>let ad altitudinem pedum ſex, novem, duodecim vel quindecim. </s> <s><lb></lb>In Mari autem <emph type="italics"></emph>Pacifico,<emph.end type="italics"></emph.end>quod profundius eſt & latius patet, Æſtus <lb></lb>dicuntur eſſe majores quam in <emph type="italics"></emph>Atlantico<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Æthiopico.<emph.end type="italics"></emph.end>Etenim <lb></lb>ut plenus ſit Æſtus, latitudo Maris ab Oriente in Occidentem non <lb></lb>minor eſſe debet quàm graduum nonaginta. </s> <s>In Mari <emph type="italics"></emph>Æthiopico,<emph.end type="italics"></emph.end><lb></lb>aſcenſus aquæ intra Tropicos minor eſt quam in Zonis tempera<lb></lb>tis, propter anguſtiam Maris inter <emph type="italics"></emph>Africam<emph.end type="italics"></emph.end>& Auſtralem partem <lb></lb><emph type="italics"></emph>Americæ.<emph.end type="italics"></emph.end>In medio Mari aqua nequit aſcendere, niſi ad littus <lb></lb>utrumque & orientale & occidentale ſimul deſcendat: cum tamen <lb></lb>vicibus alternis ad littora illa in Maribus noſtris anguſtis deſcen<lb></lb>dere debeat. </s> <s>Ea de cauſa fluxus & refluxus in Inſulis, quæ à <lb></lb>littoribus longiſſime abſunt, perexiguus eſſet ſolet. </s> <s>In Portubus <lb></lb>quibuſdam, ubi aqua cum impetu magno per loca vadoſa, ad <lb></lb>Sinus alternis vicibus implendos & evacuandos, influere & effluere <lb></lb>cogitur, fluxus & refluxus debent eſſe ſolito majores, uti ad <lb></lb><emph type="italics"></emph>Plymuthum<emph.end type="italics"></emph.end>& pontem <emph type="italics"></emph>Chepſtowæ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Anglia<emph.end type="italics"></emph.end>; ad montes S. <emph type="italics"></emph>Mi<lb></lb>chaelis<emph.end type="italics"></emph.end>& urbem <emph type="italics"></emph>Abrincatuorum<emph.end type="italics"></emph.end>(vulgo <emph type="italics"></emph>Auranches<emph.end type="italics"></emph.end>) in <emph type="italics"></emph>Normania<emph.end type="italics"></emph.end>; <lb></lb>ad <emph type="italics"></emph>Cambaiam<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Pegu<emph.end type="italics"></emph.end>in <emph type="italics"></emph>India<emph.end type="italics"></emph.end>orientali. </s> <s>His in locis mare, <lb></lb>magna cum velocitate accedendo & recedendo, littora nunc in<lb></lb>undat nunc arida relinquit ad multa milliaria. </s> <s>NeQ.E.I.petus <lb></lb>influendi & remeandi prius frangi poteſt, quam aqua attollitur <lb></lb>vel deprimitur ad pedes 30, 40, vel 50 & amplius. </s> <s>Et par eſt <lb></lb>ratio fretorum oblongorum & vadoſorum, uti <emph type="italics"></emph>Magellanici<emph.end type="italics"></emph.end>& ejus <lb></lb>quo <emph type="italics"></emph>Anglia<emph.end type="italics"></emph.end>circundatur. </s> <s>Æſtus in hujuſmodi portubus & fretis, <lb></lb>per impetum curſus & recurſus ſupra modum augetur. </s> <s>Ad littora <lb></lb>vero quæ deſcenſu præcipiti ad mare profundum & apertum <lb></lb>ſpectant, ubi aqua ſine impetu effluendi & remeandi attolli & <lb></lb>ſubſidere poteſt, magnitudo Æſtus reſpondet viribus Solis & <lb></lb>Lunæ. <pb xlink:href="039/01/458.jpg" pagenum="430"></pb><arrow.to.target n="note459"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note458"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note459"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Cum vis Lunæ ad Mare movendum, ſit ad vim gravi<lb></lb>tatis ut 1 ad 2871400, perſpicuum eſt quod vis illa ſit longe <lb></lb>minor quam quæ vel in experimentis Pendulorum, vel in Staticis <lb></lb>aut Hydroſtaticis quibuſcunque ſentiri poſſit. </s> <s>In Æſtu ſolo ma<lb></lb>rino hæc vis ſenſibilem edit effectum. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Quoniam vis Lunæ ad Mare movendum, eſt ad Solis <lb></lb>vim conſimilem ut 4,4815 ad 1, & vires illæ (per Corol. </s> <s>14. <lb></lb>Prop. </s> <s>LXVI. Lib. </s> <s>I.) ſunt ut denſitates corporum Lunæ & Solis <lb></lb>& cubi diametrorum apparentium conjunctim; denſitas Lunæ erit <lb></lb>ad denſitatem Solis, ut 4,4815 ad 1 directe & cubus diametri <lb></lb>Lunæ ad cubum diametri Solis inverſe: id eſt (cum diametri me<lb></lb>diocres apparentes Lunæ & Solis ſint 31′. </s> <s>16 1/2″ & 32′. </s> <s>12″) ut <lb></lb>4891 ad 1000. Denſitas autem Solis erat ad denſitatem Terræ, <lb></lb>ut 100 ad 396; & propterea denſitas Lunæ eſt ad denſitatem <lb></lb>Terræ, ut 4891 ad 3960 ſeu 21 ad 17. Eſt igitur corpus Lunæ <lb></lb>denſius & magis terreſtre quam Terra noſtra. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Et cum vera diameter Lunæ (ex Obſervationibus <lb></lb>Aſtronomicis) ſit ad veram diametrum Terræ, ut 100 ad 365; <lb></lb>erit maſla Lunæ ad maſſam Terræ, ut 1 ad 39,371. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>5. Et gravitas acceleratrix in ſuperficie Lunæ, erit quaſi <lb></lb>triplo minor quam gravitas acceleratrix in ſuperficie Terræ. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>6. Et diſtantia centri Lunæ a centro Terræ, erit ad di<lb></lb>ſtantiam centri Lunæ a communi gravitatis centro Terræ & Lunæ, <lb></lb>ut 40,371 ad 39,371. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>7. Et mediocris diſtantia centri Lunæ a centro Terræ, erit <lb></lb>ſemidiametrorum maximarum Terræ 60 1/4 quamproxime. </s> <s>Nam <lb></lb>ſemidiameter maxima Terræ fuit pedum Pariſienſium 19767630, <lb></lb>& mediocris diſtantia centrorum Terræ & Lunæ ex hujuſmodi <lb></lb>ſemidiametris 60 1/4 conſtans, æqualis eſt pedibus 1190999707. Et <lb></lb>hæc diſtantia (per Corollarium ſuperius) eſt ad diſtantiam centri <lb></lb>Lunæ a communi gravitatis centro Terræ & Lunæ, ut 40,371 ad <lb></lb>39,371, quæ proinde eſt pedum 1161498340. Et cum Luna re<lb></lb>volvatur reſpectu Fixarum, diebus 27, horis 7 & minutis primis 43 1/5; <lb></lb>ſinus verſus anguli quem Luna, tempore minuti unius primi motu <lb></lb>ſuo medio, circa commune gravitatis centrum Terræ & Lunæ de<lb></lb>ſcribit, eſt 1275235, exiſtente radio 100,000000,000000, Et ut <lb></lb>radius eſt ad hunc ſinum verſum, ita ſunt pedes 1161498340 ad <lb></lb>pedes 14,811833. Luna igitur vi illa qua retinetur in Orbe, ca<lb></lb>dendo in Terram, tempore minuti unius primi deſcribet pedes <lb></lb>14,811833. Et ſi hæc vis augeatur in ratione (177 29/40) ad (178 29/40), ha-<pb xlink:href="039/01/459.jpg" pagenum="431"></pb>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"></arrow.to.target>Et hac vi Luna cadendo, tempore minuti unius primi deſcribere <lb></lb>deberet pedes 14,89517. Et ad ſexageſimam partem hujus di<lb></lb>ſtantiæ, id eſt, ad diſtantiam pedum 19849995 a centro Terræ, <lb></lb>corpus grave cadendo, tempore minuti unius ſecundi deſcribere <lb></lb>deberet etiam pedes 14,89517. Diminuatur hæc diſtantia in ſub<lb></lb>duplicata ratione pedum 14,89517 ad pedes 15,12028, & habebitur <lb></lb>diſtantia pedum 19701678 a qua grave cadendo, eodem tempore <lb></lb>minuti unius ſecundi deſcribet pedes 15,12028, id eſt, pedes 15, <lb></lb>dig 1, lin. </s> <s>5,32. Et hac vi gravia cadunt in ſuperficie Terræ, in <lb></lb>Latitudine urbis <emph type="italics"></emph>Lutetiæ Pariſiorum,<emph.end type="italics"></emph.end>ut ſupra oſtenſum eſt. </s> <s>Eſt <lb></lb>autem diſtantia pedum 19701678 paulo minor quam ſemidiame<lb></lb>ter globi huic Terræ æqualis, & paulo major quam Terræ hujus <lb></lb>ſemidiameter mediocris, ut oportet. </s> <s>Sed differentiæ ſunt inſenſi<lb></lb>biles. </s> <s>Et propterea vis qua Luna retinetur in Orbe ſuo, ad di<lb></lb>ſtantiam maximarum Terræ ſemidiametrorum 60 1/4, ea eſt quam <lb></lb>vis Gravitatis in ſuperficie Terræ requirit. </s></p> <p type="margin"> <s><margin.target id="note460"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>8. Diſtantia mediocris centrorum Terræ & Lunæ, eſt me<lb></lb>diocrium Terræ ſemidiametrorum 60 1/2 quamproxime. </s> <s>Nam ſe<lb></lb>midiameter mediocris, quæ erat pedum 19688725, eſt ad ſemi<lb></lb>diametrum maximam pedum 19767630, ut 60 1/4 ad 60 1/2 quam<lb></lb>proxime. </s></p> <p type="main"> <s>In his computationibus Attractionem magneticam Terræ non <lb></lb>conſideravimus, cujus utique quantitas perparva eſt & ignotatur. </s> <s><lb></lb>Siquando vero hæc Attractio inveſtigari poterit, & menſuræ gra<lb></lb>duum in Meridiano, ac longitudines Pendulorum iſochronorum in <lb></lb>diverſis parallelis, legeſque motuum Maris, & parallaxis Lunæ <lb></lb>cum diametris apparentibus Solis & Lunæ ex Phænomenis accu<lb></lb>ratius determinatæ fuerint: licebit calculum hunc omnem accura<lb></lb>tius repetere. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXVIII. PROBLEMA XIX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire Figuram corporis Lunæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si corpus Lunare fluidum eſſet ad inſtar Maris noſtri, vis Terræ <lb></lb>ad fluidum illud in partibus & citimis & ultimis elevandum, eſſet <lb></lb>ad vim Lunæ, qua Mare noſtrum in partibus & ſub Luna & Lunæ <lb></lb>oppoſitis attollitur, ut gravitas acceleratrix Lunæ in Terram ad <lb></lb>gravitatem acceleratricem Terræ in Lunam & diameter Lunæ ad <pb xlink:href="039/01/460.jpg" pagenum="432"></pb><arrow.to.target n="note461"></arrow.to.target>diametrum Terræ conjunctim; id eſt, ut 39,371 ad 1 & 100 ad <lb></lb>365 conjunctim, ſeu 1079 ad 100. Unde cum Mare noſtrum vi <lb></lb>Lunæ attollatur ad pedes 8 2/3, fluidum Lunare vi Terræ attolli de<lb></lb>beret ad pedes 93 1/2. EaQ.E.D. cauſa Figura Lunæ Sphærois eſſet, <lb></lb>cujus maxima diameter producta tranſiret per centrum Terræ, & <lb></lb>ſuperaret diametros perpendiculares exceſſu pedum 187. Talem <lb></lb>igitur Figuram Luna affectat, eamque ſub initio induere debuit. <lb></lb><emph type="italics"></emph><expan abbr="q.">que</expan> E. I.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note461"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Inde vero fit ut eadem ſemper Lunæ facies in Terram <lb></lb>obvertatur. </s> <s>In alio enim ſitu corpus Lunare quieſcere non po<lb></lb>teſt, ſed ad hunc ſitum oſcillando ſemper redibit. </s> <s>Attamen oſcil<lb></lb>lationes, ob parvitatem virium agitantium, eſſent longè tardiſſimæ: <lb></lb>adeo ut facies illa, quæ Terram ſemper reſpicere deberet, poſſit <lb></lb>alterum orbis Lunaris umbilicum, ob rationem in Prop. </s> <s>XVII. alla<lb></lb>tam reſpicere, neque ſtatim abinde retrahi & in Terram converti. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si<emph.end type="italics"></emph.end>APEp <emph type="italics"></emph>Terram deſignet uniformiter denſam, centroque <lb></lb>C & Polis<emph.end type="italics"></emph.end>P, p <emph type="italics"></emph>& Æquatore<emph.end type="italics"></emph.end>AE <emph type="italics"></emph>delineatam; & ſi centro<emph.end type="italics"></emph.end>C <lb></lb><emph type="italics"></emph>radio<emph.end type="italics"></emph.end>CP <emph type="italics"></emph>deſcribi intelligatur Sphæra<emph.end type="italics"></emph.end>Pape; <emph type="italics"></emph>ſit autem<emph.end type="italics"></emph.end>QR <emph type="italics"></emph>pla<lb></lb>num, cui recta a centro Solis ad centrum Terræ ducta normaliter <lb></lb>inſiſtit; & Terræ totius exterioris<emph.end type="italics"></emph.end>PapAPepE, <emph type="italics"></emph>quæ Sphæra <lb></lb>modo deſcripta altior eſt, particulæ ſingulæ conentur recedere hinc <lb></lb>inde a plano<emph.end type="italics"></emph.end>QR, <emph type="italics"></emph>ſitque conatus particulæ cujuſque ut ejuſdem <lb></lb>diſtantia a plano: Dico primo, quod tota particularum omnium, in <lb></lb>Æquatoris circulo<emph.end type="italics"></emph.end>AE, <emph type="italics"></emph>extra globum uniformiter per totum cir<lb></lb>cuitum in morem annuli diſpoſitarum, vis & efficacia ad Terram <lb></lb>circum centrum ejus rotandam, ſit ad totam particularum totidem <lb></lb>in Æquatoris puncto<emph.end type="italics"></emph.end>A, <emph type="italics"></emph>quod a plano<emph.end type="italics"></emph.end>QR <emph type="italics"></emph>maxime diſtat, con<lb></lb>ſiſtentium vim & efficaciam, ad Terram conſimili motu circulari <lb></lb>circum centrum ejus movendam, ut unum ad duo. </s> <s>Et motus iſte <lb></lb>circularis circum axem, in communi ſectione Æquatoris & plani<emph.end type="italics"></emph.end><lb></lb>QR <emph type="italics"></emph>jacentem, peragetur.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam centro <emph type="italics"></emph>C<emph.end type="italics"></emph.end>diametro <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>deſcribatur ſemicirculus <lb></lb><emph type="italics"></emph>BAFDC.<emph.end type="italics"></emph.end>Dividi intelligatur ſemicircum ferentia <emph type="italics"></emph>BAD<emph.end type="italics"></emph.end>in <pb xlink:href="039/01/461.jpg" pagenum="433"></pb>partes innumeras æquales, & a partibus ſingulis <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ad diame<lb></lb><arrow.to.target n="note462"></arrow.to.target>trum <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>demittantur ſinus <emph type="italics"></emph>FY.<emph.end type="italics"></emph.end>Et ſumma quadratorum ex <lb></lb>ſinibus omnibus <emph type="italics"></emph>FY<emph.end type="italics"></emph.end>æqualis erit ſummæ quadratorum ex ſinibus <lb></lb>omnibus <emph type="italics"></emph>CY,<emph.end type="italics"></emph.end>& ſumma utraque æqualis erit ſummæ quadrato<lb></lb>rum ex totidem ſemidiametris <emph type="italics"></emph>CF<emph.end type="italics"></emph.end>; adeoque ſumma quadrato<lb></lb>rum ex omnibus <emph type="italics"></emph>FY,<emph.end type="italics"></emph.end>erit duplo minor quam ſumma quadrato<lb></lb>rum ex totidem ſemidiametris <emph type="italics"></emph>CF.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note462"></margin.target>LIBER <lb></lb>TERTIUS.</s></p><figure id="id.039.01.461.1.jpg" xlink:href="039/01/461/1.jpg"></figure> <p type="main"> <s>Jam dividatur perimeter circuli <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>in particulas totidem æ<lb></lb>quales, & ab earum unaquaque <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ad planum <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>demittatur <lb></lb>perpendiculum <emph type="italics"></emph>FG,<emph.end type="italics"></emph.end>ut & a puncto <emph type="italics"></emph>A<emph.end type="italics"></emph.end>perpendiculum <emph type="italics"></emph>AH.<emph.end type="italics"></emph.end>Et <lb></lb>vis qua particula <emph type="italics"></emph>F<emph.end type="italics"></emph.end>recedit a plano <emph type="italics"></emph>QR,<emph.end type="italics"></emph.end>erit ut perpendiculum <lb></lb>illud <emph type="italics"></emph>FG<emph.end type="italics"></emph.end>per hypotheſin, & hæc vis ducta in diſtantiam <emph type="italics"></emph>CG,<emph.end type="italics"></emph.end><lb></lb>erit efficacia particulæ <emph type="italics"></emph>F<emph.end type="italics"></emph.end>ad Terram circum centrum ejus con<lb></lb>vertendam. </s> <s>Adeoque efficacia particulæ in loco <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>erit ad effi<lb></lb>caciam particulæ in loco <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>FGXGC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AHXHC,<emph.end type="italics"></emph.end>hoc <lb></lb>eſt, ut <emph type="italics"></emph>FCq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ACq<emph.end type="italics"></emph.end>; & propterea efficacia tota particularum <lb></lb>omnium in locis ſuis <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>erit ad efficaciam particularum totidem in <lb></lb>loco <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut ſumma omnium <emph type="italics"></emph>FCq<emph.end type="italics"></emph.end>ad ſummam totidem <emph type="italics"></emph>ACq,<emph.end type="italics"></emph.end>hoc <lb></lb>eſt, (per jam demonſtrata) ut unum ad duo. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Et quoniam particulæ agunt recedendo perpendiculariter a <lb></lb>plano <emph type="italics"></emph>QR,<emph.end type="italics"></emph.end>idque æqualiter ab utraque parte hujus plani: eædem <lb></lb>convertent circumferentiam circuli Æquatoris, eiQ.E.I.hærentem <lb></lb>Terram, circum axem tam in plano illo <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>quam in plano Æqua<lb></lb>toris jacentem. <pb xlink:href="039/01/462.jpg" pagenum="434"></pb><arrow.to.target n="note463"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note463"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis: Dico ſecundo quod vis & efficacia tota parti<lb></lb>cularum omnium extra globum undique ſitarum, ad Terram cir<lb></lb>cum axem eundem rotandam, ſit ad vim totam particularum toti<lb></lb>dem, in Æquatoris circulo<emph.end type="italics"></emph.end>AE, <emph type="italics"></emph>uniformiter per totum circuitum <lb></lb>in morem annuli diſpoſitarum, ad Terram conſimili motu circulari <lb></lb>movendam, ut duo ad quinque.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Sit enim <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>circulus quilibet minor Æquatori <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>parallelus, <lb></lb>ſintque <emph type="italics"></emph>L, l<emph.end type="italics"></emph.end>particulæ duæ quævis æquales in hoc circulo extra <lb></lb>globum <emph type="italics"></emph>Pape<emph.end type="italics"></emph.end>ſitæ. </s> <s>Et ſi in planum <emph type="italics"></emph>QR,<emph.end type="italics"></emph.end>quod radio in Solem <lb></lb>ducto perpendiculare eſt, demittantur perpendicula <emph type="italics"></emph>LM, lm:<emph.end type="italics"></emph.end><lb></lb>vires totæ quibus particulæ illæ fugiunt planum <emph type="italics"></emph>QR,<emph.end type="italics"></emph.end>proporti<lb></lb>onales erunt perpendiculis illis <emph type="italics"></emph>LM, lm.<emph.end type="italics"></emph.end>Sit autem recta <emph type="italics"></emph>Ll<emph.end type="italics"></emph.end><lb></lb>plano <emph type="italics"></emph>Pape<emph.end type="italics"></emph.end>parallela & biſecetur eadem in <emph type="italics"></emph>X,<emph.end type="italics"></emph.end>& per pun<lb></lb>ctum <emph type="italics"></emph>X<emph.end type="italics"></emph.end>agatur <emph type="italics"></emph>Nn,<emph.end type="italics"></emph.end>quæ parallela ſit plano <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>& perpendi<lb></lb><figure id="id.039.01.462.1.jpg" xlink:href="039/01/462/1.jpg"></figure><lb></lb>culis <emph type="italics"></emph>LM, lm<emph.end type="italics"></emph.end>occurrat in <emph type="italics"></emph>N<emph.end type="italics"></emph.end>ac <emph type="italics"></emph>n,<emph.end type="italics"></emph.end>& in planum <emph type="italics"></emph>QR<emph.end type="italics"></emph.end>demit<lb></lb>tatur perpendiculum <emph type="italics"></emph>XT.<emph.end type="italics"></emph.end>Et particularum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>& <emph type="italics"></emph>l<emph.end type="italics"></emph.end>vires con<lb></lb>trariæ, ad Terram in contrarias partes rotandam, ſunt ut <lb></lb><emph type="italics"></emph>LMXMC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>lmXmC,<emph.end type="italics"></emph.end>hoc eſt, ut <emph type="italics"></emph>LNXMC+NMXMC<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>lnXmC-nmXmC,<emph.end type="italics"></emph.end>ſeu <emph type="italics"></emph>LNXMC+NMXMC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>LNXmC<emph.end type="italics"></emph.end><pb xlink:href="039/01/463.jpg" pagenum="435"></pb>-<emph type="italics"></emph>NMXmC<emph.end type="italics"></emph.end>: & harum differentia <emph type="italics"></emph>LNXMm-NMX—MC+mC,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note464"></arrow.to.target>eſt vis particularum ambarum ſimul ſumptarum ad Terram <lb></lb>rotandam. </s> <s>Hujus differentiæ pars affirmativa <emph type="italics"></emph>LNXMm<emph.end type="italics"></emph.end>ſeu <lb></lb>2<emph type="italics"></emph>LNXNX,<emph.end type="italics"></emph.end>eſt ad particularum duarum ejuſdem magnitudi<lb></lb>nis in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>conſiſtentium vim 2<emph type="italics"></emph>AHXHC,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>LXq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="ACq.">ACque</expan><emph.end type="italics"></emph.end><lb></lb>Et pars negativa <emph type="italics"></emph>NMX—MC+mC<emph.end type="italics"></emph.end>ſeu 2<emph type="italics"></emph>XYXCY,<emph.end type="italics"></emph.end>ad parti<lb></lb>cularum earundem in <emph type="italics"></emph>A<emph.end type="italics"></emph.end>conſiſtentium vim 2<emph type="italics"></emph>AHXHC,<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>CXq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="ACq.">ACque</expan><emph.end type="italics"></emph.end>Ac proinde partium differentia, id eſt, par<lb></lb>ticularum duarum <emph type="italics"></emph>L<emph.end type="italics"></emph.end>& <emph type="italics"></emph>l<emph.end type="italics"></emph.end>ſimul ſumptarum vis ad Terram rotan<lb></lb>dam, eſt ad vim particularum duarum iiſdem æqualium & in loco <lb></lb><emph type="italics"></emph>A<emph.end type="italics"></emph.end>conſiſtentium, ad Terram itidem rotandam, ut <emph type="italics"></emph>LXq-CXq<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph><expan abbr="ACq.">ACque</expan><emph.end type="italics"></emph.end>Sed ſi circuli <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>circumferentia <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>dividatur in par<lb></lb>ticulas innumeras æquales <emph type="italics"></emph>L,<emph.end type="italics"></emph.end>erunt omnes <emph type="italics"></emph>LXq<emph.end type="italics"></emph.end>ad totidem <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end><lb></lb>ut 1 ad 2, (per Lem. </s> <s>I.) atque ad totidem <emph type="italics"></emph>ACq,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>ad <lb></lb>2<emph type="italics"></emph>ACq<emph.end type="italics"></emph.end>; & totidem <emph type="italics"></emph>CXq<emph.end type="italics"></emph.end>ad totidem <emph type="italics"></emph>ACq<emph.end type="italics"></emph.end>ut 2<emph type="italics"></emph>CXq<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph><expan abbr="ACq.">ACque</expan><emph.end type="italics"></emph.end><lb></lb>Quare vires conjunctæ particularum omnium in circuitu circuli <lb></lb><emph type="italics"></emph>IK,<emph.end type="italics"></emph.end>ſunt ad vires conjunctas particularum totidem in loco <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>-2<emph type="italics"></emph>CXq<emph.end type="italics"></emph.end>ad 2<emph type="italics"></emph>ACq<emph.end type="italics"></emph.end>: & propterea (per Lem. </s> <s>I.) ad vires <lb></lb>conjunctas particularum totidem in circuitu circuli <emph type="italics"></emph>AE,<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>-2<emph type="italics"></emph>CXq<emph.end type="italics"></emph.end>ad <emph type="italics"></emph><expan abbr="ACq.">ACque</expan><emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note464"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Jam vero ſi Sphæræ diameter <emph type="italics"></emph>Pp<emph.end type="italics"></emph.end>dividatur in partes innume<lb></lb>ras æquales, quibus inſiſtant circuli totidem <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>; materia in peri<lb></lb>metro circuli cujuſque <emph type="italics"></emph>IK<emph.end type="italics"></emph.end>erit ut <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>: ideoque vis materiæ <lb></lb>illius ad Terram rotandam, erit ut <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>in <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>-2<emph type="italics"></emph><expan abbr="CXq.">CXque</expan><emph.end type="italics"></emph.end>Et <lb></lb>vis materiæ ejuſdem, ſi in circuli <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>perimetro conſiſteret, eſſet <lb></lb>ut <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>in <emph type="italics"></emph><expan abbr="ACq.">ACque</expan><emph.end type="italics"></emph.end>Et propterea vis particularum omnium ma<lb></lb>teriæ totius, extra globum in perimetris circulorum omnium con<lb></lb>ſiſtentis, eſt ad vim particularum totidem in perimetro circuli <lb></lb>maximi <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>conſiſtentis, ut omnia <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>in <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>-2<emph type="italics"></emph>CXq<emph.end type="italics"></emph.end>ad <lb></lb>totidem <emph type="italics"></emph>IXq<emph.end type="italics"></emph.end>in <emph type="italics"></emph>ACq,<emph.end type="italics"></emph.end>hoc eſt, ut omnia <emph type="italics"></emph>ACq-CXq<emph.end type="italics"></emph.end>in <lb></lb><emph type="italics"></emph>ACq<emph.end type="italics"></emph.end>-3<emph type="italics"></emph>CXq<emph.end type="italics"></emph.end>ad totidem <emph type="italics"></emph>ACq-CXq<emph.end type="italics"></emph.end>in <emph type="italics"></emph>ACq,<emph.end type="italics"></emph.end>id eſt, ut <lb></lb>omnia <emph type="italics"></emph>ACqq<emph.end type="italics"></emph.end>-4<emph type="italics"></emph>ACqXCXq<emph.end type="italics"></emph.end>+3<emph type="italics"></emph>CXqq<emph.end type="italics"></emph.end>ad totidem <emph type="italics"></emph>ACqq <lb></lb>-ACqXCXq,<emph.end type="italics"></emph.end>hoc eſt, ut tota quantitas fluens cujus fluxio <lb></lb>eſt <emph type="italics"></emph>ACqq<emph.end type="italics"></emph.end>-4<emph type="italics"></emph>ACqXCXq<emph.end type="italics"></emph.end>+3<emph type="italics"></emph>CXqq,<emph.end type="italics"></emph.end>ad totam quantitatem flu<lb></lb>entem cujus fluxio eſt <emph type="italics"></emph>ACqq-ACqXCXq<emph.end type="italics"></emph.end>; ac proinde per Me<lb></lb>thodum Fluxionum, ut <emph type="italics"></emph>ACqqXCX<emph.end type="italics"></emph.end>-4/3<emph type="italics"></emph>ACqxCXcub<emph.end type="italics"></emph.end>+3/5<emph type="italics"></emph>CXqc<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>ACqqXCX<emph.end type="italics"></emph.end>-1/3<emph type="italics"></emph>ACqXCXcub,<emph.end type="italics"></emph.end>id eſt, ſi pro <emph type="italics"></emph>CX<emph.end type="italics"></emph.end>ſcribatur <lb></lb>tota <emph type="italics"></emph>Cp<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>ut (4/15)<emph type="italics"></emph>ACqc<emph.end type="italics"></emph.end>ad 2/3<emph type="italics"></emph>ACqc,<emph.end type="italics"></emph.end>hoc eſt, ut duo ad <lb></lb>quinque. <emph type="italics"></emph><expan abbr="q.">que</expan> E. D.<emph.end type="italics"></emph.end><pb xlink:href="039/01/464.jpg" pagenum="436"></pb><arrow.to.target n="note465"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note465"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA III.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Iiſdem poſitis: Dico tertio quod motus Terræ totius circum axem <lb></lb>jam ante deſcriptum, ex motibus particularum omnium compoſi<lb></lb>tus, erit ad motum annuli prædicti circum axem eundem, in ra<lb></lb>tione quæ componitur ex ratione materiæ in Terra ad materiam <lb></lb>in annulo, & ratione trium quadratorum ex arcu quadrantali <lb></lb>circuli cujuſcunque ad duo quadrata ex diametro; id eſt, in ra<lb></lb>tione materiæ ad materiam & numeri<emph.end type="italics"></emph.end>925275 <emph type="italics"></emph>ad numerum<emph.end type="italics"></emph.end><lb></lb>1000000. </s></p> <p type="main"> <s>Eſt enim motus Cylindri circum axem ſuum immotum revol<lb></lb>ventis, ad motum Sphæræ inſcriptæ & ſimul revolventis, ut quæ<lb></lb>libet quatuor æqualia quadrata ad tres ex circulis ſibi inſcriptis: <lb></lb>& motus Cylindri ad motum annuli tenuiſſimi, Sphæram & Cy<lb></lb>lindrum ad communem eorum contactum ambientis, ut duplum <lb></lb>materiæ in Cylindro ad triplum materiæ in annulo; & annuli <lb></lb>motus iſte circum axem Cylindri uniformiter continuatus, ad <lb></lb>ejuſdem motum uniformem circum diametrum propriam, eodem <lb></lb>tempore periodico factum, ut circumferentia circuli ad duplum <lb></lb>diametri. </s></p> <p type="main"> <s><emph type="center"></emph>HYPOTHESIS II.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si annulus prædictus Terra omni reliqua ſublata, ſolus in Orbe <lb></lb>Terræ, motu annuo circa Solem ferretur, & interea circa axem <lb></lb>ſuum, ad planum Eclipticæ in angulo graduum<emph.end type="italics"></emph.end>23 1/2 <emph type="italics"></emph>inclinatum, <lb></lb>motu diurno revolveretur: idem foret motus Punctorum Æqui<lb></lb>noctialium ſive annulus iſte fluidus eſſet, ſive is ex materia rigida <lb></lb>& firma conſtaret.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/465.jpg" pagenum="437"></pb> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XXXIX. PROBLEMA XX.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note466"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note466"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire Præceſſionem Æquinoctiorum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Motus mediocris horarius Nodorum Lunæ in Orbe circulari, <lb></lb>ubi Nodi ſunt in Quadraturis, erat 16″. </s> <s>35′. </s> <s>16<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>36<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. </s> <s>& hujus <lb></lb>dimidium 8′. </s> <s>17′. </s> <s>38<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>18<emph type="sup"></emph>v<emph.end type="sup"></emph.end>. (ob rationes ſupra explicatas) eſt mo<lb></lb>tus medius horarius Nodorum in tali Orbe; fitque anno toto <lb></lb>ſidereo 20<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 11′. </s> <s>46″. </s> <s>Quoniam igitur Nodi Lunæ in tali Orbe <lb></lb>conficerent annuatim 20<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 11′. </s> <s>46″. </s> <s>in antecedentia; & ſi plures <lb></lb>eſſent Lunæ motus Nodorum cujuſque, per Corol. </s> <s>16. Prop. </s> <s><lb></lb>LXVI. Lib. </s> <s>I. forent ut tempora periodica; ſi Luna ſpatio <lb></lb>diei ſiderei juxta ſuperficiem Terræ revolveretur, motus annuus <lb></lb>Nodorum foret ad 20<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 11′. </s> <s>46″. </s> <s>ut dies ſidereus horarum 23. 56′. </s> <s><lb></lb>ad tempus periodicum Lunæ dierum 27. 7 hor. </s> <s>43′; id eſt, ut <lb></lb>1436 ad 39343. Et par eſt ratio Nodorum annuli Lunarum <lb></lb>Terram ambientis; ſive Lunæ illæ ſe mutuo non contingant, ſive <lb></lb>liqueſcant & in annulum continuum formentur, ſive denique an<lb></lb>nulus ille rigeſcat & inflexibilis reddatur. </s></p> <p type="main"> <s>Fingamus igitur quod annulus iſte, quoad quantitatem materiæ, <lb></lb>æqualis ſit Terræ omni <emph type="italics"></emph>PapAPepE<emph.end type="italics"></emph.end>quæ globo <emph type="italics"></emph>Pape<emph.end type="italics"></emph.end>ſuperior <lb></lb>eſt; (<emph type="italics"></emph>Vid. </s> <s>Fig. </s> <s>pag.<emph.end type="italics"></emph.end>434.) & quoniam globus iſte eſt ad Terram illam <lb></lb>ſuperiorem ut <emph type="italics"></emph>aCqu.<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ACqu.-aCqu.<emph.end type="italics"></emph.end>id eſt (cum Terræ diameter <lb></lb>minor <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>aC<emph.end type="italics"></emph.end>ſit ad diametrum majorem <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ut 229 ad 230,) <lb></lb>ut 52441 ad 459; ſi annulus iſte Terram ſecundum Æquatorem <lb></lb>cingeret & uterque ſimul circa diametrum annuli revolveretur, <lb></lb>motus annuli eſſet ad motum globi interioris (per hujus Lem. </s> <s>III.) <lb></lb>ut 459 ad 52441 & 1000000 ad 925275 conjunctim, hoc eſt, <lb></lb>ut 4590 ad 485223; ideoque motus annuli eſſet ad ſummam mo<lb></lb>tuum annuli ac globi, ut 4590 ad 489813. Unde ſi annulus glo<lb></lb>bo adhæreat, & motum ſuum quo ipſius Nodi ſeu puncta Æqui<lb></lb>noctialia regrediuntur, cum globo communicet: motus qui reſta<lb></lb>bit in annulo erit ad ipſius motum priorem, ut 4590 ad 489813; <lb></lb>& propterea motus punctorum Æquinoctialium diminuetur in <lb></lb>eadem ratione. </s> <s>Erit igitur motus annuus punctorum Æqui<lb></lb>noctialium corporis ex annulo & globo compoſiti, ad motum <pb xlink:href="039/01/466.jpg" pagenum="438"></pb><arrow.to.target n="note467"></arrow.to.target>20<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 11′. </s> <s>46″, ut 1436 ad 39343 & 4590 ad 489813 conjun<lb></lb>ctim, id eſt, ut 100 ad 292369. Vires autem quibus Nodi Lu<lb></lb>narum (ut ſupra explicui) atque adeo quibus puncta Æquinoctia<lb></lb>lia annuli regrediuntur (id eſt vires 3<emph type="italics"></emph>IT, in Fig. </s> <s>pag.<emph.end type="italics"></emph.end>403 & 404.) <lb></lb>ſunt in ſingulis particulis ut diſtantiæ particularum à plano <emph type="italics"></emph>QR,<emph.end type="italics"></emph.end><lb></lb>& his viribus particulæ illæ planum fugiunt; & propterea (per <lb></lb>Lem. </s> <s>II.) ſi materia annuli per totam globi ſuperficiem, in mo<lb></lb>rem figuræ <emph type="italics"></emph>PapAPepE,<emph.end type="italics"></emph.end>ad ſuperiorem illam Terræ partem <lb></lb>conſtituendam ſpargeretur, vis & efficacia tota particularum om<lb></lb>nium ad Terram circa quamvis Æquatoris diametrum rotandam, <lb></lb>atque adeo ad movenda puncta Æquinoctialia, evaderet minor <lb></lb>quam prius in ratione 2 ad 5. Ideoque annuus Æquinoctiorum <lb></lb>regreſſus jam eſſet ad 20<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 11′. </s> <s>46″, ut 10 ad 73092: ac proinde <lb></lb>fieret 9″. </s> <s>56′. </s> <s>50<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s></p> <p type="margin"> <s><margin.target id="note467"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Cæterum hic motus, ob inclinationem plani Æquatoris ad pla<lb></lb>num Eclipticæ, minuendus eſt, idQ.E.I. ratione ſinus 91706 (qui <lb></lb>ſinus eſt complementi graduum 23 1/2) ad Radium 100000. Qua <lb></lb>ratione motus iſte jam fiet 9″. </s> <s>7′. </s> <s>20<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>Hæc eſt annua Præceſſio <lb></lb>Æquinoctiorum a vi Solis oriunda. </s></p> <p type="main"> <s>Vis autem Lunæ ad Mare movendum erat ad vim Solis, ut <lb></lb>4,4815 ad 1 circiter. </s> <s>Et vis Lunæ ad Æquinoctia movenda, eſt <lb></lb>ad vim Soiis in eadem proportione. </s> <s>Indeque prodit annua Æ<lb></lb>quinoctiorum Præceſſio a vi Lunæ oriunda 40″. </s> <s>52′. </s> <s>52<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>; ac tota <lb></lb>Præceſſio annua a vi utraque oriunda 50″. </s> <s>00′. </s> <s>12<emph type="sup"></emph>iv<emph.end type="sup"></emph.end>. </s> <s>Et hic mo<lb></lb>tus cum Phænomenis congruit. </s> <s>Nam Præceſſio Æquinoctiorum <lb></lb>ex Obſervationibus Aſtronomicis eſt minutorum ſecundorum plus <lb></lb>minus quinquaginta. </s></p> <p type="main"> <s>Si altitudo Terræ ad Æquatorem ſuperet altitudinem ejus ad <lb></lb>Polos, milliaribus pluribus quam 17 1/6, materia ejus rarior erit ad <lb></lb>circumferentiam quam ad centrum: & Præceſſio Æquinoctiorum <lb></lb>ob altitudinem illam augeri, ob raritatem diminui debet. </s></p> <p type="main"> <s>Deſcripſimus jam Syſtema Solis, Terræ, Lunæ, & Planetarum: <lb></lb>ſupereſt ut de Cometis nonnulla adjiciantur. </s></p><pb xlink:href="039/01/467.jpg" pagenum="439"></pb> <p type="main"> <s><emph type="center"></emph>LEMMA IV.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note468"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note468"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Cometas eſſe Luna ſuperiores & in regione Planetarum verſari.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ut defectus Parallaxeos diurnæ extulit Cometas ſupra regiones <lb></lb>ſublunares, ſic ex Parallaxi annua convincitur eorum deſcenſus in <lb></lb>regiones Planetarum. </s> <s>Nam Cometæ qui progrediuntur ſecun<lb></lb>dum ordinem ſignorum ſunt omnes, ſub exitu apparitionis, aut <lb></lb>ſolito tardiores aut retrogradi, ſi Terra eſt inter ipſos & Solem; <lb></lb>at juſto celeriores ſi Terra vergit ad oppoſitionem. </s> <s>Et e contra, <lb></lb>qui pergunt contra ordinem ſignorum ſunt juſto celeriores in fine <lb></lb>apparitionis, ſi Terra verſatur inter ipſos & Solem; & juſto tar<lb></lb>diores vel retrogradi ſi Terra ſita eſt ad contrarias partes. </s> <s>Con<lb></lb>tingit hoc maxime ex motu Terræ in vario ipſius ſitu, perinde ut <lb></lb>fit in Planetis, qui, pro motu Terræ vel conſpirante vel contra<lb></lb>rio, nunc retrogradi ſunt, nunc tardius progredi videntur, nunc <lb></lb>vero celerius. </s> <s>Si Terra pergit ad eandem partem cum Cometa, <lb></lb>& motu angulari circa Solem tanto celerius fertur, ut recta per <lb></lb>Terram & Cometam perpetuo ducta convergat ad partes ultra <lb></lb>Cometam, Cometa e Terra ſpectatus, ob motum ſuum tardiorem, <lb></lb>apparet eſſe retrogradus; ſin Terra tardius fertur, motus Cometæ, <lb></lb><figure id="id.039.01.467.1.jpg" xlink:href="039/01/467/1.jpg"></figure><lb></lb>(detracto motu Terræ) fit ſaltem tardior. </s> <s>At ſi Terra pergit in <lb></lb>contrarias partes, Cometa exinde velocior apparet. </s> <s>Ex accele<lb></lb>ratione autem vel retardatione vel motu retrogrado diſtantia Co<lb></lb>metæ in hunc modum colligitur. </s> <s>Sunto <emph type="italics"></emph>r QA, r QB, r QC<emph.end type="italics"></emph.end><lb></lb>obſervatæ tres longitudines Cometæ, ſub initio motus, ſitque <lb></lb><emph type="italics"></emph>r QF<emph.end type="italics"></emph.end>longitudo ultimo obſervata, ubi Cometa videri deſinit. <pb xlink:href="039/01/468.jpg" pagenum="440"></pb><arrow.to.target n="note469"></arrow.to.target>Agatur recta <emph type="italics"></emph>ABC,<emph.end type="italics"></emph.end>cujus partes <emph type="italics"></emph>AB, BC<emph.end type="italics"></emph.end>rectis <emph type="italics"></emph>QA<emph.end type="italics"></emph.end>& <emph type="italics"></emph>QB, <lb></lb>QB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>QC<emph.end type="italics"></emph.end>interjectæ, ſint ad invicem ut tempora inter obſer<lb></lb>vationes tres primas. </s> <s>Producatur <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>G,<emph.end type="italics"></emph.end>ut ſit <emph type="italics"></emph>AG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AB<emph.end type="italics"></emph.end><lb></lb>ut tempus inter obſervationem primam & ultimam, ad tempus <lb></lb>inter obſervationem primam & ſecundam, & jungatur <emph type="italics"></emph>QG.<emph.end type="italics"></emph.end>Et <lb></lb>ſi Cometa moveretur uniformiter in linea recta, atque Terra vel <lb></lb>quieſceret, vel etiam in linea recta, uniformi cum motu, progre<lb></lb>deretur; foret angulus <emph type="italics"></emph>r QG<emph.end type="italics"></emph.end>longitudo Cometæ tempore Ob<lb></lb>ſervationis ultimæ. </s> <s>Angulus igitur <emph type="italics"></emph>FQG,<emph.end type="italics"></emph.end>qui longitudinum dif<lb></lb>ferentia eſt, oritur ab inæqualitate motuum Cometæ ac Terræ. </s> <s><lb></lb>Hic autem angulus, ſi Terra & Cometa in contrarias partes mo<lb></lb>ventur, additur angulo <emph type="italics"></emph>rQG,<emph.end type="italics"></emph.end>& ſic motum apparentem Co<lb></lb>metæ velociorem reddit: Sin Cometa pergit in eaſdem partes <lb></lb>cum Terra, eidem ſubducitur, motumque Cometæ vel tardiorem <lb></lb>reddit, vel forte retrogradum; uti modo expoſui. </s> <s>Oritur igitur <lb></lb>hic angulus præcipue ex motu Terræ, & idcirco pro parallaxi Co<lb></lb>metæ merito habendus eſt, neglecto videlicet ejus incremento vel <lb></lb>decremento nonnullo, quod a Cometæ motu inæquabili in Orbe <lb></lb>proprio oriri poſſit. </s> <s>Diſtantia vero Cometæ ex hac parallaxi ſic <lb></lb>colligitur. </s> <s>Deſignet <emph type="italics"></emph>S<emph.end type="italics"></emph.end>Solem, <emph type="italics"></emph>acT<emph.end type="italics"></emph.end>Orbem magnum, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>locum <lb></lb>Terræ in obſervatione prima, <emph type="italics"></emph>c<emph.end type="italics"></emph.end>locum <lb></lb><figure id="id.039.01.468.1.jpg" xlink:href="039/01/468/1.jpg"></figure><lb></lb>Terræ in obſervatione tertia, <emph type="italics"></emph>T<emph.end type="italics"></emph.end>locum <lb></lb>Terræ in obſervatione ultima, & <emph type="italics"></emph>Tr<emph.end type="italics"></emph.end>li<lb></lb>neam rectam verſus principium Arietis <lb></lb>ductam. </s> <s>Sumatur angulus <emph type="italics"></emph>rTV<emph.end type="italics"></emph.end>æqua<lb></lb>lis angulo <emph type="italics"></emph>rQF,<emph.end type="italics"></emph.end>hoc eſt, æqualis lon<lb></lb>gitudini Cometæ ubi Terra verſatur in <lb></lb><emph type="italics"></emph>T.<emph.end type="italics"></emph.end>Jungatur <emph type="italics"></emph>ac,<emph.end type="italics"></emph.end>& producatur ea ad <emph type="italics"></emph>g,<emph.end type="italics"></emph.end><lb></lb>ut ſit <emph type="italics"></emph>ag<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>ac<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AG<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>& <lb></lb>erit <emph type="italics"></emph>g<emph.end type="italics"></emph.end>locus quem Terra tempore obſer<lb></lb>vationis ultimæ, motu in recta <emph type="italics"></emph>ac<emph.end type="italics"></emph.end>uNI<lb></lb>formiter continuato, attingeret. </s> <s>Ideo<lb></lb>que ſi ducatur <emph type="italics"></emph>g r<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>Tr<emph.end type="italics"></emph.end>parallela, <lb></lb>& capiatur angulus <emph type="italics"></emph>rgV<emph.end type="italics"></emph.end>angulo <emph type="italics"></emph>rQG<emph.end type="italics"></emph.end><lb></lb>æqualis, erit hic angulus <emph type="italics"></emph>rgV<emph.end type="italics"></emph.end>æqualis <lb></lb>longitudini Cometæ e loco <emph type="italics"></emph>g<emph.end type="italics"></emph.end>ſpectati; <lb></lb>& angulus <emph type="italics"></emph>TVg<emph.end type="italics"></emph.end>parallaxis erit, quæ oritur a tranſlatione Terræ <lb></lb>de loco <emph type="italics"></emph>g<emph.end type="italics"></emph.end>in locum <emph type="italics"></emph>T<emph.end type="italics"></emph.end>: ac proinde <emph type="italics"></emph>V<emph.end type="italics"></emph.end>locus erit Cometæ in plano <lb></lb>Eclipticæ. </s> <s>Hic autem locus <emph type="italics"></emph>V<emph.end type="italics"></emph.end>Orbe Jovis inferior eſſe ſolet. </s></p><pb xlink:href="039/01/469.jpg" pagenum="441"></pb> <p type="margin"> <s><margin.target id="note469"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Idem colligitur ex curvatura viæ Cometarum. </s> <s>Pergunt hæc <lb></lb><arrow.to.target n="note470"></arrow.to.target>corpora propemodum in circulis maximis quamdiu moventur cele<lb></lb>rius; at in fine curſus, ubi motus apparentis pars illa quæ à pa<lb></lb>rallaxi oritur, majorem habet proportionem ad motum totum ap<lb></lb>parentem, deflectere ſolent ab his circulis, & quoties Terra mo<lb></lb>vetur in unam partem, abire in partem contrariam. </s> <s>Oritur hæc <lb></lb>deflexio maxime ex Parallaxi, propterea quod reſpondet motui <lb></lb>Terræ; & inſignis ejus quantitas, meo computo, collocavit diſpa<lb></lb>rentes Cometas ſatis longe infra Jovem. </s> <s>Unde conſequens eſt <lb></lb>quod in Perigæis & Periheliis, ubi propius adſunt, deſcendunt <lb></lb>ſæpius infra orbes Martis & inferiorum Planetarum. </s></p> <p type="margin"> <s><margin.target id="note470"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Confirmatur etiam propinquitas Cometarum ex luce capitum. </s> <s><lb></lb>Nam corporis cœleſtis a Sole illuſtrati & in regiones longinquas <lb></lb>abeuntis, diminuitur ſplendor in quadruplicata ratione diſtantiæ: <lb></lb>in duplicata ratione videlicet ob auctam corporis diſtantiam a <lb></lb>Sole, & in alia duplicata ratione ob diminutam diametrum appa<lb></lb>rentem. </s> <s>Unde ſi detur & lucis quantitas & apparens diameter <lb></lb>Cometæ, dabitur diſtantia, dicendo quod diſtantia ſit ad diſtan<lb></lb>tiam Planetæ, in ratione diametri ad diametrum directe & ratione <lb></lb>ſubduplicata lucis ad lucem inverſe. </s> <s>Sic minima capillitii Co<lb></lb>metæ anni 1682 diameter, per Tubum opticum ſexdecim pedum <lb></lb>a <emph type="italics"></emph>Flamſtedio<emph.end type="italics"></emph.end>obſervata & Micrometro menſurata, æquabat 2′. </s> <s>0″. </s> <s><lb></lb>Nucleus autem ſeu ſtella in medio capitis vix decimam partem la<lb></lb>titudinis hujus occupabat, adeoque lata erat tantum 11″ vel 12″. </s> <s><lb></lb>Luce vero & claritate capitis ſuperabat caput Cometæ anni 1680, <lb></lb>ſtellaſque primæ vel ſecundæ magnitudinis æmulabatur. </s> <s>Ponamus <lb></lb>Saturnum cum annulo ſuo quaſi quadruplo lucidiorem fuiſſe: & <lb></lb>quoniam lux annuli propemodum æquabat lucem globi inter<lb></lb>medii, & diameter apparens globi ſit quaſi 21″, adeoque lux <lb></lb>globi & annuli conjunctim æquaret lucem globi, cujus diameter <lb></lb>eſſet 30″: erit diſtantia Cometæ ad diſtantiam Saturni ut 1 ad √ 4 <lb></lb>inverſe, & 12″ ad 30″ directe, id eſt, ut 24 ad 30 ſeu 4 ad 5. <lb></lb>Rurſus Cometa anni 1665 menſe <emph type="italics"></emph>Aprili,<emph.end type="italics"></emph.end>ut author eſt <emph type="italics"></emph>Hevelius,<emph.end type="italics"></emph.end><lb></lb>claritate ſua pene Fixas omnes ſuperabat, quinetiam ipſum Satur<lb></lb>num, ratione coloris videlicet longe vividioris. </s> <s>Quippe lucidior <lb></lb>erat hic Cometa altero illo, qui in fine anni præcedentis apparu<lb></lb>erat & cum ſtellis primæ magnitudinis conferebatur. </s> <s>Latitudo <lb></lb>capillitii erat quaſi 6′, at nucleus cum Planetis ope Tubi optici <lb></lb>collatus, plane minor erat Jove, & nunc minor corpore interme-<pb xlink:href="039/01/470.jpg" pagenum="442"></pb><arrow.to.target n="note471"></arrow.to.target>dio Saturni, nunc ipſi æqualis judicabatur. </s> <s>Porro cum diameter <lb></lb>capillitii Cometarum raro ſuperet 8′ vel 12′, diameter vero nu<lb></lb>clei ſeu ſtellæ centralis ſit quaſi decima vel forte decima quinta <lb></lb>pars diametri capillitii, patet Stellas haſce ut plurimum ejuſdem <lb></lb>eſſe apparentis magnitudinis cum Planetis. </s> <s>Unde cum lux earum <lb></lb>cum luce Saturni non raro conferri poſſit, eamque aliquando ſu<lb></lb>peret; manifeſtum eſt quod Cometæ omnes in Periheliis vel in<lb></lb>fra Saturnum collocandi ſint, vel non longe ſupra. </s> <s>Errant igitur <lb></lb>toto cœlo qui Cometas in regionem Fixarum prope ablegant: qua <lb></lb>certe ratione non magis illuſtrari deberent a Sole noſtro, quam <lb></lb>Planetæ, qui hic ſunt, illuſtrantur a Stellis fixis. </s></p> <p type="margin"> <s><margin.target id="note471"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Hæc diſputavimus non conſiderando obſcurationem Cometa<lb></lb>rum per ſumum illum maxime copioſum & craſſum, quo caput <lb></lb>circundatur, quaſi per nubem obtuſe ſemper lucens. </s> <s>Nam quan<lb></lb>to obſcurius redditur corpus per hunc fumum, tanto propius ad <lb></lb>Solem accedat neceſſe eſt, ut copia lucis a ſe reflexa Planetas æmu<lb></lb>letur. </s> <s>Inde veriſimile fit Cometas longe infra ſphæram Saturni <lb></lb>deſcendere, uti ex Parallaxi probavimus. </s> <s>Idem vero quam ma<lb></lb>xime confirmatur ex Caudis. </s> <s>Hæ vel ex reflexione fumi ſparſi <lb></lb>per Æthera, vel ex luce capitis oriuntur. </s> <s>Priore caſu minuenda <lb></lb>eſt diſtantia Cometarum, ne fumus a capite ſemper ortus per <lb></lb>ſpatia nimis ampla incredibili cum velocitate & expanſione pro<lb></lb>pagetur. </s> <s>In poſteriore referenda eſt lux omnis tam caudæ quam <lb></lb>capillitii ad nucleum capitis. </s> <s>Igitur ſi concipiamus lucem hanc <lb></lb>omnem congregari & intra diſcum nuclei coarctari, nucleus ille <lb></lb>jam certe, quoties caudam maximam & fulgentiſſimam emittit, <lb></lb>Jovem ipſum ſplendore ſuo multum ſuperabit. </s> <s>Minore igitur <lb></lb>cum diametro apparente plus lucis emittens, multo magis illuſtra<lb></lb>bitur a Sole, adeoque erit Soli multo propior. </s> <s>Quinetiam capita <lb></lb>ſub Sole deliteſcentia, & caudas cum maximas tum fulgentiſſimas <lb></lb>inſtar trabium ignitarum nonnunquam emittentia, eodem argu<lb></lb>mento infra orbem Veneris collocari debent. </s> <s>Nam lux illa omnis <lb></lb>ſi in ſtellam congregari ſupponatur, ipſam Venerem ne dicam Ve<lb></lb>neres plures conjunctas quandoque ſuperaret. </s></p> <p type="main"> <s>Idem denique colligitur ex luce capitum creſcente in receſſu <lb></lb>Cometarum a Terra Solem verſus, ac decreſcente in eorum receſſu <lb></lb>a Sole verſus Terram. </s> <s>Sic enim Cometa poſterior Anni 1665 <lb></lb>(obſervante <emph type="italics"></emph>Hevelio,<emph.end type="italics"></emph.end>) ex quo conſpici cœpit, remittebat ſemper <pb xlink:href="039/01/471.jpg" pagenum="443"></pb>de motu ſuo apparente, adeoque præterierat Perigæum; Splen<lb></lb><arrow.to.target n="note472"></arrow.to.target>dor vero capitis nihilominus indies creſcebat, uſQ.E.D.m Cometa <lb></lb>radiis Solaribus obtectus deſiit apparere. </s> <s>Cometa Anni 1683, <lb></lb>obſervante eodem <emph type="italics"></emph>Hevelio,<emph.end type="italics"></emph.end>in fine Menſis <emph type="italics"></emph>Julii<emph.end type="italics"></emph.end>ubi primum con<lb></lb>ſpectus eſt, tardiſſime movebatur, minuta prima 40 vel 45 circi<lb></lb>ter ſingulis diebus in Orbe ſuo conficiens. </s> <s>Ex eo tempore motus <lb></lb>ejus diurnus perpetuo augebatur uſque ad <emph type="italics"></emph>Sept.<emph.end type="italics"></emph.end>4. quando evaſit <lb></lb>graduum quaſi quinque. </s> <s>Igitur toto hoc tempore Cometa ad <lb></lb>Terram appropinquabat. </s> <s>Id quod etiam ex diametro capitis <lb></lb>Micrometro menſurata colligitur: quippe quam <emph type="italics"></emph>Hevelius<emph.end type="italics"></emph.end>reperit <lb></lb><emph type="italics"></emph>Aug.<emph.end type="italics"></emph.end>6. eſſe tantum 6′. </s> <s>5″ incluſa coma, at <emph type="italics"></emph>Sept.<emph.end type="italics"></emph.end>2. eſſe 9′. </s> <s>7″. </s> <s><lb></lb>Caput igitur initio longe minus apparuit quam in ſine motus, at <lb></lb>initio tamen in vicinia Solis longe lucidius extitit quam circa <lb></lb>finem, ut refert idem <emph type="italics"></emph>Hevelius.<emph.end type="italics"></emph.end>Proinde toto hoc tempore, ob <lb></lb>receſſum ipſius a Sole, quoad lumen decrevit, non obſtante ac<lb></lb>ceſſu ad Terram. </s> <s>Cometa Anni 1618 circa medium Menſis <emph type="italics"></emph>De<lb></lb>cembris,<emph.end type="italics"></emph.end>& iſte Anni 1680 circa finem ejuſdem Menſis, celerrime <lb></lb>movebantur, adeoque tunc erant in Perigæis. </s> <s>Verum ſplendor <lb></lb>maximus capitum contigit ante duas fere ſeptimanas, ubi modo <lb></lb>exierant de radiis Solaribus; & ſplendor maximus caudarum <lb></lb>paulo ante, in majore vicinitate Solis. </s> <s>Caput Cometæ prioris, <lb></lb>juxta obſervationes <emph type="italics"></emph>Cyſati, Decemb.<emph.end type="italics"></emph.end>1. majus videbatur ſtellis pri<lb></lb>mæ magnitudinis, & <emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end>16. (jam in Perigæo exiſtens) mag<lb></lb>nitudine parum, ſplendore ſeu claritate luminis plurimum defe<lb></lb>cerat. <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>7. <emph type="italics"></emph>Keplerus<emph.end type="italics"></emph.end>de capite incertus finem fecit obſervandi. </s> <s><lb></lb>Die 12 menſis <emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end>conſpectum & a <emph type="italics"></emph>Flamſtedio<emph.end type="italics"></emph.end>obſervatum <lb></lb>eſt caput Cometæ poſterioris, in diſtantia novem graduum a Sole; <lb></lb>id quod ſtellæ tertiæ magnitudinis vix conceſſum fuiſſet. <emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end><lb></lb>15. & 17 apparuit idem ut ſtella tertiæ magnitudinis, diminutum <lb></lb>utique ſplendore Nubium juxta Solem occidentem. <emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end>26. <lb></lb>velociſſime motus, inque Perigæo propemodum exiſtens, cedebat <lb></lb>ori Pegaſi, Stellæ tertiæ magnitudinis. <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>3. apparebat ut Stella <lb></lb>quartæ, <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>9. ut Stella quintæ, <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>13. ob ſplendorem Lunæ <lb></lb>creſcentis diſparuit. <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>25. vix æquabat Stellas magnitudinis <lb></lb>ſeptimæ. </s> <s>Si ſumantur æqualia a Perigæo hinc inde tempora, ca<lb></lb>pita quæ temporibus illis in longinquis regionibus poſita, ob <lb></lb>æquales a Terra diſtantias, æqualiter lucere debuiſſent, in plaga <lb></lb>Solis maxime ſplenduere, ex altera Perigæi parte evanuere. </s> <s>Igi<lb></lb>tur ex magna lucis in utroque ſitu differentia, concluditur magna <lb></lb>Solis & Cometæ vicinitas in ſitu priore. </s> <s>Nam lux Cometarum <pb xlink:href="039/01/472.jpg" pagenum="444"></pb><arrow.to.target n="note473"></arrow.to.target>regularis eſſe ſolet, & maxima apparere ubi capita velociſſime <lb></lb>moventur, atque adeo ſunt in Perigæis; niſi quatenus ea major <lb></lb>eſt in vicinia Solis. </s></p> <p type="margin"> <s><margin.target id="note472"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note473"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Splendent igitur Cometæ luce Solis a ſe reflexa. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Ex dictis etiam intelligitur cur Cometæ tantopere fre<lb></lb>quentant regionem Solis. </s> <s>Si cernerentur in regionibus longe <lb></lb>ultra Saturnum, deberent ſæpius apparere in partibus Soli oppo<lb></lb>ſitis. </s> <s>Forent enim Terræ viciniores qui in his partibus verſa<lb></lb>rentur, & Sol interpoſitus obſcuraret cæteros. </s> <s>Verum percur<lb></lb>rendo hiſtorias Cometarum, reperi quod quadruplo vel quintuplo <lb></lb>plures detecti ſunt in Hemiſphærio Solem verſus, quam in He<lb></lb>miſphærio oppoſito, præter alios procul dubio non paucos quos <lb></lb>lux Solaris obtexit. </s> <s>Nimirum in deſcenſu ad regiones noſtras <lb></lb>neque caudas emittunt, neque adeo illuſtrantur a Sole, ut nudis <lb></lb>oculis ſe prius detegendos exhibeant, quam ſint ipſo Jove pro<lb></lb>piores. </s> <s>Spatii autem tantillo intervallo circa Solem deſcripti <lb></lb>pars longe major ſita eſt a latere Terræ quod Solem reſpicit; <lb></lb>inque parte illa majore Cometæ, Soli ut plurimum viciniores, <lb></lb>magis illuminari ſolent. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Hinc etiam manifeſtum eſt, quod Cœli reſiſtentia de<lb></lb>ſtituuntur. </s> <s>Nam Cometæ vias obliquas & nonnunquam curſui <lb></lb>Planetarum contrarias ſecuti, moventur omnifariam liberrime, & <lb></lb>motus ſuos etiam contra curſum Planetarum, diutiſſime conſer<lb></lb>vant. </s> <s>Fallor ni genus Planetarum ſint, & motu perpetuo in or<lb></lb>bem redeant. </s> <s>Nam quod Scriptores aliqui Meteora eſſe volunt, <lb></lb>argumentum a capitum perpetuis mutationibus ducentes, funda<lb></lb>mento carere videtur. </s> <s>Capita Cometarum Atmoſphæris ingen<lb></lb>tibus cinguntur; & Atmoſphæræ inferne denſiores eſſe debent. </s> <s><lb></lb>Unde nubes ſunt, non ipſa Cometarum corpora, in quibus muta<lb></lb>tiones illæ viſuntur. </s> <s>Sic Terra ſi e Planetis ſpectaretur, luce nu<lb></lb>bium ſuarum proculdubio ſplenderet, & corpus firmum ſub nu<lb></lb>bibus prope deliteſceret. </s> <s>Sic cingula Jovis in nubibus Planetæ <lb></lb>illius formata eſt, quæ ſitum mutant inter ſe, & firmum Jovis <lb></lb>corpus per nubes illas difficilius cernitur. </s> <s>Et multo magis cor<lb></lb>pora Cometarum ſub Atmoſphæris & profundioribus & craſſiori<lb></lb>bus abſcondi debent. </s></p><pb xlink:href="039/01/473.jpg" pagenum="445"></pb> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XL. THEOREMA XX.<emph.end type="center"></emph.end><lb></lb><arrow.to.target n="note474"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note474"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Cometas in Sectionibus Conicis umbilicos in centro Solis haben<lb></lb>tibus moveri, & radiis ad Solem ductis areas temporibus pro<lb></lb>portionales deſcribere.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Patet per Corol. </s> <s>1. Propoſ. </s> <s>XIII. </s> <s>Libri primi, collatum cum <lb></lb>Prop. </s> <s>VIII, XII & XIII. </s> <s>Libri tertii. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>1. Hinc ſi Cometæ in orbem redeunt: Orbes erunt Ellip<lb></lb>ſes, & tempora periodica erunt ad tempora periodica Planetarum <lb></lb>in axium principalium ratione ſeſquiplicata. </s> <s>Ideoque Cometæ <lb></lb>maxima ex parte ſupra Planetas verſantes, & eo nomine Orbes <lb></lb>axibus majoribus deſcribentes, tardius revolventur. </s> <s>Ut ſi axis Or<lb></lb>bis Cometæ ſit quadruplo major axe Orbis Saturni, tempus revo<lb></lb>lutionis Cometæ erit ad tempus revolutionis Saturni, id eſt, ad <lb></lb>annos 30, ut 4 √ 4 (ſeu 8) ad 1, ideoque erit annorum 240. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>2. Orbes autem erunt Parabolis adeo finitimi, ut eorum <lb></lb>vice Parabolæ, abſque erroribus ſenſibilibus, adhiberi poſſint. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>3. Et propterea, per Corol. </s> <s>7. Prop. </s> <s>XVI. Lib. </s> <s>I. velo<lb></lb>citas Cometæ omnis, erit ſemper ad velocitatem Planetæ cujuſvis <lb></lb>circa Solem in circulo revolventis, in ſubduplicata ratione duplæ <lb></lb>diſtantiæ Planetæ a centro Solis, ad diſtantiam Cometæ a centro <lb></lb>Solis quamproxime. </s> <s>Ponamus radium Orbis magni, ſeu Ellipſeos <lb></lb>in qua Terra revolvitur ſemidiametrum maximam, eſſe partium <lb></lb>100000000: & Terra motu ſuo diurno mediocri deſcribet partes <lb></lb>1720212, & motu horario partes 71675 1/2. Ideoque Cometa in <lb></lb>eadem Telluris a Sole diſtantia mediocri, ea cum velocitate quæ <lb></lb>ſit ad velocitatem Telluris ut √ 2 ad 1, deſcribet motu ſuo diurno <lb></lb>partes 2432747, & motu horario partes 10136. In majoribus <lb></lb>autem vel minoribus diſtantiis, motus tum diurnus tum horarius <lb></lb>erit ad hunc motum diurnum & horarium in ſubduplicata ratione <lb></lb>diſtantiarum reciproce, ideoQ.E.D.tur. </s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>4. Unde ſi Latus rectum Parabolæ quadruplo majus ſit <lb></lb>radio Orbis magni, & quadratum radii illius ponatur eſſe partium <lb></lb>100000000: area quam Cometa radio ad Solem ducto ſingulis die<lb></lb>bus deſcribit, erit partium 1216373 1/4, & ſingulis horis area illa <lb></lb>erit partium 50682 1/4. Sin latus rectum majus ſit vel minus in ra<lb></lb>tione quavis, erit area diurna & horaria major vel minor in ea<lb></lb>dem ratione ſubduplicata. </s></p><pb xlink:href="039/01/474.jpg" pagenum="446"></pb> <p type="main"> <s><arrow.to.target n="note475"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note475"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Invenire lineam curvam generis Parabolici, quæ per data <lb></lb>quotcunque puncta tranſibit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Sunto puncta illa <emph type="italics"></emph>A, B, C, D, E, F,<emph.end type="italics"></emph.end>&c. </s> <s>& ab iiſdem ad rectam <lb></lb>quamvis poſitione datam <emph type="italics"></emph>HN<emph.end type="italics"></emph.end>demitte perpendicula quotcunque <lb></lb><emph type="italics"></emph>AH, BI, CK, DL, EM, FN.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>1. Si punctorum <emph type="italics"></emph>H, I, K, L, M, N<emph.end type="italics"></emph.end>æqualia ſunt inter<lb></lb>valla <emph type="italics"></emph>HI, IK, KL,<emph.end type="italics"></emph.end>&c. </s> <s>collige perpendiculorum <emph type="italics"></emph>AH, BI, <lb></lb>CK,<emph.end type="italics"></emph.end>&c. </s> <s>differentias primas <emph type="italics"></emph>b,<emph.end type="italics"></emph.end>2<emph type="italics"></emph>b,<emph.end type="italics"></emph.end>3<emph type="italics"></emph>b,<emph.end type="italics"></emph.end>4<emph type="italics"></emph>b,<emph.end type="italics"></emph.end>5<emph type="italics"></emph>b,<emph.end type="italics"></emph.end>&c. </s> <s>ſecundas <emph type="italics"></emph>c,<emph.end type="italics"></emph.end>2<emph type="italics"></emph>c,<emph.end type="italics"></emph.end><lb></lb>3<emph type="italics"></emph>c,<emph.end type="italics"></emph.end>4<emph type="italics"></emph>c,<emph.end type="italics"></emph.end>&c. </s> <s>tertias <emph type="italics"></emph>d,<emph.end type="italics"></emph.end>2<emph type="italics"></emph>d,<emph.end type="italics"></emph.end>3<emph type="italics"></emph>d,<emph.end type="italics"></emph.end>&c. </s> <s>id eſt, ita ut ſit <emph type="italics"></emph>AH-BI=b, <lb></lb>BI-CK=2b, CK-DL=3b, DL+EM=4b,-EM+FN=5b,<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.474.1.jpg" xlink:href="039/01/474/1.jpg"></figure><lb></lb>&c. </s> <s>dein <emph type="italics"></emph>b-2b=c,<emph.end type="italics"></emph.end>&c. <lb></lb></s> <s>& ſic pergatur ad diffe<lb></lb>rentiam ultimam quæ hic <lb></lb>eſt <emph type="italics"></emph>f.<emph.end type="italics"></emph.end>Deinde erecta qua<lb></lb>cunque perpendiculari <lb></lb><emph type="italics"></emph>RS,<emph.end type="italics"></emph.end>quæ fuerit ordina<lb></lb>tim applicata ad curvam <lb></lb>quæſitam: ut inveniatur <lb></lb>hujus longitudo, pone <lb></lb>intervalla <emph type="italics"></emph>HI, IK, KL, <lb></lb>LM,<emph.end type="italics"></emph.end>&c. </s> <s>unitates eſſe, <lb></lb>& dic <emph type="italics"></emph>AH=a,-HS=p, <lb></lb>1/2p<emph.end type="italics"></emph.end>in -<emph type="italics"></emph>IS=q, 1/3q<emph.end type="italics"></emph.end>in <lb></lb>+<emph type="italics"></emph>SK=r, 1/4r<emph.end type="italics"></emph.end>in +<emph type="italics"></emph>SL=s, 1/5s<emph.end type="italics"></emph.end>in +<emph type="italics"></emph>SM=t<emph.end type="italics"></emph.end>; pergendo videlicet <lb></lb>ad uſque penultimum perpendiculum <emph type="italics"></emph>ME,<emph.end type="italics"></emph.end>& præponendo ſigna <lb></lb>negativa terminis <emph type="italics"></emph>HS, IS,<emph.end type="italics"></emph.end>&c. </s> <s>qui jacent ad partes puncti <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ver<lb></lb>ſus <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>& ſigna affirmativa terminis <emph type="italics"></emph>SK, SL,<emph.end type="italics"></emph.end>&c. </s> <s>qui jacent <lb></lb>ad alteras partes puncti <emph type="italics"></emph>S.<emph.end type="italics"></emph.end>Et ſignis probe obſervatis, erit <lb></lb><emph type="italics"></emph>RS=a+bp+cq+dr+es+ft,<emph.end type="italics"></emph.end>&c. </s></p> <p type="main"> <s><emph type="italics"></emph>Caſ.<emph.end type="italics"></emph.end>2. Quod ſi punctorum <emph type="italics"></emph>H, I, K, L,<emph.end type="italics"></emph.end>&c. </s> <s>inæqualia ſint inter<lb></lb>valla <emph type="italics"></emph>HI, IK,<emph.end type="italics"></emph.end>&c. </s> <s>collige perpendiculorum <emph type="italics"></emph>AH, BI, CK,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>differentias primas per intervalla perpendiculorum diviſas <emph type="italics"></emph>b,<emph.end type="italics"></emph.end>2<emph type="italics"></emph>b,<emph.end type="italics"></emph.end><lb></lb>3<emph type="italics"></emph>b,<emph.end type="italics"></emph.end>4<emph type="italics"></emph>b,<emph.end type="italics"></emph.end>5<emph type="italics"></emph>b<emph.end type="italics"></emph.end>; ſecundas per intervalla bina diviſas <emph type="italics"></emph>c,<emph.end type="italics"></emph.end>2<emph type="italics"></emph>c,<emph.end type="italics"></emph.end>3<emph type="italics"></emph>c,<emph.end type="italics"></emph.end>4<emph type="italics"></emph>c,<emph.end type="italics"></emph.end>&c. </s> <s><lb></lb>tertias per intervalla terna diviſas <emph type="italics"></emph>d,<emph.end type="italics"></emph.end>2<emph type="italics"></emph>d,<emph.end type="italics"></emph.end>3<emph type="italics"></emph>d,<emph.end type="italics"></emph.end>&c. </s> <s>quartas per <pb xlink:href="039/01/475.jpg" pagenum="447"></pb>intervalla quaterna diviſas <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>2<emph type="italics"></emph>e,<emph.end type="italics"></emph.end>&c. </s> <s>& ſic deinceps; id eſt, ita <lb></lb><arrow.to.target n="note476"></arrow.to.target>ut ſit <emph type="italics"></emph>b=(AH-BI/HI), 2b=(BI-CK/IK), 3b=(CK-DL/KL),<emph.end type="italics"></emph.end>&c. </s> <s>dein <lb></lb><emph type="italics"></emph>c=(b-2b/HK), 2c=(2b-3b/IL), 3c=(3b-4b/KM),<emph.end type="italics"></emph.end>&c. </s> <s>Poſtea <emph type="italics"></emph>d=(c-2c/HL), <lb></lb>2d=(2c-3c/IM),<emph.end type="italics"></emph.end>&c. </s> <s>Inventis differentiis, dic <emph type="italics"></emph>AH=a, -HS=p, <lb></lb>p<emph.end type="italics"></emph.end>in -<emph type="italics"></emph>IS=q, q<emph.end type="italics"></emph.end>in +<emph type="italics"></emph>SK=r, r<emph.end type="italics"></emph.end>in +<emph type="italics"></emph>SL=s, s<emph.end type="italics"></emph.end>in +<emph type="italics"></emph>SM=t<emph.end type="italics"></emph.end>; <lb></lb>pergendo ſcilicet ad uſque perpendiculum penultimum <emph type="italics"></emph>ME,<emph.end type="italics"></emph.end>& erit <lb></lb>ordinatim applicata <emph type="italics"></emph>RS=a+bp+cq+dr+es+ft,<emph.end type="italics"></emph.end>&c. </s></p> <p type="margin"> <s><margin.target id="note476"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Hinc areæ curvarum omnium inveniri poſſunt quampro<lb></lb>xime. </s> <s>Nam ſi curvæ cujuſvis quadrandæ inveniantur puncta ali<lb></lb>quot, & Parabola per eadem duci intelligatur: erit area Parabolæ <lb></lb>hujus eadem quam proxime cum area curvæ illius quadrandæ. </s> <s><lb></lb>Poteſt autem Parabola, per Methodos notiſſimas, ſemper quadrari <lb></lb>Geometrice. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA VI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Ex obſervatis aliquot locis Cometæ invenive locum ejus ad <lb></lb>tempus quodvis intermedium datum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Deſignent <emph type="italics"></emph>HI, IK, KL, LM<emph.end type="italics"></emph.end>tempora inter obſervationes, <lb></lb><emph type="italics"></emph>(in Fig. </s> <s>præced.) HA, IB, KC, LD, ME<emph.end type="italics"></emph.end>obſervatas quinque <lb></lb>longitudines Cometæ, <emph type="italics"></emph>HS<emph.end type="italics"></emph.end>tempus datum inter obſervationem pri<lb></lb>mam & longitudinem quæſitam. </s> <s>Et ſi per puncta <emph type="italics"></emph>A, B, C, D, E<emph.end type="italics"></emph.end><lb></lb>duci intelligatur curva regularis <emph type="italics"></emph>ABCDE<emph.end type="italics"></emph.end>; & per Lemma ſupe<lb></lb>rius inveniatur ejus ordinatim applicata <emph type="italics"></emph>RS,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>RS<emph.end type="italics"></emph.end>longitudo <lb></lb>quæſita. </s></p> <p type="main"> <s>Eadem methodo ex obſervatis quinque latitudinibus invenitur <lb></lb>latitudo ad tempus datum. </s></p> <p type="main"> <s>Si longitudinum obſervatarum parvæ ſint differentiæ, puta gra<lb></lb>duum tantum 4 vel 5; ſuffecerint obſervationes tres vel quatuor <lb></lb>ad inveniendam longitudinem & latitudinem novam. </s> <s>Sin majores <lb></lb>ſint differentiæ, puta graduum 10 vel 20, debebunt obſervationes <lb></lb>quinque adhiberi. <pb xlink:href="039/01/476.jpg" pagenum="448"></pb><arrow.to.target n="note477"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note477"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA VII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Per datum punctum<emph.end type="italics"></emph.end>P <emph type="italics"></emph>ducere rectam lineam<emph.end type="italics"></emph.end>BC, <emph type="italics"></emph>cujus partes<emph.end type="italics"></emph.end><lb></lb>PB, PC, <emph type="italics"></emph>rectis duabus poſitione datis<emph.end type="italics"></emph.end>AB, AC <emph type="italics"></emph>abſciſſæ, da<lb></lb>tam habeant rationem ad invicem.<emph.end type="italics"></emph.end></s></p><figure id="id.039.01.476.1.jpg" xlink:href="039/01/476/1.jpg"></figure> <p type="main"> <s>A puncto illo <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ad rectarum al<lb></lb>terutram <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ducatur recta quævis <lb></lb><emph type="italics"></emph>PD,<emph.end type="italics"></emph.end>& producatur eadem verſus <lb></lb>rectam alteram <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>uſque ad <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>ut <lb></lb>ſit <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PD<emph.end type="italics"></emph.end>in data illa ratione. </s> <s><lb></lb>Ipſi <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>parallela ſit <emph type="italics"></emph>EC<emph.end type="italics"></emph.end>; & ſi <lb></lb>agatur <emph type="italics"></emph>CPB,<emph.end type="italics"></emph.end>erit <emph type="italics"></emph>PC<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PB<emph.end type="italics"></emph.end>ut <lb></lb><emph type="italics"></emph>PE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>PD. q.E.F.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>LEMMA VIII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Sit<emph.end type="italics"></emph.end>ABC <emph type="italics"></emph>Parabola umbilicum habens<emph.end type="italics"></emph.end>S. <emph type="italics"></emph>Chorda<emph.end type="italics"></emph.end>AC <emph type="italics"></emph>biſecta <lb></lb>in<emph.end type="italics"></emph.end>I <emph type="italics"></emph>abſcindatur ſegmentum<emph.end type="italics"></emph.end>ABCI, <emph type="italics"></emph>cujus diameter ſit<emph.end type="italics"></emph.end>I <foreign lang="grc">μ</foreign> <emph type="italics"></emph>& <lb></lb>vertex<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>. <emph type="italics"></emph>In<emph.end type="italics"></emph.end>I <foreign lang="grc">μ</foreign> <emph type="italics"></emph>producta capiatur<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> O <emph type="italics"></emph>æqualis dimidio ipſius<emph.end type="italics"></emph.end><lb></lb><figure id="id.039.01.476.2.jpg" xlink:href="039/01/476/2.jpg"></figure><lb></lb>I <foreign lang="grc">μ</foreign>. <emph type="italics"></emph>Jungatur<emph.end type="italics"></emph.end>OS, <emph type="italics"></emph>& producatur ea ad <foreign lang="grc">ξ</foreign>, ut ſit<emph.end type="italics"></emph.end>S <foreign lang="grc">ξ</foreign> <emph type="italics"></emph>æqualis<emph.end type="italics"></emph.end><lb></lb>2SO. <emph type="italics"></emph>Et ſi Cometa<emph.end type="italics"></emph.end>B <emph type="italics"></emph>moveatur in arcu<emph.end type="italics"></emph.end>CBA, <emph type="italics"></emph>& agatur<emph.end type="italics"></emph.end><lb></lb><foreign lang="grc">ξ</foreign> B <emph type="italics"></emph>ſecans<emph.end type="italics"></emph.end>AC <emph type="italics"></emph>in<emph.end type="italics"></emph.end>E: <emph type="italics"></emph>dico quod punctum<emph.end type="italics"></emph.end>E <emph type="italics"></emph>abſcindet de chordo<emph.end type="italics"></emph.end><lb></lb>AC <emph type="italics"></emph>ſegmentum<emph.end type="italics"></emph.end>AE <emph type="italics"></emph>tempori proportionale quamproxime.<emph.end type="italics"></emph.end></s></p><pb xlink:href="039/01/477.jpg" pagenum="449"></pb> <p type="main"> <s>Jungatur enim <emph type="italics"></emph>EO<emph.end type="italics"></emph.end>ſecans arcum Parabolicum <emph type="italics"></emph>ABC<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Y,<emph.end type="italics"></emph.end>& aga<lb></lb><arrow.to.target n="note478"></arrow.to.target>tur <foreign lang="grc">μ</foreign><emph type="italics"></emph>X<emph.end type="italics"></emph.end>quæ tangat eundem arcum in vertice <foreign lang="grc">μ</foreign> & actæ <emph type="italics"></emph>EO<emph.end type="italics"></emph.end>occur<lb></lb>rat in <emph type="italics"></emph>X<emph.end type="italics"></emph.end>; & erit area curvilinea <emph type="italics"></emph>AEX<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>ad aream curvilineam <lb></lb><emph type="italics"></emph>ACY<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC.<emph.end type="italics"></emph.end>Ideoque cum triangulum <emph type="italics"></emph>ASE<emph.end type="italics"></emph.end>ſit <lb></lb>ad triangulum <emph type="italics"></emph>ASC<emph.end type="italics"></emph.end>in eadem ratione, erit area tota <emph type="italics"></emph>ASEX<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end><lb></lb>ad aream totam <emph type="italics"></emph>ASCY<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC.<emph.end type="italics"></emph.end>Cum autem <foreign lang="grc">ξ</foreign><emph type="italics"></emph>O<emph.end type="italics"></emph.end><lb></lb>ſit ad <emph type="italics"></emph>SO<emph.end type="italics"></emph.end>ut 3 ad 1, & <emph type="italics"></emph>EO<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>XO<emph.end type="italics"></emph.end>in eadem ratione, erit <emph type="italics"></emph>SX<emph.end type="italics"></emph.end><lb></lb>ipſi <emph type="italics"></emph>EB<emph.end type="italics"></emph.end>parallela: & propterea ſi jungatur <emph type="italics"></emph>BX,<emph.end type="italics"></emph.end>erit triangulum <lb></lb><emph type="italics"></emph>SEB<emph.end type="italics"></emph.end>triangulo <emph type="italics"></emph>XEB<emph.end type="italics"></emph.end>æquale. </s> <s>Unde ſi ad aream <emph type="italics"></emph>ASEX<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end><lb></lb>addatur triangulum <emph type="italics"></emph>EXB,<emph.end type="italics"></emph.end>& de ſumma auferatur triangulum <lb></lb><emph type="italics"></emph>SEB,<emph.end type="italics"></emph.end>manebit area <emph type="italics"></emph>ASBX<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>areæ <emph type="italics"></emph>ASEX<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>æqualis, <lb></lb>atque adeo ad aream <emph type="italics"></emph>ASCY<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>AC.<emph.end type="italics"></emph.end>Sed areæ <lb></lb><emph type="italics"></emph>ASBX<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>æqualis eſt area <emph type="italics"></emph>ASBY<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>quamproxime, & hæc <lb></lb>area <emph type="italics"></emph>ASBY<foreign lang="grc">μ</foreign>A<emph.end type="italics"></emph.end>eſt ad aream <emph type="italics"></emph>ASCY<foreign lang="grc">μ</foreign>A,<emph.end type="italics"></emph.end>ut tempus deſcripti <lb></lb>arcus <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>ad tempus deſcripti arcus totius <emph type="italics"></emph>AC.<emph.end type="italics"></emph.end>Ideoque <emph type="italics"></emph>AE<emph.end type="italics"></emph.end><lb></lb>eſt ad <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>in ratione temporum quamproxime. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note478"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Ubi punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>incidit in Parabolæ verticem <foreign lang="grc">μ</foreign>, eſt <emph type="italics"></emph>AE<emph.end type="italics"></emph.end><lb></lb>ad <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>in ratione temporum accurate. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Scholium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Si jungatur <foreign lang="grc">μξ</foreign> ſecans <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>in <foreign lang="grc">δ</foreign> & in ea capiatur <foreign lang="grc">ξ</foreign><emph type="italics"></emph>n<emph.end type="italics"></emph.end>quæ ſit <lb></lb>ad <foreign lang="grc">μ</foreign><emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut 27 <emph type="italics"></emph>MI<emph.end type="italics"></emph.end>ad 16 <emph type="italics"></emph>M<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>: acta <emph type="italics"></emph>Bn<emph.end type="italics"></emph.end>ſecabit chordam <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>in <lb></lb>ratione temporum magis accurate quam prius. </s> <s>Jaceat autem <lb></lb>punctum <emph type="italics"></emph>n<emph.end type="italics"></emph.end>ultra punctum <foreign lang="grc">ξ</foreign>, ſi punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>magis diſtat a vertice <lb></lb>principali Parabolæ quam punctum <foreign lang="grc">μ</foreign>; & citra, ſi minus diſtat ab <lb></lb>eodem vertice. </s></p> <p type="main"> <s><emph type="center"></emph>LEMMA IX.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Rectæ<emph.end type="italics"></emph.end>I<foreign lang="grc">μ</foreign> & <foreign lang="grc">μ</foreign>M <emph type="italics"></emph>& longitudo (AIC/4S<foreign lang="grc">μ</foreign>) æquantur inter ſe.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam 4<emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> eſt latus rectum Parabolæ pertinens ad verti<lb></lb>cem <foreign lang="grc">μ</foreign>. <pb xlink:href="039/01/478.jpg" pagenum="450"></pb><arrow.to.target n="note479"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note479"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA X.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si producatur<emph.end type="italics"></emph.end>S<foreign lang="grc">μ</foreign> <emph type="italics"></emph>ad<emph.end type="italics"></emph.end>N & P, <emph type="italics"></emph>ut<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>N <emph type="italics"></emph>ſit pars tertia ipſius<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>I, <lb></lb>& SP <emph type="italics"></emph>ſit ad<emph.end type="italics"></emph.end>SN <emph type="italics"></emph>ut<emph.end type="italics"></emph.end>SN <emph type="italics"></emph>ad<emph.end type="italics"></emph.end>S<foreign lang="grc">μ</foreign>. <emph type="italics"></emph>Cometa, quo tempore deſcri<lb></lb>bit arcum<emph.end type="italics"></emph.end>A<foreign lang="grc">μ</foreign>C, <emph type="italics"></emph>ſi progrederetur ea ſemper cum velocitate <lb></lb>quam habet in altitudine ipſi<emph.end type="italics"></emph.end>SP <emph type="italics"></emph>æquali, deſcriberet longitudi<lb></lb>nem æqualem chordæ<emph.end type="italics"></emph.end>AC. </s></p> <p type="main"> <s>Nam ſi Cometa velocitate quam habet in <foreign lang="grc">μ</foreign>, eodem tempore <lb></lb>progrederetur uniformiter in recta quæ Parabolam tangit in <foreign lang="grc">μ</foreign>; <lb></lb>area quam radio ad punctum <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ducto deſcriberet, æqualis eſſet <lb></lb>areæ Parabolicæ <emph type="italics"></emph>ASC<emph.end type="italics"></emph.end><foreign lang="grc">μ. </foreign></s> <s>Ideoque contentum ſub longitudine in <lb></lb>tangente deſcripta & longitudine <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>, eſſet ad contentum ſub <lb></lb>longitudinibus <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>& <emph type="italics"></emph>SM,<emph.end type="italics"></emph.end>ut area <emph type="italics"></emph>ASC<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> ad triangulum <lb></lb><emph type="italics"></emph>ASCM,<emph.end type="italics"></emph.end>id eſt, ut <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>SM.<emph.end type="italics"></emph.end>Quare <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>eſt ad longitudi<lb></lb>nem in tangente deſcriptam, ut <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> ad <emph type="italics"></emph>SN.<emph.end type="italics"></emph.end>Cum autem velocitas <lb></lb><figure id="id.039.01.478.1.jpg" xlink:href="039/01/478/1.jpg"></figure><lb></lb>Cometæ in altitudine <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ſit (per Corol. </s> <s>6. Prop. </s> <s>XVI. Lib. </s> <s>I.) <lb></lb>ad velocitatem in altitudine <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>, in ſubduplicata ratione <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ad <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> inverſe, id eſt, in ratione <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> ad <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>; longitudo hac velo<lb></lb>citate eodem tempore deſcripta, erit ad longitudinem in tangente <lb></lb>deſcriptam, ut <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> ad <emph type="italics"></emph>SN,<emph.end type="italics"></emph.end>Igitur <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>& longitudo hac nova ve<lb></lb>locitate deſcripta, cum ſint ad longitudinem in tangente deſcrip<lb></lb>tam in eadem ratione, æquantur inter ſe. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Corol.<emph.end type="italics"></emph.end>Cometa igitur ea cum velocitate, quam habet in altitudine <lb></lb><emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>+2/3<emph type="italics"></emph>I<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>, eodem tempore deſcriberet chordam <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>quamproxime. <pb xlink:href="039/01/479.jpg" pagenum="451"></pb><arrow.to.target n="note480"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note480"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph>LEMMA XI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Si Cometa motu omni privatus de altitudine<emph.end type="italics"></emph.end>SN <emph type="italics"></emph>ſeu<emph.end type="italics"></emph.end>S<foreign lang="grc">μ</foreign>+1/3I<foreign lang="grc">μ</foreign><lb></lb><emph type="italics"></emph>demitteretur, ut caderet in Solem, & ea ſemper vi uniformiter <lb></lb>continuata urgeretur in Solem, qua urgetur ſub initio; idem ſe<lb></lb>miſſe temporis quo in Orbe ſuo deſcribat arcum<emph.end type="italics"></emph.end>AC, <emph type="italics"></emph>deſcenſu <lb></lb>ſuo deſcriberet ſpatium longitudini<emph.end type="italics"></emph.end>I<foreign lang="grc">μ</foreign> <emph type="italics"></emph>æquale.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Nam Cometa quo tempore deſcribat arcum Parabolicum <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end><lb></lb>eodem tempore ea cum velocitate quam habet in altitudine <emph type="italics"></emph>SP<emph.end type="italics"></emph.end><lb></lb>(per Lemma noviſſimum) deſcribet chordam <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>adeoque (per <lb></lb>Corol. </s> <s>7. Prop. </s> <s>XVI. Lib. </s> <s>I.) eodem tempore in Circulo cujus ſemi<lb></lb>diameter eſſet <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>vi gravitatis ſuæ revolvendo, deſcriberet arcum <lb></lb>cujus longitudo eſſet ad arcus Parabolici chordam <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>in ſubdu<lb></lb>plicata ratione unius ad duo. </s> <s>Et propterea eo cum pondere quod <lb></lb>habet in Solem in altitudine <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>cadendo de altitudine illa in <lb></lb>Solem, deſcriberet ſemiſſe temporis illius (per Corol.9. Prop. </s> <s>IV. <lb></lb>Lib. </s> <s>I.) ſpatium æquale quadrato ſemiſſis chordæ illius applicato <lb></lb>ad quadruplum altitudinis <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>id eſt, ſpatium (<emph type="italics"></emph>AIq/4SP<emph.end type="italics"></emph.end>). Unde cum <lb></lb>pondus Cometæ in Solem in altitudine <emph type="italics"></emph>SN,<emph.end type="italics"></emph.end>ſit ad ipſius pondus <lb></lb>in Solem in altitudine <emph type="italics"></emph>SP,<emph.end type="italics"></emph.end>ut <emph type="italics"></emph>SP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>: Cometa pondere <lb></lb>quod habet in altitudine <emph type="italics"></emph>SN<emph.end type="italics"></emph.end>eodem tempore, in Solem caden<lb></lb>do, deſcribet ſpatium (<emph type="italics"></emph>AIq/4S<foreign lang="grc">μ</foreign><emph.end type="italics"></emph.end>), id eſt, ſpatium longitudini <emph type="italics"></emph>I<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> vel <lb></lb><emph type="italics"></emph>M<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> æquale. <emph type="italics"></emph>Q.E.D.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLI. PROBLEMA XXI.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Cometæ in Parabola moti Trajectoriam ex datis tribus <lb></lb>Obſervationibus determinare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Problema hocce longe difficillimum multimode aggreſſus, com<lb></lb>poſui Problemata quædam in Libro primo quæ ad ejus ſolutio<lb></lb>nem ſpectant. </s> <s>Poſtea ſolutionem ſequentem paulo ſimpliciorem <lb></lb>excogitavi. </s></p> <p type="main"> <s>Seligantur tres obſervationes æqualibus temporum intervallis ab <lb></lb>invicem quamproxime diſtantes. </s> <s>Sit autem temporis intervallum <lb></lb>illud ubi Cometa tardius movetur paulo majus altero, ita videlicet <pb xlink:href="039/01/480.jpg" pagenum="452"></pb><arrow.to.target n="note481"></arrow.to.target>ut temporum differentia ſit ad ſummam temporum, ut ſumma tem<lb></lb>porum ad dies plus minus ſexcentos; vel ut punctum <emph type="italics"></emph>E<emph.end type="italics"></emph.end>incidat in <lb></lb>punctum <emph type="italics"></emph>M<emph.end type="italics"></emph.end>quamproxime, & inde aberret verſus <emph type="italics"></emph>I<emph.end type="italics"></emph.end>potius quam <lb></lb>verſus <emph type="italics"></emph>A.<emph.end type="italics"></emph.end>Si tales obſervationes non præſto ſint, inveniendus eſt <lb></lb>novus Cometæ locus per Lemma ſextum. </s></p> <p type="margin"> <s><margin.target id="note481"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Deſignent <emph type="italics"></emph>S<emph.end type="italics"></emph.end>Solem, <emph type="italics"></emph>T, t,<emph.end type="italics"></emph.end><foreign lang="grc">τ</foreign> tria loca Terræ in Orbe magno, <lb></lb><emph type="italics"></emph>TA, tB, <foreign lang="grc">τ</foreign>C<emph.end type="italics"></emph.end>obſervatas tres longitudines Cometæ, V tempus in<lb></lb>ter obſervationem primam & ſecundam, W tempus inter ſecun<lb></lb>dam ac tertiam, X longitudinem quam Cometa toto illo tempore, <lb></lb>ea cum velocitate quam habet in mediocri Telluris à Sole diſtan<lb></lb>tia, deſcribere poſſet, quæque per Corol. </s> <s>3. Prop. </s> <s>XL, Lib. </s> <s>III. <lb></lb>invenienda eſt, & <emph type="italics"></emph>tV<emph.end type="italics"></emph.end>perpendiculum in chordam <emph type="italics"></emph>T<emph.end type="italics"></emph.end><foreign lang="grc">τ. </foreign></s> <s>In longi<lb></lb><figure id="id.039.01.480.1.jpg" xlink:href="039/01/480/1.jpg"></figure><lb></lb>tudine media <emph type="italics"></emph>tB<emph.end type="italics"></emph.end>ſumatur utcunque punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>pro loco Co<lb></lb>metæ in plano Eclipticæ, & inde verſus Solem <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ducatur linea <lb></lb><emph type="italics"></emph>BE,<emph.end type="italics"></emph.end>quæ ſit ad ſagittam <emph type="italics"></emph>tV,<emph.end type="italics"></emph.end>ut contentum ſub <emph type="italics"></emph>SB<emph.end type="italics"></emph.end>& <emph type="italics"></emph>St quad.<emph.end type="italics"></emph.end><lb></lb>ad cubum hypotenuſæ trianguli rectanguli, cujus latera ſunt <emph type="italics"></emph>SB<emph.end type="italics"></emph.end>& <lb></lb>tangens latitudinis Cometæ in obſervatione ſecunda ad radium <emph type="italics"></emph>tB.<emph.end type="italics"></emph.end><pb xlink:href="039/01/481.jpg" pagenum="453"></pb>Et per punctum <emph type="italics"></emph>E<emph.end type="italics"></emph.end>agatur (per hujus Lem. </s> <s>VII.) recta <emph type="italics"></emph>AEC,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note482"></arrow.to.target>cujus partes <emph type="italics"></emph>AE, EC<emph.end type="italics"></emph.end>ad rectas <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>& <foreign lang="grc">τ</foreign><emph type="italics"></emph>C<emph.end type="italics"></emph.end>terminatæ, ſint ad <lb></lb>invicem ut tempora V & W: & erunt <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>C<emph.end type="italics"></emph.end>loca Cometæ in <lb></lb>plano Eclipticæ in obſervatione prima ac tertia quamproxime, ſi <lb></lb>modo <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ſit locus ejus recte aſſumptus in obſervatione ſecunda. </s></p> <p type="margin"> <s><margin.target id="note482"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Ad <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>biſectam in <emph type="italics"></emph>I<emph.end type="italics"></emph.end>erige perpendiculum <emph type="italics"></emph>Ii.<emph.end type="italics"></emph.end>Per punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end><lb></lb>age occultam <emph type="italics"></emph>Bi<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>parallelam. </s> <s>Junge occultam <emph type="italics"></emph>Si<emph.end type="italics"></emph.end>ſecan<lb></lb>tem <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>in <foreign lang="grc">λ</foreign>, & comple parallelogrammum <emph type="italics"></emph>iI<emph.end type="italics"></emph.end><foreign lang="grc">λμ. </foreign></s> <s>Cape <emph type="italics"></emph>I<emph.end type="italics"></emph.end><foreign lang="grc">σ</foreign> æqua<lb></lb>lem 3<emph type="italics"></emph>I<emph.end type="italics"></emph.end><foreign lang="grc">λ</foreign>, & per Solem <emph type="italics"></emph>S<emph.end type="italics"></emph.end>age occultam <foreign lang="grc">σξ</foreign> æqualem 3<emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">σ</foreign>+3<emph type="italics"></emph>i<emph.end type="italics"></emph.end><foreign lang="grc">λ</foreign>, <lb></lb>Et deletis jam literis <emph type="italics"></emph>A, E, C, I,<emph.end type="italics"></emph.end>a puncto <emph type="italics"></emph>B<emph.end type="italics"></emph.end>verſus punctum <foreign lang="grc">ξ</foreign><lb></lb>duc occultam novam <emph type="italics"></emph>BE,<emph.end type="italics"></emph.end>quæ ſit ad priorem <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>in duplicata <lb></lb>ratione diſtantiæ <emph type="italics"></emph>BS<emph.end type="italics"></emph.end>ad quantitatem <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>+1/3<emph type="italics"></emph>i<emph.end type="italics"></emph.end><foreign lang="grc">λ. </foreign></s> <s>Et per punctum <lb></lb><emph type="italics"></emph>E<emph.end type="italics"></emph.end>iterum duc rectam <emph type="italics"></emph>AEC<emph.end type="italics"></emph.end>eadem lege ac prius, id eſt, ita ut ejus <lb></lb>partes <emph type="italics"></emph>AE<emph.end type="italics"></emph.end>& <emph type="italics"></emph>EC<emph.end type="italics"></emph.end>ſint ad invicem, ut tempora inter obſervationes <lb></lb>V & W. </s> <s>Et erunt <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>C<emph.end type="italics"></emph.end>loca Cometæ magis accurate. </s></p> <p type="main"> <s>Ad <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>biſectam in <emph type="italics"></emph>1<emph.end type="italics"></emph.end>erigantur perpendicula <emph type="italics"></emph>AM, CN, IO,<emph.end type="italics"></emph.end><lb></lb>quarum <emph type="italics"></emph>AM<emph.end type="italics"></emph.end>& <emph type="italics"></emph>CN<emph.end type="italics"></emph.end>ſint tangentes latitudinum in obſervatione <lb></lb>prima ac tertia ad radios <emph type="italics"></emph>TA<emph.end type="italics"></emph.end>& <foreign lang="grc">τ</foreign><emph type="italics"></emph>C.<emph.end type="italics"></emph.end>Jungatur <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>ſecans <emph type="italics"></emph>IO<emph.end type="italics"></emph.end><lb></lb>in <emph type="italics"></emph>O.<emph.end type="italics"></emph.end>Conſtituatur rectangulum <emph type="italics"></emph>iI<emph.end type="italics"></emph.end><foreign lang="grc">λμ</foreign> ut prius. </s> <s>In <emph type="italics"></emph>IA<emph.end type="italics"></emph.end>pro<lb></lb>ducta capiatur <emph type="italics"></emph>ID<emph.end type="italics"></emph.end>æqualis <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign>+2/3<emph type="italics"></emph>i<emph.end type="italics"></emph.end><foreign lang="grc">λ</foreign>, & agatur occulta <emph type="italics"></emph>OD.<emph.end type="italics"></emph.end><lb></lb>Deinde in <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>verſus <emph type="italics"></emph>N<emph.end type="italics"></emph.end>capiatur <emph type="italics"></emph>MP,<emph.end type="italics"></emph.end>quæ ſit ad longitudinem <lb></lb>ſupra inventam X, in ſubduplicata ratione mediocris diſtantiæ Tel<lb></lb>luris a Sole (ſeu ſemidiametri Orbis magni) ad diſtantiam <emph type="italics"></emph>OD.<emph.end type="italics"></emph.end><lb></lb>Si punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>incidat in punctum <emph type="italics"></emph>N<emph.end type="italics"></emph.end>; erunt <emph type="italics"></emph>A, B, C<emph.end type="italics"></emph.end>tria loca Co<lb></lb>metæ, per quæ Orbis ejus in plano Eclipticæ deſcribi debet. </s> <s>Sin <lb></lb>punctum <emph type="italics"></emph>P<emph.end type="italics"></emph.end>non incidat in punctum <emph type="italics"></emph>N<emph.end type="italics"></emph.end>; in recta <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>capiatur <lb></lb><emph type="italics"></emph>CG<emph.end type="italics"></emph.end>ipſi <emph type="italics"></emph>NP<emph.end type="italics"></emph.end>æqualis, ita ut puncta <emph type="italics"></emph>G<emph.end type="italics"></emph.end>& <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ad eaſdem partes <lb></lb>rectæ <emph type="italics"></emph>NC<emph.end type="italics"></emph.end>jaceant. </s></p> <p type="main"> <s>Eadem methodo qua puncta <emph type="italics"></emph>E, A, C, G,<emph.end type="italics"></emph.end>ex aſſumpto puncto <lb></lb><emph type="italics"></emph>B<emph.end type="italics"></emph.end>inventa ſunt, inveniantur ex aſſumptis utcunque punctis aliis <lb></lb><emph type="italics"></emph>b<emph.end type="italics"></emph.end>& <foreign lang="grc">β</foreign> puncta nova <emph type="italics"></emph>e, a, c, g,<emph.end type="italics"></emph.end>& <foreign lang="grc">ε, α, χ, γ. </foreign></s> <s>Deinde ſi per <emph type="italics"></emph>G, g,<emph.end type="italics"></emph.end><foreign lang="grc">γ</foreign><lb></lb>ducatur circumferentia circuli <emph type="italics"></emph>Gg<emph.end type="italics"></emph.end><foreign lang="grc">γ</foreign>, ſecans rectam <foreign lang="grc">τ</foreign><emph type="italics"></emph>C<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>: erit <lb></lb><emph type="italics"></emph>Z<emph.end type="italics"></emph.end>locus Cometæ in plano Eclipticæ. </s> <s>Et ſi in <emph type="italics"></emph>AC, ac,<emph.end type="italics"></emph.end><foreign lang="grc">αχ</foreign> capi<lb></lb>antur <emph type="italics"></emph>AF, af,<emph.end type="italics"></emph.end><foreign lang="grc">αφ</foreign> ipſis <emph type="italics"></emph>CG, eg,<emph.end type="italics"></emph.end><foreign lang="grc">χγ</foreign> reſpective æquales, & per <lb></lb>puncta <emph type="italics"></emph>F, f,<emph.end type="italics"></emph.end><foreign lang="grc">φ</foreign> ducatur circumferentia circuli <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end><foreign lang="grc">φ</foreign>, ſecans rectam <lb></lb><emph type="italics"></emph>AT<emph.end type="italics"></emph.end>in <emph type="italics"></emph>X;<emph.end type="italics"></emph.end>erit punctum <emph type="italics"></emph>X<emph.end type="italics"></emph.end>alius Cometæ locus in plano Eclipticæ. </s> <s><lb></lb>Ad puncta <emph type="italics"></emph>X<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Z<emph.end type="italics"></emph.end>erigantur tangentes latitudinum Cometæ ad ra<lb></lb>dios <emph type="italics"></emph>TX<emph.end type="italics"></emph.end>& <foreign lang="grc">τ</foreign><emph type="italics"></emph>Z<emph.end type="italics"></emph.end>; & habebuntur loca duo Cometæ in Orbe proprio. </s> <s><lb></lb>Denique (per Prop. </s> <s>XIX. Lib. </s> <s>I.) umbilico <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>per loca illa duo de<lb></lb>ſcribatur Parabola, & hæc erit Trajectoria Cometæ. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end><pb xlink:href="039/01/482.jpg" pagenum="454"></pb><arrow.to.target n="note483"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note483"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Conſtructionis hujus demonſtratio ex Lemmatibus conſequitur: <lb></lb>quippe cum recta <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>ſecetur in <emph type="italics"></emph>E<emph.end type="italics"></emph.end>in ratione temporum, per <lb></lb>Lemma VII, ut oportet per Lem. </s> <s>VIII: & <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>per Lem. </s> <s>XI. <lb></lb>ſit pars rectæ <emph type="italics"></emph>BS<emph.end type="italics"></emph.end>vel <emph type="italics"></emph>B<emph.end type="italics"></emph.end><foreign lang="grc">ξ</foreign> in plano Eclipticæ arcui <emph type="italics"></emph>ABC<emph.end type="italics"></emph.end>& <lb></lb>chordæ <emph type="italics"></emph>AEC<emph.end type="italics"></emph.end>interjecta; & <emph type="italics"></emph>MP<emph.end type="italics"></emph.end>(per Corol. </s> <s>Lem. </s> <s>X.) longi<lb></lb>tudo ſit chordæ arcus, quem Cometa in Orbe proprio inter ob<lb></lb>ſervationem primam ac tertiam deſcribere debet, ideoQ.E.I.ſi <lb></lb><emph type="italics"></emph>MN<emph.end type="italics"></emph.end>æqualis fuerit, ſi modo <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ſit verus Cometæ locus in plano <lb></lb>Eclipticæ. </s></p><figure id="id.039.01.482.1.jpg" xlink:href="039/01/482/1.jpg"></figure> <p type="main"> <s>Cæterum puncta <emph type="italics"></emph>B, b,<emph.end type="italics"></emph.end><foreign lang="grc">β</foreign> non quælibet, ſed vero proxima eli<lb></lb>gere convenit. </s> <s>Si angulus <emph type="italics"></emph>AQt,<emph.end type="italics"></emph.end>in quo veſtigium Orbis in <lb></lb>plano Eclipticæ deſcriptum ſecat rectam <emph type="italics"></emph>tB,<emph.end type="italics"></emph.end>præterpropter in<lb></lb>noteſcat; in angulo illo ducenda erit recta occulta <emph type="italics"></emph>AC,<emph.end type="italics"></emph.end>quæ ſit <lb></lb>ad 4/3<emph type="italics"></emph>T<emph.end type="italics"></emph.end><foreign lang="grc">τ</foreign> in ſubduplicata ratione <emph type="italics"></emph>SQ<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>St.<emph.end type="italics"></emph.end>Et agendo rectam <lb></lb><emph type="italics"></emph>SEB<emph.end type="italics"></emph.end>cujus pars <emph type="italics"></emph>EB<emph.end type="italics"></emph.end>æquetur longitudini <emph type="italics"></emph>Vt,<emph.end type="italics"></emph.end>determinabitur <lb></lb>punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>quod prima vice uſurpare licet. </s> <s>Tum recta <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>de<lb></lb>leta & ſecundum præcedentem conſtructionem iterum ducta, & <pb xlink:href="039/01/483.jpg" pagenum="455"></pb>inventa inſuper longitudine <emph type="italics"></emph>MP<emph.end type="italics"></emph.end>; in <emph type="italics"></emph>tB<emph.end type="italics"></emph.end>capiatur punctum <emph type="italics"></emph>b,<emph.end type="italics"></emph.end></s></p> <p type="main"> <s><arrow.to.target n="note484"></arrow.to.target>ea lege, ut ſi <emph type="italics"></emph>TA, <foreign lang="grc">τ</foreign>C<emph.end type="italics"></emph.end>ſe mutuo ſecuerint in <emph type="italics"></emph>Y,<emph.end type="italics"></emph.end>ſit diſtantia <emph type="italics"></emph>Yb<emph.end type="italics"></emph.end><lb></lb>ad diſtantiam <emph type="italics"></emph>YB,<emph.end type="italics"></emph.end>in ratione compoſita ex ratione <emph type="italics"></emph>MP<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>MN<emph.end type="italics"></emph.end><lb></lb>& ratione ſubduplicata <emph type="italics"></emph>SB<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>Sb.<emph.end type="italics"></emph.end>Et eadem methodo inveNI<lb></lb>endum erit punctum tertium <foreign lang="grc">β</foreign>, ſi modo operationem tertio repe<lb></lb>tere lubet. </s> <s>Sed hac methodo operationes duæ ut plurimum ſuf<lb></lb>fecerint. </s> <s>Nam ſi diſtantia <emph type="italics"></emph>Bb<emph.end type="italics"></emph.end>perexigua obvenerit; poſtquam <lb></lb>inventa ſunt puncta <emph type="italics"></emph>F, f<emph.end type="italics"></emph.end>& <emph type="italics"></emph>G, g,<emph.end type="italics"></emph.end>actæ rectæ <emph type="italics"></emph>Ff<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Gg<emph.end type="italics"></emph.end>ſecabunt <lb></lb><emph type="italics"></emph>TA<emph.end type="italics"></emph.end>& <foreign lang="grc">τ</foreign><emph type="italics"></emph>C<emph.end type="italics"></emph.end>in punctis quæſitis <emph type="italics"></emph>X<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Z.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note484"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Exemplum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Proponatur Cometa anni 1680. Hujus motum a <emph type="italics"></emph>Flamſtedio<emph.end type="italics"></emph.end><lb></lb>obſervatum Tabula ſequens exhibet. <lb></lb><arrow.to.target n="table9"></arrow.to.target></s></p><table><table.target id="table9"></table.target><row><cell></cell><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></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"></emph>Dec.<emph.end type="italics"></emph.end></cell><cell>12</cell><cell>4.</cell><cell>46</cell><cell>4.</cell><cell>46.</cell><cell>0</cell><cell> 1.</cell><cell>51.</cell><cell>23</cell><cell> 6.</cell><cell>31.</cell><cell>21</cell><cell>8.</cell><cell>26.</cell><cell>0</cell></row><row><cell></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> 5.</cell><cell>7.</cell><cell>38</cell><cell>21.</cell><cell>45.</cell><cell>30</cell></row><row><cell></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><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><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> 13.</cell><cell>11.</cell><cell>45</cell><cell>28.</cell><cell>10.</cell><cell>5</cell></row><row><cell></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"></emph>Jan.<emph.end type="italics"></emph.end></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> 8.</cell><cell>49.</cell><cell>10</cell><cell>26.</cell><cell>15.</cell><cell>26</cell></row><row><cell></cell><cell>9</cell><cell>6.</cell><cell>49</cell><cell>7.</cell><cell>0.</cell><cell>53</cell><cell> 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><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><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><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> 9.</cell><cell>35.</cell><cell>48</cell><cell>17.</cell><cell>56.</cell><cell>54</cell></row><row><cell></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"></emph>Feb.<emph.end type="italics"></emph.end></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><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 Obſervationes quaſdam e noſtris. <lb></lb><arrow.to.target n="table10"></arrow.to.target></s></p><table><table.target id="table10"></table.target><row><cell></cell><cell></cell><cell>Temp. appar.</cell><cell>Cometæ Longit.</cell><cell>Com. Lat.</cell></row><row><cell><emph type="italics"></emph>Febr.<emph.end type="italics"></emph.end></cell><cell>25</cell><cell>8<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.</cell><cell>30′</cell><cell> 26<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>.</cell><cell>18′.</cell><cell>17″</cell><cell>12<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>.</cell><cell>46′ 7/8</cell></row><row><cell></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"></emph>Mart.<emph.end type="italics"></emph.end></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><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><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><cell>9</cell><cell>8.</cell><cell>30</cell><cell> 0.</cell><cell>43.</cell><cell>4</cell><cell>11.</cell><cell>45 7/8</cell></row></table> <p type="main"> <s>Hæ Obſervationes Teleſcopio ſeptupedali, & Micrometro filiſ<lb></lb>Q.E.I. ſoco Teleſcopii locatis peractæ ſunt: quibus inſtrumentis <pb xlink:href="039/01/484.jpg" pagenum="456"></pb><arrow.to.target n="note485"></arrow.to.target>& poſitiones fixarum inter ſe & poſitiones Cometæ ad fixas de<lb></lb>terminavimus. </s> <s>Deſignet <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ſtellam in ſiniſtro calcaneo Perſei <lb></lb><emph type="italics"></emph>(Bayero o) B<emph.end type="italics"></emph.end>ſtellam ſequentem in ſiniſtro pede (<emph type="italics"></emph>Bayero<emph.end type="italics"></emph.end><foreign lang="grc">ζ</foreign>) & <lb></lb><emph type="italics"></emph>C, D, E, F, G, H, I, K, L, M, N, O<emph.end type="italics"></emph.end>ſtellas alias minores in eo<lb></lb>dem pede. </s> <s>Sintque <emph type="italics"></emph>P, Q, R, S, T<emph.end type="italics"></emph.end>loca Cometæ in obſervati<lb></lb>onibus ſupra deſcriptis: & exiſtente diſtantia <emph type="italics"></emph>AB<emph.end type="italics"></emph.end>partium (80 7/12), <lb></lb>erat <emph type="italics"></emph>AC<emph.end type="italics"></emph.end>partium 52 1/4, <emph type="italics"></emph>BC<emph.end type="italics"></emph.end>58 5/6, <emph type="italics"></emph>AD<emph.end type="italics"></emph.end>(57 5/12), <emph type="italics"></emph>BD<emph.end type="italics"></emph.end>(82 6/11), <emph type="italics"></emph>CD<emph.end type="italics"></emph.end>23 2/3, <lb></lb><emph type="italics"></emph>AE<emph.end type="italics"></emph.end>29 4/7, <emph type="italics"></emph>CE<emph.end type="italics"></emph.end>57 1/2, <emph type="italics"></emph>DE<emph.end type="italics"></emph.end>(49 11/12), <emph type="italics"></emph>AI<emph.end type="italics"></emph.end>(27 7/12), <emph type="italics"></emph>BI<emph.end type="italics"></emph.end>52 1/6, <emph type="italics"></emph>CI<emph.end type="italics"></emph.end>(36 7/12), <lb></lb><figure id="id.039.01.484.1.jpg" xlink:href="039/01/484/1.jpg"></figure><lb></lb><emph type="italics"></emph>DI<emph.end type="italics"></emph.end>(53 5/11), <emph type="italics"></emph>AK<emph.end type="italics"></emph.end>38 2/3, <emph type="italics"></emph>BK<emph.end type="italics"></emph.end>43, <emph type="italics"></emph>CK<emph.end type="italics"></emph.end>31 5/9, <emph type="italics"></emph>FK<emph.end type="italics"></emph.end>29, <emph type="italics"></emph>FB<emph.end type="italics"></emph.end>23, <emph type="italics"></emph>FC<emph.end type="italics"></emph.end>36 1/4, <lb></lb><emph type="italics"></emph>AH<emph.end type="italics"></emph.end>18 6/7, <emph type="italics"></emph>DH<emph.end type="italics"></emph.end>50 7/8, <emph type="italics"></emph>BN<emph.end type="italics"></emph.end>(46 5/12), <emph type="italics"></emph>CN<emph.end type="italics"></emph.end>31 1/3, <emph type="italics"></emph>BL<emph.end type="italics"></emph.end>(45 5/12), <emph type="italics"></emph>NL<emph.end type="italics"></emph.end>31 5/7. <lb></lb><emph type="italics"></emph>HO<emph.end type="italics"></emph.end>erat ad <emph type="italics"></emph>HI<emph.end type="italics"></emph.end>ut 7 ad 6 & producta tranſibat inter ſtellas <lb></lb>D & <emph type="italics"></emph>E,<emph.end type="italics"></emph.end>ſic ut diſtantia ſtellæ <emph type="italics"></emph>D<emph.end type="italics"></emph.end>ab hac recta eſſet 1/6<emph type="italics"></emph>CD. LM<emph.end type="italics"></emph.end><lb></lb>erat ad <emph type="italics"></emph>LB<emph.end type="italics"></emph.end>ut 2 ad 9 & producta tranſibat per ſtellam <emph type="italics"></emph>H.<emph.end type="italics"></emph.end>His <lb></lb>interminabantur poſitiones fixarum inter ſe. </s></p> <p type="margin"> <s><margin.target id="note485"></margin.target>E MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Die Veneris <emph type="italics"></emph>Feb.<emph.end type="italics"></emph.end>25. St. </s> <s>vet. </s> <s>Hor. </s> <s>8 1/2 P. M. </s> <s>Cometæ in <emph type="italics"></emph>p<emph.end type="italics"></emph.end>ex<lb></lb>iſtentis diſtantia a ſtella <emph type="italics"></emph>E<emph.end type="italics"></emph.end>erat minor quam (3/13) <emph type="italics"></emph>AE,<emph.end type="italics"></emph.end>major quam <lb></lb>3/5 <emph type="italics"></emph>AE,<emph.end type="italics"></emph.end>adeoque æqualis (3/14)<emph type="italics"></emph>AE<emph.end type="italics"></emph.end>proxime; & angulus <emph type="italics"></emph>ApE<emph.end type="italics"></emph.end>non<lb></lb>nihil obtuſus erat, ſed fere rectus. </s> <s>Nempe ſi demitteretur ad <lb></lb><emph type="italics"></emph>pE<emph.end type="italics"></emph.end>perpendiculum ab <emph type="italics"></emph>A,<emph.end type="italics"></emph.end>diſtantiæ Cometæ a perpendiculo illo <lb></lb>erat 1/5<emph type="italics"></emph>pE.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Eadem nocte, hora 9 1/2, Cometæ in <emph type="italics"></emph>P<emph.end type="italics"></emph.end>exiſtentis diſtantia a ſtella <lb></lb><emph type="italics"></emph>E<emph.end type="italics"></emph.end>erat major quam (1/(4 1/2))<emph type="italics"></emph>AE,<emph.end type="italics"></emph.end>minor quam (1/(5 1/4))<emph type="italics"></emph>AE,<emph.end type="italics"></emph.end>adeoque æqua-<pb xlink:href="039/01/485.jpg" pagenum="457"></pb>lis (1/(4 7/8))<emph type="italics"></emph>AE,<emph.end type="italics"></emph.end>ſeu (1/39)<emph type="italics"></emph>AE<emph.end type="italics"></emph.end>quamproxime. </s> <s>A perpendiculo autem a <lb></lb><arrow.to.target n="note486"></arrow.to.target>ſtella <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ad rectam <emph type="italics"></emph>PE<emph.end type="italics"></emph.end>demiſſo, diſtantia Cometæ erat 4/5<emph type="italics"></emph>PE.<emph.end type="italics"></emph.end></s></p> <p type="margin"> <s><margin.target id="note486"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Die <emph type="sup"></emph>is<emph.end type="sup"></emph.end>, <emph type="italics"></emph>Feb.<emph.end type="italics"></emph.end>27. hor. </s> <s>8 1/4 P.M. </s> <s>Cometæ in <emph type="italics"></emph>Q<emph.end type="italics"></emph.end>exiſtentis di<lb></lb>ſtantia a ſtella <emph type="italics"></emph>O<emph.end type="italics"></emph.end>æquabat diſtantiam ſtellarum <emph type="italics"></emph>O<emph.end type="italics"></emph.end>& <emph type="italics"></emph>H,<emph.end type="italics"></emph.end>& recta <lb></lb><emph type="italics"></emph>QO<emph.end type="italics"></emph.end>producta tranſibat inter ſtellas <emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B.<emph.end type="italics"></emph.end>Poſitionem hujus <lb></lb>rectæ ob nubes intervenientes, magis accurate definire non potui. </s></p> <p type="main"> <s>Die <emph type="sup"></emph>tis<emph.end type="sup"></emph.end>, <emph type="italics"></emph>Mart<emph.end type="italics"></emph.end>1, hor. </s> <s>11. P.M. </s> <s>Cometa in <emph type="italics"></emph>R<emph.end type="italics"></emph.end>exiſtens, ſtellis <lb></lb><emph type="italics"></emph>K<emph.end type="italics"></emph.end>& <emph type="italics"></emph>C<emph.end type="italics"></emph.end>accurate interjacebat, & rectæ <emph type="italics"></emph>CRK<emph.end type="italics"></emph.end>pars <emph type="italics"></emph>CR<emph.end type="italics"></emph.end>paulo <lb></lb>major erat quam 1/3<emph type="italics"></emph>CK,<emph.end type="italics"></emph.end>& paulo minor quam 1/3<emph type="italics"></emph>CK<emph.end type="italics"></emph.end>+1/8<emph type="italics"></emph>CR,<emph.end type="italics"></emph.end><lb></lb>adeoque æqualis 1/3<emph type="italics"></emph>CK<emph.end type="italics"></emph.end>+(1/16)<emph type="italics"></emph>CR<emph.end type="italics"></emph.end>ſeu (16/45)<emph type="italics"></emph>CK.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Die <emph type="sup"></emph>ii<emph.end type="sup"></emph.end>, <emph type="italics"></emph>Mart.<emph.end type="italics"></emph.end>2. hor. </s> <s>8. P.M. </s> <s>Cometæ exiſtentis in <emph type="italics"></emph>S,<emph.end type="italics"></emph.end>di<lb></lb>ſtantia a ſtella <emph type="italics"></emph>C<emph.end type="italics"></emph.end>erat 4/9<emph type="italics"></emph>FC<emph.end type="italics"></emph.end>quamproxime. </s> <s>Diſtantia ſtellæ <emph type="italics"></emph>F<emph.end type="italics"></emph.end>a <lb></lb>recta <emph type="italics"></emph>CS<emph.end type="italics"></emph.end>producta erat (1/24)<emph type="italics"></emph>FC<emph.end type="italics"></emph.end>; & diſtantia ſtellæ <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ab eadem recta, <lb></lb>erat quintuplo major quam diſtantia ſtellæ <emph type="italics"></emph>F.<emph.end type="italics"></emph.end>Item recta <emph type="italics"></emph>NS<emph.end type="italics"></emph.end><lb></lb>producta tranſibat inter ſtellas <emph type="italics"></emph>H<emph.end type="italics"></emph.end>& <emph type="italics"></emph>I,<emph.end type="italics"></emph.end>quintuplo vel ſextuplo pro<lb></lb>pior exiſtens ſtellæ <emph type="italics"></emph>H<emph.end type="italics"></emph.end>quam ſtellæ <emph type="italics"></emph>I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Die <emph type="sup"></emph>ni<emph.end type="sup"></emph.end>, <emph type="italics"></emph>Mart.<emph.end type="italics"></emph.end>5. hor. </s> <s>11 1/2. P. M. </s> <s>Cometa exiſtente in <emph type="italics"></emph>T,<emph.end type="italics"></emph.end><lb></lb>recta <emph type="italics"></emph>MT<emph.end type="italics"></emph.end>æqualis erat 1/2<emph type="italics"></emph>ML,<emph.end type="italics"></emph.end>& recta <emph type="italics"></emph>LT<emph.end type="italics"></emph.end>producta tranſibat <lb></lb>inter <emph type="italics"></emph>B<emph.end type="italics"></emph.end>& <emph type="italics"></emph>F,<emph.end type="italics"></emph.end>quadruplo vel quintuplo propior <emph type="italics"></emph>F<emph.end type="italics"></emph.end>quam <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>au<lb></lb>ferens a <emph type="italics"></emph>BF<emph.end type="italics"></emph.end>quintam vel ſextam ejus partem verſus <emph type="italics"></emph>F.<emph.end type="italics"></emph.end>Et <emph type="italics"></emph>MT<emph.end type="italics"></emph.end><lb></lb>producta tranſibat extra ſpatium <emph type="italics"></emph>BF<emph.end type="italics"></emph.end>ad partes ſtellæ <emph type="italics"></emph>B,<emph.end type="italics"></emph.end>quadru<lb></lb>plo propior exiſtens ſtellæ <emph type="italics"></emph>B<emph.end type="italics"></emph.end>quam ſtellæ <emph type="italics"></emph>F.<emph.end type="italics"></emph.end>Erat <emph type="italics"></emph>M<emph.end type="italics"></emph.end>ſtella pere<lb></lb>xigua quæ per Teleſcopium videri vix potuit, & <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ſtella major <lb></lb>quaſi magnitudinis octavæ. </s></p> <p type="main"> <s>Ex hujuſmodi obſervationibus per conſtructiones figurarum & <lb></lb>computationes (poſito quod ſtellarum <emph type="italics"></emph>A<emph.end type="italics"></emph.end>& <emph type="italics"></emph>B<emph.end type="italics"></emph.end>diſtantia eſſet <lb></lb>2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 6′. </s> <s>46″, & ſtellæ <emph type="italics"></emph>A<emph.end type="italics"></emph.end>longitudo 26<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 41′. </s> <s>50″ & latitudo <lb></lb>borealis 12<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 8′ 1/2, ſtellæque <emph type="italics"></emph>B<emph.end type="italics"></emph.end>longitudo 28<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 40′. </s> <s>24″ & lati<lb></lb>tudo borealis 11<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> (17′ 9/10);) derivabam longitudines & latitudines <lb></lb>Cometæ. </s> <s>Micrometro parum affabre conſtructo uſus ſum, ſed <lb></lb>longitudinum tamen & latitudinum errores (quatenus ab ob<lb></lb>ſervationibus noſtris oriantur) dimidium minuti unius primi vix <lb></lb>ſuperant, præterquam in obſervatione ultima <emph type="italics"></emph>Mart.<emph.end type="italics"></emph.end>9. ubi poſi<lb></lb>tiones ſtellarum minus accurate determinare potui. <emph type="italics"></emph>Caſſinus<emph.end type="italics"></emph.end>qui <lb></lb>aſcenſionem rectam Cometæ eodem tempore obſervavit, decli<lb></lb>nationem ejus tanquam invariatam manentem parum diligenter <lb></lb>definivit. </s> <s>Nam Cometa (juxta obſervationes noſtras) in fine <pb xlink:href="039/01/486.jpg" pagenum="458"></pb><arrow.to.target n="note487"></arrow.to.target>motus ſui notabiliter deflectere cœpit boream verſus, a paral<lb></lb>lelo quem in fine Menſis <emph type="italics"></emph>Februarii<emph.end type="italics"></emph.end>tenuerat. </s></p> <p type="margin"> <s><margin.target id="note487"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Jam ad Orbem Cometæ determinandum; ſelegi ex obſervatio<lb></lb>nibus hactenus deſcriptis tres, quas <emph type="italics"></emph>Flamſtedius<emph.end type="italics"></emph.end>habuit <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>21, <lb></lb><emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>5, & <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>25. Ex his inveni <emph type="italics"></emph>St<emph.end type="italics"></emph.end>partium 9842,1 & <emph type="italics"></emph>Vt<emph.end type="italics"></emph.end>par<lb></lb>tium 455, quales 10000 ſunt ſemidiameter Orbis magni. </s> <s>Tum <lb></lb>ad operationem primam aſſumendo <emph type="italics"></emph>tB<emph.end type="italics"></emph.end>partium 5657, inveni <lb></lb><emph type="italics"></emph>SB<emph.end type="italics"></emph.end>9747, <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>prima vice 412, <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">μ</foreign> 9503, <emph type="italics"></emph>i<emph.end type="italics"></emph.end><foreign lang="grc">λ</foreign> 413: <emph type="italics"></emph>BE<emph.end type="italics"></emph.end>ſecun<lb></lb>da vice 421, <emph type="italics"></emph>OD<emph.end type="italics"></emph.end>10186, X 8528,4, <emph type="italics"></emph>MP<emph.end type="italics"></emph.end>8450, <emph type="italics"></emph>MN<emph.end type="italics"></emph.end>8475, <lb></lb><emph type="italics"></emph>NP<emph.end type="italics"></emph.end>25. Unde ad operationem ſecundam collegi diſtantiam <lb></lb><emph type="italics"></emph>tb<emph.end type="italics"></emph.end>5640. Et per hanc operationem inveni tandem diſtantias <lb></lb><emph type="italics"></emph>TX<emph.end type="italics"></emph.end>4775 & <foreign lang="grc">τ</foreign><emph type="italics"></emph>Z<emph.end type="italics"></emph.end>11322. Ex quibus Orbem definiendo, inveni <lb></lb>Nodos ejus deſcendentem in & aſcendentem in 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 53′; <lb></lb>Inclinationem plani ejus ad planum Eclipticæ 61<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 20′ 2/3; verti<lb></lb>cem ejus (ſeu Perihelium Cometæ) diſtare a Nodo 8<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 38′, & <lb></lb>eſſe in 27<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 43′ cum latitudine auſtrali 7<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 34′; & ejus latus <lb></lb>rectum eſſe 236,8, areamque radio ad Solem ducto ſingulis diebus <lb></lb>deſcriptam 93585, quadrato ſemidiametri Orbis magni poſito <lb></lb>100000000; Cometam vero in hoc Orbe ſecundum ſeriem ſigno<lb></lb>rum proceſſiſſe, & <emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end>8<emph type="sup"></emph>d<emph.end type="sup"></emph.end>. </s> <s>0<emph type="sup"></emph>h<emph.end type="sup"></emph.end>. </s> <s>4′. </s> <s>P. M. in vertice Orbis ſeu <lb></lb>Perihelio fuiſſe. </s> <s>Hæc omnia per ſcalam partium æqualium & <lb></lb>chordas angulorum ex Tabula ſinuum naturalium collectas, deter<lb></lb>minavi Graphice; conſtruendo Schema ſatis amplum, in quo vide<lb></lb>licet ſemidiameter Orbis magni (partium 10000) æqualis eſſet <lb></lb>digitis 16 2/3 pedis Anglicani. </s></p> <p type="main"> <s>Tandem ut conſtaret an Cometa in Orbe ſic invento vere mo<lb></lb>veretur, collegi per operationes partim Arithmeticas partim Gra<lb></lb>phicas, loca Cometæ in hoc Orbe ad obſervationum quarundam <lb></lb>tempora: uti in Tabula ſequente videre licet. <lb></lb><arrow.to.target n="table11"></arrow.to.target> </s></p><table><table.target id="table11"></table.target><row><cell></cell><cell></cell><cell>Diſtant.Co<lb></lb>metæ a Sole</cell><cell>Long.Collect.</cell><cell>Lat. Collect.</cell><cell>Long. Obſ.</cell><cell>Lat. Obſ.</cell><cell>Differ <lb></lb> Long.</cell><cell>Differ. <lb></lb> Lat.</cell></row><row><cell></cell><cell></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"></emph>Dec.<emph.end type="italics"></emph.end></cell><cell>12</cell><cell>2792</cell><cell> 6.</cell><cell>32</cell><cell>8.</cell><cell>18 1/2</cell><cell> 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> 13.</cell><cell>13 2/3</cell><cell>28.</cell><cell>0</cell><cell> 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"></emph>Febr.<emph.end type="italics"></emph.end></cell><cell>5</cell><cell>16669</cell><cell> 17.</cell><cell>0</cell><cell>15.</cell><cell>29 2/3</cell><cell> 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"></emph>Mar.<emph.end type="italics"></emph.end></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>Poſtea vero <emph type="italics"></emph>Halleius<emph.end type="italics"></emph.end>noſter Orbitam, per calculum Arithmeti<lb></lb>cum, accuratius determinavit quam per deſeriptiones linearum <lb></lb>fieri licuit; & retinuit quidem locum Nodorum in & 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 53′, <lb></lb>& Inclinationem plani Orbitæ ad Eclipticam 61<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 20′ 1/3, ut & tem<lb></lb>pus Perihelii Cometæ <emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end>8<emph type="sup"></emph>d<emph.end type="sup"></emph.end>. </s> <s>O<emph type="sup"></emph>h<emph.end type="sup"></emph.end>. </s> <s>4′: diſtantiam vero Peri-<pb xlink:href="039/01/487.jpg" pagenum="459"></pb>helii a Nodo aſcendente, in Orbita Cometæ menſuratam, invenit <lb></lb><arrow.to.target n="note488"></arrow.to.target>eſſe 9<emph type="sup"></emph>gr<emph.end type="sup"></emph.end> 20′, & Latus rectum Parabolæ eſſe 243 partium, ex<lb></lb>iſtente mediocri Solis a Terra diſtantia partium 10000. Et ex his <lb></lb>datis, calculo itidem Arithmetico accurate inſtituto, loca Cometæ <lb></lb>ad obſervationum tempora computavit, ut ſequitur. <lb></lb><arrow.to.target n="table12"></arrow.to.target> </s></p> <p type="margin"> <s><margin.target id="note488"></margin.target>LIBER <lb></lb>TERTIUS.</s></p><table><table.target id="table12"></table.target><row><cell>Tempus verum</cell><cell>Diſtantia</cell><cell>Long. comp.</cell><cell>Lat. comp.</cell><cell>Errores in</cell></row><row><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell>Cometæ a </cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell>Long.</cell><cell>Lat.</cell></row><row><cell></cell><cell>d.</cell><cell>h.</cell><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>″</cell></row><row><cell><emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end></cell><cell>12.</cell><cell>4.</cell><cell>46.</cell><cell>0</cell><cell>28028</cell><cell> 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><cell>21.</cell><cell>6.</cell><cell>36.</cell><cell>59</cell><cell>61076</cell><cell> 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><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><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><cell>29.</cell><cell>8.</cell><cell>3.</cell><cell>2</cell><cell>84021</cell><cell> 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><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"></emph>Jan.<emph.end type="italics"></emph.end></cell><cell>5.</cell><cell>6.</cell><cell>1.</cell><cell>38</cell><cell>101440</cell><cell> 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><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><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><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><cell>25.</cell><cell>7.</cell><cell>58.</cell><cell>42</cell><cell>145370</cell><cell> 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><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"></emph>Feb.<emph.end type="italics"></emph.end></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><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><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"></emph>Mar.<emph.end type="italics"></emph.end></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 menſe <emph type="italics"></emph>Novembri<emph.end type="italics"></emph.end>præcedente, & <lb></lb>die undecimo hujus menſis ſtylo veteri, ad horam quintam ma<lb></lb>tutinam, <emph type="italics"></emph>Cantuariæ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Anglia,<emph.end type="italics"></emph.end>viſus fuit in 12 1/2 cum latitudine <lb></lb>boreali 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> circiter. </s> <s>Craſſiſſima fuit hæc Obſervatio: meliores ſunt <lb></lb>quæ ſequuntur. </s></p> <p type="main"> <s><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>17, ſt. </s> <s>vet. <emph type="italics"></emph>Pontbæus<emph.end type="italics"></emph.end>& ſocii hora ſexta matutina <emph type="italics"></emph>Romæ<emph.end type="italics"></emph.end><lb></lb>(id eſt, hora 5, 10′ <emph type="italics"></emph>Londini<emph.end type="italics"></emph.end>) filis ad fixas applicatis Cometam <lb></lb>obſervarunt in 8. 30′, cum latitudine auſtrali 0<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 40′. </s> <s>Extant <lb></lb>eorum Obſervationes in tractatu quem <emph type="italics"></emph>Penthæus,<emph.end type="italics"></emph.end>de hoc Cometa, <lb></lb>in lucem edidit. <emph type="italics"></emph>Cellius<emph.end type="italics"></emph.end>qui aderat & obſervationes ſuas in Epi<lb></lb>ſtola ad <emph type="italics"></emph>D. Caſſinum<emph.end type="italics"></emph.end>miſit, Cometam eadem hora vidit in 8 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end><lb></lb>30′ cum latitudine auſtrali 0<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′. </s> <s>Eadem hora <emph type="italics"></emph>Galletius<emph.end type="italics"></emph.end>etiam <lb></lb>Cometam vidit in 8<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> ſine latitudine. </s></p> <p type="main"> <s><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>18. hora matutina 6. 30′ <emph type="italics"></emph>Romæ<emph.end type="italics"></emph.end>(id eſt, hora 5, 40′ <emph type="italics"></emph>Lon<lb></lb>dini) Ponthæus<emph.end type="italics"></emph.end>Cometam vidit in 13<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′ cum latitudine au<lb></lb>ſtrali 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 20′. <emph type="italics"></emph>Cellius<emph.end type="italics"></emph.end>in 13<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 00′, cum latitudine auſtrali <lb></lb>1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 00′. <emph type="italics"></emph>Galletius<emph.end type="italics"></emph.end>autem hora matutina 5. 30′ <emph type="italics"></emph>Romæ,<emph.end type="italics"></emph.end>Cometam <lb></lb>vidit in 13<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 00′, cum latitudine auſtrali 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 00′. </s> <s>Et <emph type="italics"></emph>R. P. <lb></lb>Ango<emph.end type="italics"></emph.end>in Academia <emph type="italics"></emph>Flexienſi<emph.end type="italics"></emph.end>apud <emph type="italics"></emph>Galles,<emph.end type="italics"></emph.end>hora quinta matutina <lb></lb>(id eſt, hora 5, 9′ <emph type="italics"></emph>Londini<emph.end type="italics"></emph.end>) Cometam vidit in medio inter ſtellas <pb xlink:href="039/01/488.jpg" pagenum="460"></pb><arrow.to.target n="note489"></arrow.to.target>duas parvas, quarum una media eſt trium in recta linea in Virgi<lb></lb>nis auſtrali manu, & altera eſt extrema alæ. </s> <s>Unde Cometa tunc <lb></lb>fuit in 12. 46′, cum latitudine auſtrali 50′. </s> <s>Eodem die <emph type="italics"></emph>Bo<lb></lb>ſtoniæ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Nova-Anglia<emph.end type="italics"></emph.end>in Latitudine 42 1/2 graduum, hora quinta <lb></lb>matutina, (id eſt <emph type="italics"></emph>Londini<emph.end type="italics"></emph.end>hora matutina 9. 44′) Cometa viſus <lb></lb>eſt prope 14, cum latitudine auſtrali 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′, uti a <emph type="italics"></emph>Cl. </s> <s>Hal<lb></lb>leio<emph.end type="italics"></emph.end>accepi. </s></p> <p type="margin"> <s><margin.target id="note489"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>19. hora mat. </s> <s>4 1/2 <emph type="italics"></emph>Cantabrigiæ,<emph.end type="italics"></emph.end>Cometa (obſervante ju<lb></lb>vene quodam) diſtabat a Spica quaſi 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> Boreazephyrum <lb></lb>verſus. </s> <s>Eodem die hor. </s> <s>5. mat. <emph type="italics"></emph>Boſtoniæ<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Nova-Anglia,<emph.end type="italics"></emph.end>Co<lb></lb>meta diſtabat a Spica gradu uno, differentia latitudinum ex<lb></lb>iſtente 40′. </s> <s>Eodem die in Inſula <emph type="italics"></emph>Jamaica,<emph.end type="italics"></emph.end>Cometa diſtabat a Spica <lb></lb>intervallo quaſi gradus unius. </s> <s>Et ex his obſervationibus inter ſe <lb></lb>collatis colligo, quod hora 9. 44′. <emph type="italics"></emph>Londini,<emph.end type="italics"></emph.end>Cometa erat in 18 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end><lb></lb>40′, cum latitudine auſtrali 1 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 18′ circiter. </s> <s>Eodem die D. <emph type="italics"></emph>Ar<lb></lb>thurus Storer<emph.end type="italics"></emph.end>ad fluvium <emph type="italics"></emph>Patuxent,<emph.end type="italics"></emph.end>prope <emph type="italics"></emph>Hunting-Creek<emph.end type="italics"></emph.end>in <emph type="italics"></emph>Mary<lb></lb>Land,<emph.end type="italics"></emph.end>in confinio <emph type="italics"></emph>Virginiæ<emph.end type="italics"></emph.end>in Lat. </s> <s>38 1/2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> hora quinta matutina <lb></lb>(id eſt, hora 10<emph type="sup"></emph>2<emph.end type="sup"></emph.end> <emph type="italics"></emph>Londini<emph.end type="italics"></emph.end>) Cometam vidit ſupra Spicam , & <lb></lb>cum Spica propemodum conjunctum, exiſtente diſtantia inter eoſ<lb></lb>dem quaſi 3/4<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>. </s> <s>Obſervator idem, eadem hora diei ſequentis, <lb></lb>Cometam vidit quaſi 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> inferiorem Spica. </s> <s>Congruent hæ ob<lb></lb>ſervationes cum obſervationibus in <emph type="italics"></emph>Nova-Anglia<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Jamaica<emph.end type="italics"></emph.end>factis, <lb></lb>ſi modo diſtantiæ (pro motu diurno Cometæ) nonnihil augean<lb></lb>tur, ita ut Cometa die priore ſuperior eſſet Spica , altitudine <lb></lb>1 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> circiter, ac die poſteriore inferior eadem ſtella, altitudine per<lb></lb>pendiculari 3 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 40′. </s></p> <p type="main"> <s><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>20. D. <emph type="italics"></emph>Montenarus<emph.end type="italics"></emph.end>Aſtronomiæ Profeſſor <emph type="italics"></emph>Paduenſis,<emph.end type="italics"></emph.end>hora <lb></lb>ſexta matutina <emph type="italics"></emph>Venetiis<emph.end type="italics"></emph.end>(id eſt, hora 5. 10′ <emph type="italics"></emph>Londini<emph.end type="italics"></emph.end>) Cometam <lb></lb>vidit in 23 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>, cum latitudine auſtrali 1 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′. </s> <s>Eodem die <lb></lb><emph type="italics"></emph>Boſtoniæ,<emph.end type="italics"></emph.end>diſtabat Cometa a Spica , 4<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> longitudinis in orien<lb></lb>tem, adeoque erat in 23 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 24′ circiter. </s></p> <p type="main"> <s><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>21. <emph type="italics"></emph>Ponthæus<emph.end type="italics"></emph.end>& ſocii hor. </s> <s>mat. </s> <s>7 1/4 Cometam obſerva<lb></lb>runt in 27<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 50′, cum latitudine auſtrali 1 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 16′; <emph type="italics"></emph>Ango<emph.end type="italics"></emph.end>hora <lb></lb>quinta matutina in 27<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 45′, <emph type="italics"></emph>Montenarus<emph.end type="italics"></emph.end>in 27<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 51′. </s> <s>Eo<lb></lb>dem die in Inſula <emph type="italics"></emph>Jamaica,<emph.end type="italics"></emph.end>Cometa viſus eſt prope principium <lb></lb>Scorpii, eandemque circiter latitudinem habuit cum Spica Virgi<lb></lb>nis, id eſt, 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 2′. </s></p> <p type="main"> <s><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>22. Cometa viſus eſt a <emph type="italics"></emph>Montenaro<emph.end type="italics"></emph.end>in 2. 33′. <emph type="italics"></emph>Boſtoniæ<emph.end type="italics"></emph.end><lb></lb>autem in <emph type="italics"></emph>Nova-Anglia<emph.end type="italics"></emph.end>apparuit in 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> circiter, eadem fere <lb></lb>cum latitudine ac prius, id eſt, 1 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′. </s> <s>Eodem die <emph type="italics"></emph>Londini,<emph.end type="italics"></emph.end><pb xlink:href="039/01/489.jpg" pagenum="461"></pb>hora mat. </s> <s>6 1/2 <emph type="italics"></emph>Hookius<emph.end type="italics"></emph.end>noſter Cometam vidit in 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′ cir<lb></lb><arrow.to.target n="note490"></arrow.to.target>citer, idQ.E.I. linea recta quæ tranſit per Spicam Virginis & <lb></lb>Cor Leonis, non exacte quidem, ſed a linea illa paululum defle<lb></lb>ctentem ad boream. <emph type="italics"></emph>Montenarus<emph.end type="italics"></emph.end>itidem notavit quod linea a <lb></lb>Cometa per Spicam ducta, hoc die & ſequentibus tranſibat per <lb></lb>auſtrale latus Cordis Leonis, interpoſito perparvo intervallo inter <lb></lb>Cor Leonis & hanc lineam. </s> <s>Linea recta per Cor Leonis & <lb></lb>Spicam Virginis tranſiens, Eclipticam ſecuit in 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 46′, in an<lb></lb>gulo 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 51′. </s> <s>Et ſi Cometa locatus fuiſſet in hac linea in 3 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>, <lb></lb>ejus latitudo fuiſſet 2 <emph type="sup"></emph>gr<emph.end type="sup"></emph.end> 26′. </s> <s>Sed cum Cometa conſentientibus <lb></lb><emph type="italics"></emph>Hookio<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Montenaro,<emph.end type="italics"></emph.end>nonnihil diſtaret ab hac linea boream ver<lb></lb>ſus, latitudo ejus fuit paulo minor. </s> <s>Die 20. ex obſervatione <emph type="italics"></emph>Mon<lb></lb>tenari,<emph.end type="italics"></emph.end>latitudo ejus propemodum æquabat latitudinem Spicæ , <lb></lb>eratque 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′ circiter, & conſentientibus <emph type="italics"></emph>Hookio, Montenaro<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>Angone<emph.end type="italics"></emph.end>perpetuo augebatur, ideoque jam ſenſibiliter major erat <lb></lb>quam 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′. </s> <s>Inter limites autem jam conſtitutos 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 26′ & <lb></lb>1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′, magnitudine mediocri latitudo erit 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 58′ circiter. </s> <s><lb></lb>Cauda Cometæ, conſentientibus <emph type="italics"></emph>Hookio<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Montenaro,<emph.end type="italics"></emph.end>dirigebatur <lb></lb>ad Spicam , declinans aliquantulum a Stella iſta, juxta <emph type="italics"></emph>Hookium<emph.end type="italics"></emph.end><lb></lb>in auſtrum, juxta <emph type="italics"></emph>Montenarum<emph.end type="italics"></emph.end>in boream; ideoQ.E.D.clinatio illa <lb></lb>vix fuit ſenſibilis, & Cauda Æquatori fere parallela exiſtens, ali<lb></lb>quantulum deflectebatur ab oppoſitione Solis boream verſus. </s></p> <p type="margin"> <s><margin.target id="note490"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>24. Ante ortum Solis Cometa viſus eſt a <emph type="italics"></emph>Montenaro<emph.end type="italics"></emph.end><lb></lb>in 12<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 52′, ad boreale latus rectæ quæ per Cor Leonis & Spicam <lb></lb>Virginis ducebatur, ideoque latitudinem habuit paulo minorem <lb></lb>quam 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 38′. </s> <s>Hæc latitudo uti diximus, ex obſervationibus <lb></lb><emph type="italics"></emph>Montenari, Angonis<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Hookii,<emph.end type="italics"></emph.end>perpetuo augebatur; ideoque jam <lb></lb>paulo major erat quam 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 58′; & magnitudine mediocri, abſque <lb></lb>notabili errore, ſtatui poteſt 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 18′. </s> <s>Latitudinem <emph type="italics"></emph>Ponthæus<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>Galletius<emph.end type="italics"></emph.end>jam decreviſſe volunt, & <emph type="italics"></emph>Cellius<emph.end type="italics"></emph.end>& Obſervator in <emph type="italics"></emph>Nova<lb></lb>Anglia<emph.end type="italics"></emph.end>eandem fere magnitudinem retinuiſſe, ſcilicet gradus unius <lb></lb>vel unius cum ſemiſſe. </s> <s>Craſſiores ſunt obſervationes <emph type="italics"></emph>Ponthæi<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>Cellii,<emph.end type="italics"></emph.end>eæ præſertim quæ per Azimuthes & Altitudines capieban<lb></lb>tur, ut & eæ <emph type="italics"></emph>Galletii<emph.end type="italics"></emph.end>: meliores ſunt eæ quæ per poſitiones Co<lb></lb>metæ ad fixas a <emph type="italics"></emph>Montenaro, Hookio, Angone<emph.end type="italics"></emph.end>& Obſervatore in <lb></lb><emph type="italics"></emph>Nova-Anglia,<emph.end type="italics"></emph.end>& nonnunquam a <emph type="italics"></emph>Ponthæo<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Cellio<emph.end type="italics"></emph.end>ſunt factæ. </s></p> <p type="main"> <s>Jam collatis Obſervationibus inter ſe, colligere videor quod <lb></lb>Cometa hoc menſe circulum fere maximum deſcripſit, ſecantem <lb></lb>Eclipticam in 25. 12′, idQ.E.I. angulo 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 12′ quamproxime. </s> <s><lb></lb>Nam & <emph type="italics"></emph>Montenarus<emph.end type="italics"></emph.end>Orbitam ab Ecliptica in auſtrum, tribus ſal-<pb xlink:href="039/01/490.jpg" pagenum="462"></pb><arrow.to.target n="note491"></arrow.to.target>tem gradibus declinaſſe dicit. </s> <s>Et cognita curſus poſitione, lon<lb></lb>gitudines Cometæ ex obſervationibus collectæ, ad incudem jam <lb></lb>revocari poſſunt & melius nonnunquam determinari, ut ſit in ſe<lb></lb>quentibus. <emph type="italics"></emph>Cellius<emph.end type="italics"></emph.end>Novemb. </s> <s>17. obſervavit diſtantiam Cometæ a <lb></lb>Spica , æqualem eſſe diſtantiæ ejus a ſtella lucida in dextra ala <lb></lb>Corvi: & hinc locandus eſt Cometa in interſectione hujus circuli <lb></lb>quem Cometa motu apparente deſcripſit, cum circulo maximo <lb></lb>qui a fixis illis duabus æqualiter diſtat, atque adeo in 7<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 54′, <lb></lb>cum latitudine auſtrali 43′. </s> <s>Præterea <emph type="italics"></emph>Montenarus, Novemb.<emph.end type="italics"></emph.end>20. <lb></lb>hora ſexta matutina <emph type="italics"></emph>Venetiis,<emph.end type="italics"></emph.end>Cometam vidit non totis quatuor <lb></lb>gradibus diſtantiam a Spica; dicitque hanc diſtantiam, vix æquaſſe <lb></lb>diſtantiam ſtellarum duarum lucidarum in alis Corvi, vel duarum <lb></lb>in juba Leonis, hoc eſt 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> & 30′ vel 32′. </s> <s>Sit igitur diſtantia <lb></lb>Cometæ a Spica 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′, & Cometa locabitur in 22<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 48′, cum <lb></lb>latitudine auſtrali 1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′. </s> <s>Adhæc <emph type="italics"></emph>Montenarus, Novemb.<emph.end type="italics"></emph.end>21, 22, <lb></lb>24 & 25 ante ortum Solis, Sextante æneo quintupedali ad mi<lb></lb>nuta prima & ſemiminuta diviſo & vitris Teleſcopicis armato, <lb></lb>diſtantias menſuravit Cometæ a Spica 8<emph type="sup"></emph>gr<emph.end type="sup"></emph.end> 28′, 13<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 10′, 23<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end><lb></lb>30′, & 28<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 13′: & has diſtantias, per refractionem nondum cor<lb></lb>rectas, addendo longitudini Spicæ, collegit Cometam his tempo<lb></lb>ribus fuiſſe in 27<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 51′, 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 33′, 12<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 52′ & 17<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 45′. </s> <s><lb></lb>Si diſtantiæ illæ per refractiones corrigantur, & ex diſtantiis cor<lb></lb>rectis differentiæ longitudinum inter Spicam & Cometam probe <lb></lb>deriventur, locabitur Cometa his temporibus in 27<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 52′, <lb></lb> 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 36′, 12<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 58′ & 17<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 53′ circiter. </s> <s>Latitudines au<lb></lb>tem ad has longitudines in via Cometæ captas, prodeunt 1 <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 45′, <lb></lb>1<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 58′, 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 22′ & 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 31′. </s> <s>Harum quatuor obſervationum ho<lb></lb>ras matutinas <emph type="italics"></emph>Montenarus<emph.end type="italics"></emph.end>non poſuit. </s> <s>Priores duæ ante ho<lb></lb>ram ſextam, poſteriores (ob viciniam Solis) poſt ſextam factæ <lb></lb>videntur. </s> <s>Die 22, ubi Cometa ex obſervatione <emph type="italics"></emph>Montenari<emph.end type="italics"></emph.end>loca<lb></lb>tur in 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 36′, <emph type="italics"></emph>Hookius<emph.end type="italics"></emph.end>noſter eundem locavit in 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 30′ <lb></lb>ut ſupra. <emph type="italics"></emph>Montenarus<emph.end type="italics"></emph.end>in defectu, <emph type="italics"></emph>Hookius<emph.end type="italics"></emph.end>in exceſſu erraſſe viden<lb></lb>tur. </s> <s>Nam Cometa, ex ſerie obſervationum, jam fuit in 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 56′ <lb></lb>vel 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> circiter. </s></p> <p type="margin"> <s><margin.target id="note491"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Obſervationum ſuarum ultimam inter vapores & diluculum <lb></lb>captam, <emph type="italics"></emph>Montenarus<emph.end type="italics"></emph.end>ſuſpectam habebat. </s> <s>Et <emph type="italics"></emph>Cellius<emph.end type="italics"></emph.end>eodem tem<lb></lb>pore (id eſt, <emph type="italics"></emph>Novem.<emph.end type="italics"></emph.end>25) Cometam per ejus Altitudinem & Azi<lb></lb>muthum locavit in 15<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 47′, cum latitudine auſtrali quaſi gra<lb></lb>dus unius Sed <emph type="italics"></emph>Cellius<emph.end type="italics"></emph.end>obſervavit etiam eodem tempore, quod <lb></lb>Cometa erat in linea recta cum ſtella lucida in dextro ſemore<pb xlink:href="039/01/491.jpg" pagenum="463"></pb>Virginis & cum Lance auſtrali Libræ, & hæc linea ſecat viam <lb></lb><arrow.to.target n="note492"></arrow.to.target>Cometæ in 18<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 36′. <emph type="italics"></emph>Ponthæus<emph.end type="italics"></emph.end>etiam eodem tempore obſer<lb></lb>vavit, quod Cometa erat in recta tranſeunte per Chelam auſtri <lb></lb>nam Scorpii & per ſtellam quæ Lancem borealem ſequitur: & <lb></lb>hæc recta ſecat viam Cometæ in 16<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 34′. </s> <s>Obſervavit etiam, <lb></lb>quod Cometa erat in recta tranſeunte per ſtellam ſupra Lancem <lb></lb>auſtralem Libræ & ſtellam in principio pedis ſecundi Scorpii: & <lb></lb>hæc recta ſecat viam Cometæ in 17<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 55′. </s> <s>Et inter longitu<lb></lb>dines ex his tribus Obſervationibus ſic derivatas, longitudo me<lb></lb>diocris eſt 17<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 42′, quæ cum obſervatione <emph type="italics"></emph>Montenari<emph.end type="italics"></emph.end>ſatis <lb></lb>congruit. </s></p> <p type="margin"> <s><margin.target id="note492"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Erravit igitur <emph type="italics"></emph>Cellius<emph.end type="italics"></emph.end>jam locando Cometam in 15<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 47′, <lb></lb>per ejus Azimuthum & Altitudinem. </s> <s>Et ſimilibus Azimuthorum <lb></lb>& Altitudinum obſervationibus, <emph type="italics"></emph>Cellius<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Ponthæus<emph.end type="italics"></emph.end>non minus <lb></lb>erraverunt locando Cometam in 20 & 24 diebus duobus <lb></lb>ſequentibus, ubi ſtellæ fixæ ob diluculum vix aut ne vix quidem <lb></lb>apparuere. </s> <s>Et corrigendæ ſunt hæ obſervationes per additionem <lb></lb>duorum graduum, vel duorum cum ſemiſſe. </s></p> <p type="main"> <s>Ex omnibus autem Obſervationibus inter ſe collatis & ad meri<lb></lb>dianum <emph type="italics"></emph>Londini<emph.end type="italics"></emph.end>reductis, colligo Cometam hujuſmodi curſum <lb></lb>quamproxime deſcripſiſſe. <lb></lb><arrow.to.target n="table13"></arrow.to.target> </s></p><table><table.target id="table13"></table.target><row><cell>Temp. med. ſt. vet.</cell><cell>Long. Cometæ</cell><cell>Lat. Cometæ</cell></row><row><cell></cell><cell>d.</cell><cell>h.</cell><cell>′</cell><cell>gr.</cell><cell>′</cell><cell>gr.</cell><cell>′</cell><cell></cell></row><row><cell><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end></cell><cell>15.</cell><cell>17.</cell><cell>10</cell><cell> 8.</cell><cell>0</cell><cell>0.</cell><cell>44</cell><cell>Auſt.</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> 2.</cell><cell>56</cell><cell>1.</cell><cell>38</cell></row><row><cell>27.</cell><cell>17 1/4</cell><cell>ſere</cell><cell>12.</cell><cell>58</cell><cell>2.</cell><cell>20</cell></row><row><cell>24.</cell><cell>17 1/2</cell><cell>ſere</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"></emph>gr.<emph.end type="sup"></emph.end></cell><cell>2.</cell><cell>42</cell></row></table> <p type="main"> <s>Loca autem Cometæ in Orbe Parabolice computata, ita ſe habent. <lb></lb> <arrow.to.target n="table14"></arrow.to.target> <pb xlink:href="039/01/492.jpg" pagenum="464"></pb><arrow.to.target n="note493"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note493"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Congruunt igitur Obſervationes Aſtronomicæ, tam menſe <emph type="italics"></emph>No<lb></lb>vembri<emph.end type="italics"></emph.end>quam menſibus quatuor ſequentibus, cum motu Cometæ <lb></lb>circum Solem in Trajectoria hacce Parabolica, atque adeo unum <lb></lb>& cundem Cometam fuiſſe, qui menſe <emph type="italics"></emph>Novembri<emph.end type="italics"></emph.end>ad Solem deſcen<lb></lb>dir, & menſibus ſequentibus ab vodem aſcendit, abunde confir<lb></lb>mant, ut & hunc Cometam in Trajectoria hacce Parabolica dela<lb></lb>tum fuiſſe quamproxime. </s> <s>Menſibas <emph type="italics"></emph>Decembri, Januario, Fe<lb></lb>bruario<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Martio,<emph.end type="italics"></emph.end>ubi Obſervationes hujus Cometæ ſunt ſatis ac<lb></lb>curatæ, congruunt eædem cum motu ejus in hac Trajectoria, non <lb></lb>minus accurate quam obſervationes Planetarum congruere ſolent <lb></lb>cum eorum Theoriis. </s> <s>Menſe <emph type="italics"></emph>Novembri,<emph.end type="italics"></emph.end>ubi obſervationes ſunt <lb></lb>craſſæ, errores non ſunt majores quam qui craſſitudini obſerva<lb></lb>tionum tribuantur. </s> <s>Trajectoria Cometæ bis ſecuit planum Eclip<lb></lb>ticæ, & propterea non fuit rectilinea. </s> <s>Eclipticam ſecuit non in <lb></lb>oppoſitis cœli partibus, ſed in fine Virginis & principio Capri<lb></lb>corni, intervallo graduum 98 circiter; ideoque curſus Cometæ <lb></lb>plurimum deflectebatur a Circulo maximo. </s> <s>Nam & menſe <emph type="italics"></emph>No<lb></lb>vembri<emph.end type="italics"></emph.end>curſus ejus tribus ſaltem gradibus ab Ecliptica in auſtrum <lb></lb>declinabat, & poſtea menſe <emph type="italics"></emph>Decembri<emph.end type="italics"></emph.end>gradibus 29 vergebat ab <lb></lb>Ecliptica in ſeptentrionem, partibus duabus Orbitæ in quibus <lb></lb>Cometa tendebat in Solem & redibat a Sole, angulo apparente <lb></lb>graduum plus triginta ab invicem declinantibus, ut obſervavit <lb></lb><emph type="italics"></emph>Montenarus.<emph.end type="italics"></emph.end>Pergebat hic Cometa per ſigna fere novem, a Vir<lb></lb>ginis ſcilicet duodecimo gradu ad principium Geminorum, præ<lb></lb>ter ſignum Leonis per quod pergebat antequam videri cœpit: & <lb></lb>nulla alia extat Theoria, qua Cometa tantam Cœli partem motu <lb></lb>regulari percurrat. </s> <s>Motus ejus fuit maxime inæquabilis. </s> <s>Nam <lb></lb>circa diem vigeſimum <emph type="italics"></emph>Novembris,<emph.end type="italics"></emph.end>deſcripſit gradus circiter quin<lb></lb>que ſingulis diebus; dein motu retardato inter <emph type="italics"></emph>Novemb.<emph.end type="italics"></emph.end>26 & <lb></lb><emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end>12, ſpatio ſcilicet dierum quindecim cum ſemiſſe, de<lb></lb>ſcripſit gradus tantum 40; poſtea vero motu iterum accelerato, <lb></lb>deſcripſit gradus fere quinque ſingulis diebus, antequam motus <lb></lb>iterum retardari cœpir. </s> <s>Et Theoria quæ motui tam inæquabili <lb></lb>per maximam cœli partem probe reſpondet, quæque eaſdem ob<lb></lb>ſervat leges cum Theoria Planetarum, & cum accuratis obſerva<lb></lb>tionibus Aſtronomicis accurate congruit, non poteſt non eſſe vera. </s> <s><lb></lb>Cometa tamen ſub finem motus deviabat aliquantulum ab hac <lb></lb>Trajectoria Parabolica verſus axem Parabolæ, ut ex erroribus mi<lb></lb>nuti unius primi duorumve in latitudinem menſe <emph type="italics"></emph>Februario<emph.end type="italics"></emph.end>& <lb></lb><emph type="italics"></emph>Martio<emph.end type="italics"></emph.end>conſpirantibus, colligere videor; & propterea in Orbe El-<pb xlink:href="039/01/493.jpg"></pb><pb xlink:href="039/01/494.jpg"></pb><pb xlink:href="039/01/495.jpg"></pb><figure id="id.039.01.495.1.jpg" xlink:href="039/01/495/1.jpg"></figure><pb xlink:href="039/01/496.jpg" pagenum="465"></pb>liptico circum Solem movebatur, ſpatio annorum pluſquam quin</s></p> <p type="main"> <s><arrow.to.target n="note494"></arrow.to.target>gentorum, quantum ex erroribus illis judicare licuit, revolutio<lb></lb>nem peragens. </s></p> <p type="margin"> <s><margin.target id="note494"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Cæterum Trajectoriam quam Cometa deſcripſit, & Caudam <lb></lb>veram quam ſingulis in locis projecit, viſum eſt annexo ſchemate <lb></lb>in plano Trajectoriæ optice delineatas exhibere: Obſervationibus <lb></lb>ſequentibus in Cauda definienda adhibitis. </s></p> <p type="main"> <s><emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>17 Cauda gradus amplius quindecim longa <emph type="italics"></emph>Ponthæo<emph.end type="italics"></emph.end>ap<lb></lb>paruit. <emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>18 Cauda 30<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> longa, SoliQ.E.D.recte oppoſita in <lb></lb><emph type="italics"></emph>Nova-Anglia<emph.end type="italics"></emph.end>cernebatur, & protendebatur uſque ad ſtellam , <lb></lb>quæ tunc erat in 9<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 54′. <emph type="italics"></emph>Nov.<emph.end type="italics"></emph.end>19 in <emph type="italics"></emph>Mary-Land<emph.end type="italics"></emph.end>cauda viſa <lb></lb>fuit gradus 15 vel 20 longa. <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>10 Cauda (obſervante <emph type="italics"></emph>Flamſtedio<emph.end type="italics"></emph.end>) <lb></lb>tranſibat per medium diſtantiæ inter caudam ſerpentis Ophiuchi & <lb></lb>ſtellam <foreign lang="grc">δ</foreign> in Aquilæ auſtrali ala, & deſinebat prope ſtellas <emph type="italics"></emph>A, <foreign lang="grc">ω</foreign>, b<emph.end type="italics"></emph.end>in <lb></lb>Tabulis <emph type="italics"></emph>Bayeri.<emph.end type="italics"></emph.end>Terminus igitur erat in <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 19 1/2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> cum latitudine <lb></lb>boreali 34 1/4<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> circiter. <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>11 ſurgebat ad uſque caput Sagittæ <lb></lb>(<emph type="italics"></emph>Bayero,<emph.end type="italics"></emph.end><foreign lang="grc">α, β</foreign>,) deſinens in <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 26<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 43′, cum latitudine boreali <lb></lb>38<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 34′. <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>13 tranſibat per medium Sagittæ, nec longe ultra <lb></lb>protendebatur, deſinens in=4<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>, cum latitudine boreali 42 1/2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> circi<lb></lb>ter. </s> <s>Intelligenda ſunt hæc de longitudine caudæ clarioris. </s> <s>Nam luce <lb></lb>obſcuriore, in cœlo forſan magis ſereno, cauda <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>12, hora 5, 40′ <lb></lb><emph type="italics"></emph>Romæ<emph.end type="italics"></emph.end>(obſervante <emph type="italics"></emph>Ponthæo<emph.end type="italics"></emph.end>) ſupra Cygni Uropygium ad gradus 10 <lb></lb>ſeſe extulit; atque ab hac ſtella ejus latus ad occaſum & boream <lb></lb>min. </s> <s>45 deſtitit. </s> <s>Lata autem erat cauda his diebus gradus 3, juxta <lb></lb>terminum ſuperiorem, ideoque medium ejus diſtabat a Stella illa <lb></lb>2<emph type="sup"></emph>gr<emph.end type="sup"></emph.end> 15′ auſtrum verſus, & terminus ſuperior erat in <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 22<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> cum <lb></lb>latitudine boreali 61<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>. <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>21 ſurgebat fere ad cathedram <emph type="italics"></emph>Caſſio<lb></lb>peiæ,<emph.end type="italics"></emph.end>æqualiter diſtans a <foreign lang="grc">β</foreign> & <emph type="italics"></emph>Schedir,<emph.end type="italics"></emph.end>& diſtantiam ab utraque <lb></lb>diſtantiæ earum ab invicem æqualem habens, adeoQ.E.D.ſinens <lb></lb>in <emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 24<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> cum latitudine 47 1/2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>. <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>29 tangebat <emph type="italics"></emph>Scheat<emph.end type="italics"></emph.end>ſitam ad <lb></lb>ſiniſtram, & intervallum ſtellarum duarum in pede boreali <emph type="italics"></emph>Andro<lb></lb>medæ<emph.end type="italics"></emph.end>accurate complebat, & longa erat 54<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> adeoQ.E.D.ſinebat <lb></lb>in 8 19<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> cum latitudine 35<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>. <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>5 tetigit ſtellam <foreign lang="grc">π</foreign> in pectore <lb></lb><emph type="italics"></emph>Andromedæ,<emph.end type="italics"></emph.end>ad latus ſuum dextrum, & ſtellam <foreign lang="grc">μ</foreign> in ejus cingulo <lb></lb>ad latus ſiniſtrum; & (juxta Obſervationes noſtras) longa erat <lb></lb>40<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>; curva autem erat & convexo latere ſpectabat ad auſtrum. </s> <s><lb></lb>Cum circulo per Solem & caput Cometæ tranſeunte angulum <lb></lb>confecit graduum 4 juxta caput Cometæ; at juxta terminum al<lb></lb>terum inclinabatur ad circulum illum in angulo 10 vel 11 graduum, <lb></lb>& chorda caudæ cum circulo illo continebat angulum graduum <pb xlink:href="039/01/497.jpg" pagenum="466"></pb><arrow.to.target n="note495"></arrow.to.target>octo. <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>13 Cauda luce ſatis ſenſibili terminabatur inter <emph type="italics"></emph>Ala<lb></lb>mech<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Algol,<emph.end type="italics"></emph.end>& luce tenuiſſima deſinebat e regione ſtellæ <foreign lang="grc">χ</foreign> in <lb></lb>latere <emph type="italics"></emph>Perſei.<emph.end type="italics"></emph.end>Diſtantia termini caudæ a circulo Solem & Come<lb></lb>tam ungente erat 3<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 50′, & inclinatio chordæ caudæ ad circu<lb></lb>lum illum 8 1/2<emph type="sup"></emph>gr<emph.end type="sup"></emph.end>. <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>25 & 26 luce tenui micabat ad longitu<lb></lb>dinem graduum 6 vel 7; & ubi cœlum valde ſerenum erat, luce <lb></lb>tenuiſſima & ægerrime ſenſibili attingebat longitudinem graduum <lb></lb>duodecim & paulo ultra. </s> <s>Dirigebatur autem ejus axis ad Luci<lb></lb>dam in humero orientali Aurigæ accurate, adeoQ.E.D.clinabat ab <lb></lb>oppoſitione Solis boream verſus in angulo graduum decem. </s> <s>De<lb></lb>nique <emph type="italics"></emph>Feb.<emph.end type="italics"></emph.end>10 Caudam oculis armatis aſpexi gradus duos lon<lb></lb>gam. </s> <s>Nam lux prædicta tenuior per vitra non apparuit. <emph type="italics"></emph>Pon<lb></lb>thæus<emph.end type="italics"></emph.end>autem <emph type="italics"></emph>Feb.<emph.end type="italics"></emph.end>7 ſe caudam ad longitudinem graduum 12 <lb></lb>vidiſſe ſcribit. </s></p> <p type="margin"> <s><margin.target id="note495"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Orbem jam deſcriptum ſpectanti & reliqua Cometæ hujus Phæ<lb></lb>nomena in animo revolventi, haud difficulter conſtabit quod cor<lb></lb>pora Cometarum ſunt ſolida, compacta, fixa ac durabilia ad in<lb></lb>ſtar corporum Planetarum. </s> <s>Nam ſi nihil aliud eſſent quam vapo<lb></lb>res vel exhalationes Terræ, Solis & Planetarum, Cometa hicce in <lb></lb>tranſitu ſuo per viciniam Solis ſtatim diſſipari debuiſſet. </s> <s>Eſt enim <lb></lb>calor Solis ut radiorum denſitas, hoc eſt, reciproce ut quadratum <lb></lb>diſtantiæ loeorum a Sole. </s> <s>Ideoque cum diſtantia Cometæ a cen<lb></lb>tro Solis <emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end>8 ubi in Perihelio verſabatur, eſſet ad diſtan<lb></lb>tiam Terræ a centro Solis ut 6 ad 1000 circiter, calor Solis apud <lb></lb>Cometam eo tempore erat ad calorem Solis æſtivi apud nos ut <lb></lb>1000000 ad 36, ſeu 28000 ad 1. Sed calor aquæ ebullientis eſt <lb></lb>quaſi triplo major quam calor quem terra arida concipit ad æſti<lb></lb>vum Solem, ut expertus ſum: & calor ferri candentis (ſi recte <lb></lb>conjector) quaſi triplo vel quadruplo major quam calor aquæ ebul<lb></lb>lientis; adeoque calor quem terra arida apud Cometam in Peri<lb></lb>helio verſantem ex radiis Solaribus concipere poſſet, quaſi 2000 <lb></lb>vicibus major quam calor ferri candentis. </s> <s>Tanto autem calore <lb></lb>vapores & exhalationes, omniſque materia volatilis itatim conſumi <lb></lb>ac diſſipari debuiſſent. </s></p> <p type="main"> <s>Cometa igitur in Perihelio ſuo calorem immenſum ad Solem <lb></lb>concepit, & calorem illum diutiſſime conſervare poteſt. </s> <s>Nam <lb></lb>globus ferri candentis digitum unum latus, calorem ſuum omnem <lb></lb>ſpatio horæ unius in aere conſiſtens vix amitteret. </s> <s>Globus autem <lb></lb>major calorem diutius conſervaret in ratione diametri, propterea <lb></lb>quod ſuperficies (ad cujus menſuram per contactum aeris ambi-<pb xlink:href="039/01/498.jpg" pagenum="467"></pb>entis refrigeratur) in illa ration minor eſt pro quantitate mate<lb></lb><arrow.to.target n="note496"></arrow.to.target>riæ ſuæ calidæ incluſæ. </s> <s>Ideoque globus ferri candentis huic <lb></lb>Terræ æqualis, id eſt, pedes plus minus 40000000 latus, diebus <lb></lb>totidem, & idcirco annis 50000, vix refrigeſceret. </s> <s>Suſpicor ta<lb></lb>men quod duratio Caloris, ob cauſas latentes, augeatur in minore <lb></lb>ratione quam ea diametri: & optarim rationem veram per experi<lb></lb>menta inveſtigari. </s></p> <p type="margin"> <s><margin.target id="note496"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Porro notandum eſt quod Cometa Menſe <emph type="italics"></emph>Decembri,<emph.end type="italics"></emph.end>ubi ad <lb></lb>Solem modo incaluerat, caudam emittebat longe majorem & <lb></lb>ſplendidiorem quam antea Menſe <emph type="italics"></emph>Novembri,<emph.end type="italics"></emph.end>ubi Periheliunt non<lb></lb>dum attigerat. </s> <s>Et univerſaliter caudæ omnes maximæ & fulgen<lb></lb>tiſſimæ e Cometis oriuntur, ſtatim poſt tranſitum eorum per regi<lb></lb>onem Solis. </s> <s>Conducit igitur calefactio Cometæ ad magnitudi<lb></lb>nem caudæ. </s> <s>Et inde colligere videor quod cauda nihil aliud fit <lb></lb>quam vapor longe tenuiſſimus, quem caput ſeu nucleus Cometæ <lb></lb>per calorem ſuum emittit. </s></p> <p type="main"> <s>Cæterum de Cometarum caudis triplex eſt opinio; eas vel jubar <lb></lb>eſſe Solis per tranſlucida Cometarum capita propagatum, vel oriri <lb></lb>ex refractione lucis in progreſſu ipſius a capite Comeræ in Ter<lb></lb>ram, vel denique nubem eſſe ſeu vaporem a capite Comeræ jugi<lb></lb>ter ſurgentem & abeuntem in partes a Sole averſas. </s> <s>Opinio pri<lb></lb>ma eorum eſt qui nondum imbuti ſunt ſcientia rerum Opticarum. </s> <s><lb></lb>Nam jubar Solis in cubiculo tenebroſo non cernitur, niſi quatenus <lb></lb>lux reflectitur e pulverum & fumorum particulis per aerem ſem<lb></lb>per volitantibus: adeoQ.E.I. aere fumis craſſioribus infecto ſplen<lb></lb>didius eſt, & ſenſum fortius ferit; in aere clariore tenuius eſt & <lb></lb>ægrius ſentitur: in cœlis autem abſque materia reflectente nullum <lb></lb>eſſe poteſt. </s> <s>Lux non cernitur quatenus in jubare eſt, ſed quatenus <lb></lb>inde reſtectitur ad oculos noſtros. </s> <s>Nam viſio non ſit niſi per radios <lb></lb>qui in oculos impingunt. </s> <s>Requiritur igitur materia aliqua reflectens <lb></lb>in regione caudæ, ne cœlum totum luce Solis illuſtratum unifor<lb></lb>miter ſplendeat. </s> <s>Opinio ſecunda multis premitur difficultatibus. </s> <s><lb></lb>Caudæ nunquam variegantur coloribus: qui tamen refractionum <lb></lb>ſolent eſſe comites inſeparabiles. </s> <s>Lux Fixarum & Planetarum di<lb></lb>ſtincte ad nos tranſmiſſa, demonſtrat medium cœleſte nulla vi re<lb></lb>fractiva pollere. </s> <s>Nam quod dicitur Fixas ab <emph type="italics"></emph>Ægyptiis<emph.end type="italics"></emph.end>comatas <lb></lb>nonnunquam viſas fuiſſe, id quoniam rariſſime contingit, aſcri<lb></lb>bendum eſt nubium refractioni fortuitæ. </s> <s>Fixarum quoque radia<lb></lb>tio & ſcintillatio ad refractiones tum Oculorum tum Aeris tre<lb></lb>muli referendæ ſunt: quippe quæ admotis oculo Teleſcopiis <pb xlink:href="039/01/499.jpg" pagenum="468"></pb><arrow.to.target n="note497"></arrow.to.target>evaneſcunt, Aeris & aſcendentium vaporum tremore fit ut radii <lb></lb>facile de anguſto pupillæ ſpatio per vices detorqueantur, de lati<lb></lb>ore autem vitri objectivi apertura neutiquam. </s> <s>Inde eſt quod <lb></lb>ſcintillatio in priori caſa generetur, in poſteriore autem ceſſet: <lb></lb>& ceſſatio in poſteriore caſu demonſtrat regularem tranſmiſſionem <lb></lb>lucis per cœlos abſque omni refractione ſenſibili. </s> <s>Nequis con<lb></lb>tendat quod caudæ non ſoleant videri in Cometis cum eorum lux <lb></lb>non eſt ſatis fortis, quia tunc radii ſecundarii non habent ſitis vi<lb></lb>rium ad oculos movendos, & propterea caudas Fixarum non cerni: <lb></lb>ſciendum eſt quod lux Fixarum plus centum vicibus augeri poteſt <lb></lb>mediantibus Teleſcopiis, nec tamen caudæ cernuntur Planeta<lb></lb>rum quoque lux copioſior eſt, caudæ vero nunæ: Comeræ autem <lb></lb>ſæpe caudatiſſimi ſunt, ubi capitum lux tenuis eſt & valde obtuſa: <lb></lb>ſic enim Cometa Anni 1680, Menſe <emph type="italics"></emph>Decembri,<emph.end type="italics"></emph.end>quo tempore ca<lb></lb>put luce ſua vix æquabat ſtellas ſecundæ magnitudinis, caudam <lb></lb>emittebat ſplendore notabili uſque ad gradus 40, 50, 60 longi<lb></lb>tudinis & ultra: poſtea <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>27 & 28 caput apparebat ut ſtella <lb></lb>ſeptimæ tantum magnitudinis, cauda vero luce quidem pertenui <lb></lb>ſed ſatis ſenſibili longa erat 6 vel 7 gradus, & luce obſcuriſſima, <lb></lb>quæ cerni vix poſſet, porrigebatur ad gradum uſQ.E.D.odecimum <lb></lb>vel paulo ultra: ut ſupra dictum eſt. </s> <s>Sed & <emph type="italics"></emph>F<foreign lang="grc">ε</foreign>b.<emph.end type="italics"></emph.end>9 & 10 ubi <lb></lb>caput nudis oculis videri deſierat, caudam gradus duos longam <lb></lb>per Teleſcopium contemplatus ſum. </s> <s>Porro ſi cauda oriretur ex <lb></lb>refractione materiæ cœleſtis, & pro figura cœlorum deflecteretur <lb></lb>de Solis oppoſitione, deberet deflexio illa in iiſdem cœli regioNI<lb></lb>bus in eandem ſemper partem fieri. </s> <s>Atqui Cometa Anni 1680 <lb></lb><emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end>28. hora 8 1/2 P.M. <emph type="italics"></emph>Londini,<emph.end type="italics"></emph.end>verſabatur in 8<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 41′ cum <lb></lb>latitudine boreali 28<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 6′, Sole exiſtente in 18<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 26′. </s> <s>Et Co<lb></lb>meta Anni 1577, <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>29 verſabatur in 8<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 41′ cum latitu<lb></lb>dine boreali 28<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 40′, Sole etiam exiſtente in 18<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 26′ circi<lb></lb>ter. </s> <s>UtroQ.E.I. caſu Terra verſabatur in eodem loco, & Co<lb></lb>meta apparebat in eadem cœli parte: in priori tamen caſu cauda <lb></lb>Cometæ (ex meis & aliorum Obſervationibus) declinabat angulo <lb></lb>graduum 4 1/2 ab oppoſitione Solis aquilonem verſus; in poſte<lb></lb>riore vero (ex Obſervationibus <emph type="italics"></emph>Tychonis<emph.end type="italics"></emph.end>) declinatio erat gra<lb></lb>duum 21 in auſtrum. </s> <s>Igitur repudiata cœlorum refractione, <lb></lb>ſupereſt ut Phænomena Caudarum ex materia aliqua reflectente <lb></lb>deriventur. </s></p> <p type="margin"> <s><margin.target id="note497"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Caudas autem a capitibus oriri & in regiones a Sole averſas <lb></lb>aſcendere confirmatur ex legibus quas obſervant. </s> <s>Ut quod in <pb xlink:href="039/01/500.jpg" pagenum="469"></pb>planis Orbium Cometarum per Solem tranſeuntibus jacentes, de<lb></lb><arrow.to.target n="note498"></arrow.to.target>viant ab oppoſitione Solis in eas ſemper partes, quas capita in <lb></lb>Orbibus iilis progredientia relinquunt. </s> <s>Quod ſpectatori in his <lb></lb>planis conſtituto apparent in partibus a Sole directe averſis; di<lb></lb>grediente autem ſpeſtatore de his planis, deviatio paulatim ſen<lb></lb>titur, & indies apparet major. </s> <s>Quod deviatio cæteris paribus <lb></lb>minor eſt ubi cauda obliquior eſt ad Orbem Cometæ, ut & ubi <lb></lb>caput Cometæ ad Solem propius accedit; præſertim ſi ſpectetur <lb></lb>deviationis angulus juxta caput Cometæ. </s> <s>Præterea quod caudæ <lb></lb>non deviantes apparent rectæ, deviantes autem incurvantur. </s> <s>Quod <lb></lb>curvatura major eſt ubi major eſt deviatio, & magis ſenſibilis ubi <lb></lb>cauda cæteris paribus longior eſt: nam in brevioribus curvatura <lb></lb>ægre animadvertitur. </s> <s>Quod deviationis angulus minor eſt juxta <lb></lb>caput Cometæ, major juxta caudæ extremitatem alteram, atque <lb></lb>adeo quod cauda convexo ſui latere partes reſpicit a quibus ſit <lb></lb>deviatio, quæQ.E.I. recta ſunt linea a Sole per caput Cometæ in <lb></lb>infinitum ducta. </s> <s>Et quod caudæ quæ prolixiores ſunt & latiores, <lb></lb>& luce vegetiore micant, ſint ad latera convexa paulo ſplendi<lb></lb>diores & limite minus indiſtincto terminatæ quam ad concava. </s> <s><lb></lb>Pendent igitur Phænomena caudæ a motu capitis, non autem a <lb></lb>regione cœli in qua caput conſpicitur; & propterea non fiunt per <lb></lb>refractionem cœlorum, ſed a capite ſuppeditante materiam ori<lb></lb>untur. </s> <s>Etenim ut in Aere noſtro fumus corporis cujuſvis igniti <lb></lb>petit ſuperiora, idque vel perpendiculariter ſi corpus quieſcat, <lb></lb>vel oblique ſi corpus moveatur in latus: ita in Cœlis ubi corpora <lb></lb>gravitant in Solem, fumi & vapores aſcendere debent à Sole (uti <lb></lb>jam dictum eſt) & ſuperiora vel recta petere, ſi corpus fumans <lb></lb>quieſcit; vel oblique, ſi corpus progrediendo loca ſemper deſerit <lb></lb>a quibus ſuperiores vaporis partes aſcenderant. </s> <s>Et obliquitas iſta <lb></lb>minor erit ubi aſcenſus vaporis velocior eſt: nimirum in vicinia <lb></lb>Solis & juxta corpus fumans. </s> <s>Ex obliquitatis autem diverſitate <lb></lb>incurvabitur vaporis columna: & quia vapor in columnæ latere <lb></lb>præcedente paulo recentior eſt, ideo etiam is ibidem aliquanto <lb></lb>denſior erit, lucemque propterea copioſius reflectet, & limite mi<lb></lb>nus indiſtincto terminabitur. </s> <s>De Caudarum agitionibus ſubita<lb></lb>neis & incertis, deque earum figuris irregularibus, quas nonnulli <lb></lb>quandoQ.E.D.ſcribunt hic nihil adjicio; propterea quod vel a <lb></lb>mutationibus Aeris neſtri, & motibus nubium caudas aliqua ex <lb></lb>parte obſcurantium oriantur; vel forte a partibus Viæ Lacteæ, <lb></lb>quæ cum caudis prætereuntibus confundi poſſint, ac tanquam ea<lb></lb>rum partes ſpectari. <pb xlink:href="039/01/501.jpg" pagenum="470"></pb><arrow.to.target n="note499"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note498"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note499"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Vapores autem, qui ſpatiis tam immenſis implendis ſufficiant, <lb></lb>ex Cometarum Atmoſphæris oriri poſſe, intelligetur ex ratitate <lb></lb>Aeris noſtri. </s> <s>Nam Aer juxta ſuperficiem Terræ ſpatium occupat <lb></lb>quaſi 850 partibus majus quam Aqua ejuſdem ponderis, ideoque <lb></lb>Aeris columna cylindrica pedes 850 alta, ejuſdem eſt ponderis <lb></lb>cum Aquæ columna pedali latitudinis ejuſdem. </s> <s>Columna autem <lb></lb>Aeris ad ſummitatem Atmoſphæræ aſſurgens æquat pondere ſuo <lb></lb>colurnnam Aquæ pedes 33 altam circiter; & propterea ſi colum<lb></lb>næ totius Aereæ pars inferior pedum 850 altitudinis dematur, <lb></lb>pars reliqua ſuperior æquabit pondere ſuo columnam Aquæ altam <lb></lb>pedes 32. Inde vero (ex Hypotheſi multis experimentis confir<lb></lb>mata, quod compreſſio Aeris ſit ut pondus Atmoſphæræ incum<lb></lb>bentis, quodque gravitas ſit reciproce ut quadratum diſtantiæ lo<lb></lb>eorum a centro Terræ) computationem per Corol. </s> <s>Prop. </s> <s>XXII. <lb></lb>Lib. </s> <s>II. ineundo, inveni quod Aer, ſi aſcendatur a ſuperficie <lb></lb>Terræ ad altitudinem ſemidiametri unius terreſtris, rarior ſit quam <lb></lb>apud nos in ratione longe majori, quam ſpatii omnis infra Or<lb></lb>bem Saturni ad globum diametro digiti unius deſcriptum. </s> <s>Ideo<lb></lb>que globus Aeris noſtri digitum unum latus, ea cum raritate <lb></lb>quam haberet in altitudine ſemidiametri unius terreſtris, impleret <lb></lb>omnes Planetarum regiones ad uſque ſphæram Saturni & longe <lb></lb>ultra. </s> <s>Proinde cum Aer adhuc altior in immenſum rareſcat; & <lb></lb>coma ſeu Atmoſphæra Cometæ, aſcendendo ab illius centro, quaſi <lb></lb>decuplo altior ſit quam ſuperficies nuclei, deinde cauda adhuc <lb></lb>altius aſcendat, debebit cauda eſſe quam rariſſima. </s> <s>Et quamvis, <lb></lb>ob longe craſſiorem Cometarum Atmoſphæram, magnamque cor<lb></lb>porum gravitationem Solem verſus, & gravitationem particula<lb></lb>rum Aeris & vaporum in ſe mutuo, fieri poſſit ut Aer in ſpatiis <lb></lb>cœleſtibus inque Cometarum caudis non adeo rareſcat; perexi<lb></lb>guam tamen quantitatem Aeris & vaporum, ad omnia illa cauda<lb></lb>rum Phœnomena abunde ſufficere, ex hac computatione perſpi<lb></lb>cuum eſt. </s> <s>Nam & caudarum inſignis raritas colligitur ex aſtris <lb></lb>pes eas tranſlucentibus. </s> <s>Atmoſphæra terreſtris luce Solis ſplen<lb></lb>dens, craſſitudine ſua paueorum milliarium, & aſtra omnia & ip<lb></lb>ſam Lunam obſcurat & extinguit penitus: per immenſam vero <lb></lb>caudarum craſſitudinem, luce pariter Solari illuſtratam, aſtra mi<lb></lb>nima abſque claritatis detrimento tranſlucere noſcuntur. </s> <s>Neque <lb></lb>major eſſe ſolet caudarum plurimarum ſplendor, quam Aeris no<lb></lb>ſtri in tenebroſo cubiculo latitudine digiti unius duorumve, lucem <lb></lb>Solis in jubare reflectentis. </s></p><pb xlink:href="039/01/502.jpg" pagenum="471"></pb> <p type="main"> <s>Quo temporis ſpatio vapor a capite ad terminum caudæ aſcen</s></p> <p type="main"> <s><arrow.to.target n="note500"></arrow.to.target>dit, cognoſci fere poteſt ducendo rectam a termino caudæ ad So<lb></lb>lem, & notando locum ubi recta illa Trajectoriam ſecat. </s> <s>Nam <lb></lb>vapor in termino caudæ, ſi recta aſcendat a Sole, aſcendere cœpit <lb></lb>a capite quo tempore caput erat in loco interſectionis. </s> <s>At vapor <lb></lb>non recta aſcendit à Sole, ſed motum Cometæ, quem aute aſcen<lb></lb>ſum ſuum habebat, retinendo, & cum motu aſcenſus ſui eundem <lb></lb>componendo, aſcendit oblique. </s> <s>Unde verior erit Problematis <lb></lb>ſolutio, ut recta illa quæ Orbem ſecat, parallela ſit longitudini <lb></lb>caudæ, vel potius (ob motum curvilineum Cometæ) ut eadem a <lb></lb>linea caudæ divergat. </s> <s>Hoc pacto inveni quod vapor qui erat in <lb></lb>termino caudæ <emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end>25, aſcendere cœperat a capite ante <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>11, <lb></lb>adeoque aſcenſu ſuo toto dies plus 45 conſumpſerat. </s> <s>At cauda <lb></lb>illa omnis quæ <emph type="italics"></emph>Dec.<emph.end type="italics"></emph.end>10 apparuit, aſcenderat ſpatio dierum illo<lb></lb>rum duorum, qui a tempore Perihelii Cometæ elapſi fuerant. </s> <s><lb></lb>Vapor igitur ſub initio in vicinia Solis celerrime aſcendebat, & <lb></lb>poſtea cum motu per gravitatem ſuam ſemper retardato aſcen<lb></lb>dere pergebat; & aſcendendo augebat longitudinem caudæ: cauda <lb></lb>autem quamdiu apparuit ex vapore fere omni conſtabat qui a <lb></lb>tempore Perihelii aſcenderat; & vapor, qui primus aſcendit, & <lb></lb>terminum caudæ compoſuit, non prius evanuit quam ob nimiam <lb></lb>ſuam tam a Sole illuſtrante quam ab oculis noſtris diſtantiam vi<lb></lb>deri deſiit. </s> <s>Unde etiam caudæ Cometarum aliorum quæ breves <lb></lb>ſunt, non aſcendunt motu celeri & perpetuo a capitibus & mox <lb></lb>evaneſcunt, ſed ſunt permanentes vaporum & exhalationum co<lb></lb>lumnæ, a capitibus lentiſſimo multorum dierum motu propagatæ, <lb></lb>quæ, participando motum illum capitum quem habuere ſub initio, <lb></lb>per cœlos una cum capitibus moveri pergunt. </s> <s>Et hinc rurſus col<lb></lb>ligitur ſpatia cœleſtia vi reſiſtendi deſtitui; utpote in quibus non <lb></lb>ſolum ſolida Planetarum & Cometarum corpora, ſed etiam rariſ<lb></lb>ſimi caudarum vapores motus ſuos velociſſimos liberrime peragunt <lb></lb>ac diutiſſime conſervant. </s></p> <p type="margin"> <s><margin.target id="note500"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Aſcenſum caudarum ex Atmoſphæris capitum & progreſſum in <lb></lb>partes a Sole averſas <emph type="italics"></emph>Keplerus<emph.end type="italics"></emph.end>aſcribit actioni radiorum lucis ma<lb></lb>teriam caudæ ſecum rapientium. </s> <s>Et auram longe tenuiſſimam in <lb></lb>ſpatiis liberrimis actioni radiorum cedere, non eſt a ratione pror<lb></lb>ſus alienum, non obſtante quod ſubſtantiæ craſſæ, impeditiſſimis <lb></lb>in regionibus noſtris, a radiis Solis ſenſibiliter propelli nequeant. </s> <s><lb></lb>Alius particulas tam leves quam graves dari poſſe exiſtimat, & <lb></lb>materiam caudarum levitare, perque levitatem ſuam a Sole aſcen-<pb xlink:href="039/01/503.jpg" pagenum="472"></pb><arrow.to.target n="note501"></arrow.to.target>dere. </s> <s>Cum autem gravitas corporum terreſtrium ſit ut materia <lb></lb>in corporibus, ideoque ſervata quantitate materiæ intendi & re<lb></lb>mitti nequeat, ſuſpicor aſcenſum illum ex rarefactione materiæ <lb></lb>caudarum potius oriri. </s> <s>Aſcendit fumus in camino impulſu Aeris <lb></lb>cui innatat. </s> <s>Aer ille per calorem rarefactus aſcendit, ob diminu<lb></lb>tam ſuam gravitatem ſpecificam, & fumum implicatum rapit ſe<lb></lb>cum. </s> <s>Quidni cauda Cometæ ad eundem modum aſcenderit a <lb></lb>Sole? </s> <s>Nam radii Solares non agitant Media quæ permeant, niſi <lb></lb>in reflexione & refractione. </s> <s>Particulæ reflectentes ea actione cale<lb></lb>factæ calefacient auram ætheream cui implicantur. </s> <s>Illa calore ſibi <lb></lb>communicato rarefiet, & ob diminutam ea raritate gravitatem <lb></lb>ſuam ſpecificam qua prius tendebat in Solem, aſcendet & ſecum <lb></lb>rapiet particulas reflectentes ex quibus cauda componitur: Ad <lb></lb>aſcenſum vaporum conducit etiam quod hi gyrantur circa Solem <lb></lb>& ea actione conantur a Sole recedere, at Solis Atmoſphæra & <lb></lb>materia cœlorum vel plane quieſcit, vel motu ſolo quem a Solis <lb></lb>rotatione acceperint, tardius gyratur. </s> <s>Hæ ſunt cauſæ aſcenſus <lb></lb>caudarum in vicinia Solis, ubi Orbes curviores ſunt, & Cometæ <lb></lb>intra denſiorem & ea ratione graviorem Solis Atmoſphæram con<lb></lb>ſiſtunt, & caudas quam longiſſimas mox emittunt. </s> <s>Nam caudæ <lb></lb>quæ tunc naſcuntur, conſervando motum ſuum & interea verſus <lb></lb>Solem gravitando, movebuntur circa Solem in Ellipſibus pro <lb></lb>more capitum, & per motum illum capita ſemper comitabuntur <lb></lb>& iis liberrime adhærebunt. </s> <s>Gravitas enim vaporum in Solem <lb></lb>non magis efficiet ut caudæ poſtea decidant a capitibus Solem ver<lb></lb>ſus, quam gravitas capitum efficere poſſit ut hæc decidant a cau<lb></lb>dis. </s> <s>Communi gravitate vel ſimul in Solem cadunt, vel ſimul in <lb></lb>aſcenſu ſuo retardabuntur; adeoque gravitas illa non impedit, <lb></lb>quo minus caudæ & capita poſitionem quamcunque ad invicem a <lb></lb>cauſis jam deſcriptis, aut aliis quibuſcunque, facillime accipiant & <lb></lb>poſtea liberrime ſervent. </s></p> <p type="margin"> <s><margin.target id="note501"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Caudæ igitur quæ in Cometarum Periheliis naſcuntur, in regi<lb></lb>ones longinquas cum eorum capitibus abibunt, & vel inde poſt <lb></lb>longam annorum ſeriem cum iiſdem ad nos redibunt, vel potius <lb></lb>ibi rarefactæ paulatim evaneſcent. </s> <s>Nam poſtea in deſcenſu capi<lb></lb>tum ad Solem caudæ novæ breviuſculæ lento motu a capitibus <lb></lb>propagari debebunt, & ſubinde, in Periheliis Cometarum illorum <lb></lb>qui aduſque Atmoſphæram Solis deſcendunt, in immenſum au<lb></lb>geri. </s> <s>Vapor enim in ſpatiis illis liberrimis perpetuo rareſcit ac <lb></lb>dilatatur. </s> <s>Qua ratione fit ut cauda omnis ad extremitatem ſupe-<pb xlink:href="039/01/504.jpg" pagenum="473"></pb>riorem latior ſit quam juxta caput Cometæ. </s> <s>Ea autem rarefacti<lb></lb><arrow.to.target n="note502"></arrow.to.target>one vaporem perpetuo dilatatum diffundi tandem & ſpargi per <lb></lb>cœlos univerſos, deinde paulatim in Planetas per gravitatem ſuam <lb></lb>attrahi & cum eorum Atmoſphæris miſceri, rationi conſentaneum <lb></lb>videtur. </s> <s>Nam quemadmodum Maria ad conſtitutionem Terræ <lb></lb>hujus omnino requiruntur, idque ut ex iis per calorem Solis va<lb></lb>pores copioſe ſatis excitentur, qui vel in nubes coacti decidant <lb></lb>in pluviis, & terram omnem ad procreationem vegetabilium irri<lb></lb>gent & nutriant; vel in frigidis montium verticibus condenſati <lb></lb>(ut aliqui cum ratione philoſophantur) decurrant in fontes & <lb></lb>flumina ſic ad conſervationem marium & humorum in Planetis, <lb></lb>requiri videntur Cometæ, ex quorum exhalationibus & vapori<lb></lb>bus condenſatis, quicquid liquoris per vegetationem & putre<lb></lb>factionem conſumitur & in terram aridam convertitur, continuo <lb></lb>ſuppleri & refici poſſit. </s> <s>Nam vegetabilia omnia ex liquoribus <lb></lb>omnino creſcunt, dein magna ex parte in terram aridam per pu<lb></lb>trefactionem abeunt, & limus ex liquoribus putrefactis perpetuo <lb></lb>decidit. </s> <s>Hinc moles Terræ aridæ indies augetur, & liquores, niſi <lb></lb>aliunde augmentum ſumerent, perpetuo decreſcere deberent, ac <lb></lb>tandem deficere. </s> <s>Porro ſuſpicor Spiritum illum, qui Aeris noſtri <lb></lb>pars minima eſt ſed ſubtiliſſima & optima, & ad rerum omnium <lb></lb>vitam requiritur, ex Cometis præcipue venire. </s></p> <p type="margin"> <s><margin.target id="note502"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Atmoſphæræ Cometarum in deſcenſu eorum in Solem, excur<lb></lb>rendo in caudas, diminuuntur, & (ea certe in parte quæ Solem <lb></lb>reſpicit) anguſtiores redduntur: & viciſſim in receſſu eorum a <lb></lb>Sole, ubi jam minus excurrunt in caudas, ampliantur; ſi modo <lb></lb>Phænomena eorum <emph type="italics"></emph>Hevelius<emph.end type="italics"></emph.end>recte notavit. </s> <s>Minimæ autem ap<lb></lb>parent ubi capita jam modo ad Solem calefacta in caudas maximas <lb></lb>& fulgentiſſimas abiere, & nuclei fumo forſan craſſiore & nigriore <lb></lb>in Atmoſphærarum partibus infimis circundantur. </s> <s>Nam fumus <lb></lb>omnis ingenti calore excitatus, craſſior & nigrior eſſe ſolet. </s> <s>Sic <lb></lb>caput Cometæ de quo egimus, in æqualibus a Sole ac Terra di<lb></lb>ſtantiis, obſcurius apparuit poſt Perihelium ſuum quam antea. </s> <s><lb></lb>Menſe enim <emph type="italics"></emph>Decembri<emph.end type="italics"></emph.end>cum ſtellis tertiæ magnitudinis conferri ſole<lb></lb>bat, at Menſe <emph type="italics"></emph>Novembri<emph.end type="italics"></emph.end>cum ſtellis primæ & ſecundæ. </s> <s>Et qui <lb></lb>utrumque viderant, majorem deſcribunt Cometam priorem. </s> <s>Nam <lb></lb>Juveni cuidam <emph type="italics"></emph>Cantabrigienſi, Novemb.<emph.end type="italics"></emph.end>19, Cometa hicce luce ſua <lb></lb>quantumvis plumbea & obtuſa, æquabat Spicam Virginis, & cla<lb></lb>rius micabat quam poſtea. </s> <s>Et <emph type="italics"></emph>D. Storer<emph.end type="italics"></emph.end>literis quæ in manus no<lb></lb>ſtras incidere, ſcripſit caput ejus Menſe <emph type="italics"></emph>Decembri,<emph.end type="italics"></emph.end>ubi caudam <pb xlink:href="039/01/505.jpg" pagenum="474"></pb><arrow.to.target n="note503"></arrow.to.target>maximam & fulgentiſſimam emittebat, parvum eſſe & magnitu<lb></lb>dine viſibili longe cedere Cometæ, qui Menſe <emph type="italics"></emph>Novembri<emph.end type="italics"></emph.end>ante <lb></lb>Solis ortum apparuerat. </s> <s>Cujus rei rationem eſſe conjectabatur, <lb></lb>quod materia capitis ſub initio copioſior eſſet, & paulatim con<lb></lb>ſumeretur. </s></p> <p type="margin"> <s><margin.target id="note503"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Eodem ſpectare videtur quod capita Cometarum aliorum, qui <lb></lb>caudas maximas & fulgentiſſimas emiſerunt, apparuerint ſubob<lb></lb>ſcura & exigua. </s> <s>Nam Anno 1668 <emph type="italics"></emph>Mart.<emph.end type="italics"></emph.end>5. St. </s> <s>nov. </s> <s>hora ſeptima <lb></lb>veſpertina <emph type="italics"></emph>R. P. </s> <s>Vaientinus Eſtancius, Braſiliæ<emph.end type="italics"></emph.end>agens, Cometam <lb></lb>vidit Horizonti proximum ad occaſum Solis brumalem, capite <lb></lb>minimo & vix conſoicuo, cauda vero ſupra modum fulgente, ut <lb></lb>ſtantes in littore ſpeciem ejus e mari reflexam facile cernerent. </s> <s><lb></lb>Speciem utique habebat trabis ſplendentis longitudine 23 gra<lb></lb>duum, ab occidente in auſtrum vergens, & Horizonti fere para<lb></lb>lela. </s> <s>Tantus autem ſplendor tres ſolum dies durabat, ſubinde <lb></lb>notabiliter decreſcens; & interea decreſcente ſplendore aucta eſt <lb></lb>magnitudine cauda. </s> <s>Unde etiam in <emph type="italics"></emph>Portugallia<emph.end type="italics"></emph.end>quartam fere <lb></lb>cœli partem (id eſt, gradus 45) occupaſſe dicitur, ab occidente in <lb></lb>orientem ſplendore cum inſigni protenſa; nec tamen tota apparuit, <lb></lb>capite ſemper in his regionibus infra Horizontem deliteſcente. </s> <s><lb></lb>Ex incremento caudæ & decremento ſplendoris manifeſtum eſt <lb></lb>quod caput a Sole receſſit, eique proximum fuit ſub initio, pro <lb></lb>more Cometæ anni 1680. Et ſimilis legitur Cometa anni 1101 <lb></lb>vel 1106, <emph type="italics"></emph>cujus Steila erat parva & obſcura<emph.end type="italics"></emph.end>(ut ille anni 1680) <lb></lb><emph type="italics"></emph>ſed ſplendor qui ex ea exivit valde clarus & quaſi ingens trabs ad <lb></lb>Orientem & Aquilonem tendebat,<emph.end type="italics"></emph.end>ut habet <emph type="italics"></emph>Hevelius<emph.end type="italics"></emph.end>ex <emph type="italics"></emph>Simeone <lb></lb>Dunelmenſi<emph.end type="italics"></emph.end>Monacho. </s> <s>Apparuit initio Menſis <emph type="italics"></emph>Februarii,<emph.end type="italics"></emph.end>circa ve<lb></lb>ſperam, ad occaſum Solis brumalem. </s> <s>Inde vero & ex ſitu caudæ col<lb></lb>ligitur caput fuiſſe Soli vicinum. <emph type="italics"></emph>A Sole,<emph.end type="italics"></emph.end>inquit Matthæus Pari<lb></lb>ſienſis, <emph type="italics"></emph>diſtabat quaſi cubito uno, ab hora tertia<emph.end type="italics"></emph.end>[rectius ſexta] <emph type="italics"></emph>uſ<lb></lb>que ad horam nonam radium ex ſe longum emittens.<emph.end type="italics"></emph.end>Talis etiam <lb></lb>erat ardentiſſimus ille Cometa ab <emph type="italics"></emph>Ariſtotele<emph.end type="italics"></emph.end>deſcriptus Lib. </s> <s>l. <lb></lb></s> <s>Meteor. </s> <s>6. <emph type="italics"></emph>cujus caput primo die non conſpectum eſt, eo quod ante <lb></lb>Solem vel ſaltem ſub radiis ſolaribus oceidiſſet, ſequente vero die <lb></lb>quantum potuit viſum eſt. </s> <s>Nam quam minima fieri poteſt diſtantia <lb></lb>Solem reliquit, & mox occubuit. </s> <s>Ob nimium ardorem<emph.end type="italics"></emph.end>[caudæ ſcili<lb></lb>cet] <emph type="italics"></emph>nondum apparebat capitis ſparſus ignis, ſed procedente tem<lb></lb>pore<emph.end type="italics"></emph.end>(ait Ariſtoreles) <emph type="italics"></emph>cum<emph.end type="italics"></emph.end>[cauda] <emph type="italics"></emph>jam minus flagraret, reddita <lb></lb>eſt<emph.end type="italics"></emph.end>[capiti] <emph type="italics"></emph>Cometæ ſua facies. </s> <s>Et ſplendorem ſuum ad tertiam <lb></lb>uſque cæli partem<emph.end type="italics"></emph.end>[id eſt, ad 60<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end>] <emph type="italics"></emph>extendit. </s> <s>Apparuit autem<emph.end type="italics"></emph.end><pb xlink:href="039/01/506.jpg" pagenum="475"></pb><emph type="italics"></emph>tempore hyberno, & aſcendens uſque ad cingulum Orionis ibi evanuit.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note504"></arrow.to.target>Cometa ille anni 1618, qui c radiis Solaribus caudatiſſimus emerſit, <lb></lb>ſtellas primæ magnitudinis æquare vel paulo ſuperare videbatur, <lb></lb>ſed majores apparuere Cometæ non pauci qui caudas breviores <lb></lb>habuere. </s> <s>Horum aliqui Jovem, alii Venerem vel etiam Lunam <lb></lb>æquaſſe traduntur. </s></p> <p type="margin"> <s><margin.target id="note504"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Diximus Cometas eſſe genus Planetarum in Orbibus valde ec<lb></lb>centricis circa Solem revolventium. </s> <s>Et quemadmodum e Plane<lb></lb>tis non caudatis, minores eſſe ſolent qui in Orbibus minoribus & <lb></lb>Soli propioribus gyrantur, ſic etiam Cometas, qui in Perihcliis <lb></lb>ſuis ad Solem propius accedunt, ut plurimum minores eſſe, ne<lb></lb>Solem attractione ſua nimis agitent, rationi conſentaneum videtur. </s> <s><lb></lb>Orbium vero tranſverſas diametros & revolutionum tempora <lb></lb>periodica, ex collatione Cometarum in iiſdem Orbibus poſt longa <lb></lb>temporum intervalla redeuntium, determinanda relinquo. </s> <s>Interea <lb></lb>huic negotio Propoſitio ſequens lumen accendere poteſt. </s></p> <p type="main"> <s><emph type="center"></emph>PROPOSITIO XLII. PROBLEMA XXII.<emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>Trajectoriam Cometæ Graphice inventam corrigere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s><emph type="italics"></emph>Oper.<emph.end type="italics"></emph.end>1. Aſſumatur poſitio plani Trajectoriæ, per Propoſitio<lb></lb>nem ſuperiorem Graphice inventa; & ſeligantur tria loca Cometæ <lb></lb>obſervationibus accuratiſſimis deſinita, & ab invicem quam ma<lb></lb>xime diſtantia; ſitque A tempus inter primam & ſecundam, ac <lb></lb>B tempus inter ſecundam ac tertiam. </s> <s>Cometam autem in eorum <lb></lb>aliquo in Perigæo verſari convenit, vel ſaltem non longe a Peri<lb></lb>gæo abeſſe. </s> <s>Ex his locis apparentibus inveniantur, per opera<lb></lb>tiones Trigonometricas, loca tria vera Cometæ in aſſumpto illo <lb></lb>plano Trajectoriæ. </s> <s>Deinde per loca illa inventa, circa centrum <lb></lb>Solis ceu umbilicum, per operationes Arithmeticas, ope Prop. </s> <s><lb></lb>XXI. Lib. </s> <s>I. inſtitutas, deſcribatur Sectio Conica: & ejus areæ, <lb></lb>radiis a Sole ad loca inventa ductis terminatæ, ſunto D & E; <lb></lb>nempe D area inter obſervationem primam & ſecundam, & E <lb></lb>area inter ſecundam ac tertiam. </s> <s>Sitque T tempus totum quo <lb></lb>area tota D+E, velocitate Cometæ per Prop. </s> <s>XVI. Lib. </s> <s>I. in<lb></lb>venta, ceſcribi debet. </s></p> <p type="main"> <s><emph type="italics"></emph>Oper.<emph.end type="italics"></emph.end>2. Augeatur longitudo Nodorum Plani Trajectoriæ, ad<lb></lb>ditis ad longitudinem illam 20′ vel 30′, quæ dicantur P; & ſer<lb></lb>vetur plani illius inclinatio ad planum Eclipticæ. </s> <s>Deinde ex <pb xlink:href="039/01/507.jpg" pagenum="476"></pb><arrow.to.target n="note505"></arrow.to.target>prædictis tribus Cometæ locis obſervatis, inveniantur in hoc novo <lb></lb>plano loca tria vera (at ſupra:) deinde etiam Orbis per loca <lb></lb>illa tranſiens, & ejuſdem areæ duæ inter obſervationes deſcriptæ, <lb></lb>quæ ſint <emph type="italics"></emph>d<emph.end type="italics"></emph.end>& <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>nec non tempus totum <emph type="italics"></emph>t<emph.end type="italics"></emph.end>quo area tota <emph type="italics"></emph>d+e<emph.end type="italics"></emph.end>de<lb></lb>ſcribi debeat. </s></p> <p type="margin"> <s><margin.target id="note505"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"></emph>Oper.<emph.end type="italics"></emph.end>3. Servetur Longitudo Nodorum in operatione prima, & <lb></lb>augeatur inclinatio Plani Trajectoriæ ad planum Eclipticæ, addi<lb></lb>tis ad inclinationem illam 20′ vel 30′, quæ dicantur <expan abbr="q.">que</expan> Deinde <lb></lb>ex obſervatis prædictis tribus Cometæ locis apparentibus, inve<lb></lb>niantur in hoc novo Plano loca tria vera, Orbiſque per loca <lb></lb>illa tranſiens, ut & ejuſdem areæ duæ inter obſervationes de<lb></lb>ſcriptæ, quæ ſint <foreign lang="grc">δ</foreign> & <foreign lang="grc">ε</foreign>, & tempus totum <foreign lang="grc">τ</foreign> quo area tota <foreign lang="grc">δ</foreign>+<foreign lang="grc">ε</foreign><lb></lb>deſcribi debeat. </s></p> <p type="main"> <s>Jam ſit C ad I ut A ad B, & G ad 1 ut D ad E, & <emph type="italics"></emph>g<emph.end type="italics"></emph.end>ad 1 ut <lb></lb><emph type="italics"></emph>d<emph.end type="italics"></emph.end>ad <emph type="italics"></emph>e,<emph.end type="italics"></emph.end>& <foreign lang="grc">γ</foreign> ad 1 ut <foreign lang="grc">δ</foreign> ad <foreign lang="grc">ε</foreign>; ſitque S tempus verum inter obſerva<lb></lb>tionem primam ac tertiam; & ſignis + & -probe obſervatis <lb></lb>quærantur numeri <emph type="italics"></emph>m<emph.end type="italics"></emph.end>& <emph type="italics"></emph>n,<emph.end type="italics"></emph.end>ea lege, ut ſit 2G-2C=<emph type="italics"></emph>m<emph.end type="italics"></emph.end>G-<emph type="italics"></emph>mg+ <lb></lb>n<emph.end type="italics"></emph.end>G-<emph type="italics"></emph>n<emph.end type="italics"></emph.end><foreign lang="grc">γ</foreign>, & 2T-2S æquale <emph type="italics"></emph>m<emph.end type="italics"></emph.end>T-<emph type="italics"></emph>mt+n<emph.end type="italics"></emph.end>T-<emph type="italics"></emph>n<emph.end type="italics"></emph.end><foreign lang="grc">τ. </foreign></s> <s>Et ſi, in <lb></lb>operatione prima, I deſignet inclinationem plani Trajectoriæ ad <lb></lb>planum Eclipticæ, & K longitudinem Nodi alterutrius, erit <lb></lb>I+<emph type="italics"></emph>n<emph.end type="italics"></emph.end>Q vera inclinatio Plani Trajectoriæ ad Planum Eclipticæ, & <lb></lb>K+<emph type="italics"></emph>m<emph.end type="italics"></emph.end>P vera longitudo Nodi. </s> <s>Ac denique ſi in operatione <lb></lb>prima, ſecunda ac tertia, quantitates R, <emph type="italics"></emph>r<emph.end type="italics"></emph.end>& <foreign lang="grc">ρ</foreign> deſignent Latera <lb></lb>recta Trajectoriæ, & quantitates 1/L, 1/<emph type="italics"></emph>l,<emph.end type="italics"></emph.end>1/<foreign lang="grc">λ</foreign> ejuſdem Latera tranſ<lb></lb>verſa reſpective: erit R+<emph type="italics"></emph>mr-m<emph.end type="italics"></emph.end>R+<emph type="italics"></emph>n<foreign lang="grc">ρ</foreign>-n<emph.end type="italics"></emph.end>R verum Latus re<lb></lb>ctum, & (1/L+<emph type="italics"></emph>ml-m<emph.end type="italics"></emph.end>L+<emph type="italics"></emph>n<foreign lang="grc">λ</foreign>-n<emph.end type="italics"></emph.end>L) verum Latus tranſverſum Tra<lb></lb>jectoriæ quam Cometa deſcribit. </s> <s>Dato autem Latere tranſverſo <lb></lb>datur etiam tempus periodicum Cometæ. <emph type="italics"></emph>Q.E.I.<emph.end type="italics"></emph.end></s></p> <p type="main"> <s>Cæterum Cometarum revolventium tempora periodica, & Or<lb></lb>bium latera tranſverſa, haud ſatis accurate determinabuntur, niſi <lb></lb>per collationem Cometarum inter ſe, qui diverſis temporibus ap<lb></lb>parent. </s> <s>Si plures Cometæ, poſt æqualia temporum intervalla, <lb></lb>eundem Orbem deſcripſiſſe reperiantur, concludendum erit hos <lb></lb>omnes eſſe unum & eundem Cometam, in eodem Orbe revolven<lb></lb>tem. </s> <s>Et tum demum ex revolutionum temporibus, dabuntur Or<lb></lb>bium latera tranſverſa, & ex his lateribus determinabuntur Or<lb></lb>bes Elliptici. </s></p><pb xlink:href="039/01/508.jpg" pagenum="477"></pb> <p type="main"> <s>In hunc finem computandæ ſunt igitur Cometarum plurium <lb></lb><arrow.to.target n="note506"></arrow.to.target>Traiectoriæ, ex hypotheſi quod ſint Parabolicæ. </s> <s>Nam hujuſ<lb></lb>modi Trajectoriæ cum Phænomenis ſemper congruent quam<lb></lb>proxime. </s> <s>Id liquet, non tantum ex Trajectoria Parabolica Co<lb></lb>metæ anni 1680, quam cum obſervationibus ſupra contuli, ſed <lb></lb>etiam ex ea Cometæ illius inſignis, qui annis 1664 & 1665 appa<lb></lb>ruit, & ab <emph type="italics"></emph>Hevelio<emph.end type="italics"></emph.end>obſervatus fuit. </s> <s>Is ex obſervationibus ſuis <lb></lb>longitudines & latitudines hujus Cometæ computavit, ſed minus <lb></lb>accurate. </s> <s>Ex iiſdem obſervationibus, <emph type="italics"></emph>Halleius<emph.end type="italics"></emph.end>noſter loca Co<lb></lb>metæ hujus denuo computavit, & tum demum ex locis ſic inven<lb></lb>tis Trajectoriam Cometæ determinavit. </s> <s>Invenit autem ejus No<lb></lb>dum aſcendentem in II 21<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 13′. </s> <s>55″, Inclinationem Orbitæ ad <lb></lb>planum Eclipticæ 21<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 18′. </s> <s>40″, diſtantiam Perihelii a Nodo in<lb></lb>Orbita 49<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 27′. </s> <s>30″. </s> <s>Perihelium in 8<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 40′. </s> <s>30′ cum Lati<lb></lb>tudine auſtrina heliocentrica 16<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 1′. </s> <s>45″. </s> <s>Cometam in Perihelio <lb></lb><emph type="italics"></emph>Novemb.<emph.end type="italics"></emph.end>24<emph type="sup"></emph>d<emph.end type="sup"></emph.end>. </s> <s>11<emph type="sup"></emph>h<emph.end type="sup"></emph.end>. </s> <s>52′. </s> <s>P. M. tempore æquato <emph type="italics"></emph>Londini,<emph.end type="italics"></emph.end>vel 13<emph type="sup"></emph>h<emph.end type="sup"></emph.end>. </s> <s>8′ <lb></lb><emph type="italics"></emph>Gedani,<emph.end type="italics"></emph.end>ſtylo veteri, & Latus rectum Parabolæ 410286, exiſtente <lb></lb>mediocri Terræ a Sole diſtantia 100000. Quam probe loca <lb></lb>Cometæ in hoc Orbe computata, congruunt cum obſervationibus, <lb></lb>patebit ex Tabula ſequente ab <emph type="italics"></emph>Halleio<emph.end type="italics"></emph.end>ſupputata. <lb></lb><arrow.to.target n="table15"></arrow.to.target> <pb xlink:href="039/01/509.jpg" pagenum="478"></pb><arrow.to.target n="note507"></arrow.to.target><arrow.to.target n="table16"></arrow.to.target> </s></p> <p type="margin"> <s><margin.target id="note507"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p><table><table.target id="table15"></table.target><row><cell>Temp. Appar. <lb></lb> <emph type="italics"></emph>Gedani<emph.end type="italics"></emph.end></cell><cell>Obſervata Cometæ diſtantia</cell><cell>Loca obſervata</cell><cell>Loca compu<lb></lb>tata in Orbe</cell></row><row><cell></cell><cell></cell><cell></cell><cell></cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell></cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell></cell><cell>gr.</cell><cell>′</cell><cell>″</cell></row><row><cell><emph type="italics"></emph>Decemb.<emph.end type="italics"></emph.end></cell><cell>a Corde Leonis</cell><cell>46.</cell><cell>24.</cell><cell>20</cell><cell>Long. </cell><cell>7.</cell><cell>1.</cell><cell>0</cell><cell></cell><cell>7.</cell><cell>1.</cell><cell>29</cell></row><row><cell>3d.</cell><cell>18<emph type="sup"></emph>h<emph.end type="sup"></emph.end>.</cell><cell>29 1/2</cell><cell>a Spica Virginis</cell><cell>22.</cell><cell>52.</cell><cell>10</cell><cell>L<gap></gap>auſt.</cell><cell>21.</cell><cell>39.</cell><cell>0</cell><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. </cell><cell>6.</cell><cell>15.</cell><cell>0</cell><cell></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><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. </cell><cell>3.</cell><cell>6.</cell><cell>0</cell><cell></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><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. </cell><cell>2.</cell><cell>56.</cell><cell>0</cell><cell></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><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><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><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><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. </cell><cell>24.</cell><cell>24.</cell><cell>0</cell><cell></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><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. </cell><cell>9.</cell><cell>0.</cell><cell>0</cell><cell></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><cell>12.</cell><cell>34.</cell><cell>13</cell></row></table><table><table.target id="table16"></table.target><row><cell>Temp. Appar. <lb></lb> <emph type="italics"></emph>Gedani<emph.end type="italics"></emph.end></cell><cell>Obſervata Cometæ diſtantia</cell><cell>Loca obſervata</cell><cell>Loca compu<lb></lb>tata in Orbe.</cell></row><row><cell>d.</cell><cell>h.</cell><cell>′</cell><cell></cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell></cell><cell>gr.</cell><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. </cell><cell>7.</cell><cell>5.</cell><cell>40</cell><cell></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><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. </cell><cell>5.</cell><cell>24.</cell><cell>45</cell><cell></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><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. </cell><cell>2.</cell><cell>7.</cell><cell>40</cell><cell></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><cell>4.</cell><cell>16.</cell><cell>25</cell></row><row><cell><emph type="italics"></emph>Jan.<emph.end type="italics"></emph.end></cell><cell>a Cing. Androm.</cell><cell>25.</cell><cell>11.</cell><cell>0</cell><cell>Long. </cell><cell>28.</cell><cell>24.</cell><cell>47</cell><cell></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><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. </cell><cell>26.</cell><cell>29.</cell><cell>15</cell><cell></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><cell>5.</cell><cell>26.</cell><cell>0</cell></row><row><cell><emph type="italics"></emph>Mar.<emph.end type="italics"></emph.end></cell><cell>Cometa ab <emph type="italics"></emph>Hookio<emph.end type="italics"></emph.end>prope ſecundam <lb></lb> Arictis obſervabatur, <emph type="italics"></emph>Mar.<emph.end type="italics"></emph.end>1<emph type="sup"></emph>d.<emph.end type="sup"></emph.end> 7<emph type="sup"></emph>h.<emph.end type="sup"></emph.end> 0′ <lb></lb> <emph type="italics"></emph>Loudini,<emph.end type="italics"></emph.end>cum</cell><cell>Long. </cell><cell>29.</cell><cell>17.</cell><cell>20</cell><cell></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><cell>8.</cell><cell>36.</cell><cell>12</cell></row></table> <p type="main"> <s>Apparuit hic Cometa per menſes tres, ſignaque fere ſex de<lb></lb>ſcripſit, & uno die gradus fere viginti confecit. </s> <s>Curſus ejus <lb></lb>a circulo maximo plurimum deflexit, in boream incurvatus; & <lb></lb>motus ejus ſub finem ex retrogrado factus eſt directus. </s> <s>Et non <lb></lb>obſtante curſu tam inſolito, Theoria a principio ad finem cum <lb></lb>obſervationibus non minus accurate congruit, quam Theoriæ <lb></lb>Planetarum cum eorum obſervationibus congruere ſolent, ut in<lb></lb>ſpicienti Tabulam patebit. </s> <s>Subducenda tamen ſunt minuta duo <lb></lb>prima circiter, ubi Cometa velociſſimus fuit; id quod fiet au<lb></lb>ferendo duodecim minuta ſecunda. </s> <s>prima ab angulo inter Nodum aſcen<lb></lb>dentem & Perihelium, ſeu conſtituendo angulum illum 49<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end><lb></lb>27′. </s> <s>18″. </s> <s>Cometæ utriuſque (& hujus & ſuperioris) parallaxis <lb></lb>annua inſignis fuit, & inde demonſtratur motus annuus Terræ in <lb></lb>Orbe magno. </s></p> <p type="main"> <s>Confirmatur etiam Theoria per motum Cometæ qui apparuit <lb></lb>anno 1683. Hic fuit retrogradus in Orbe cujus planum cum <lb></lb>plano Eclipticæ angulum fere rectum continebat. </s> <s>Hujus Nodus <lb></lb>aſcendens (computante <emph type="italics"></emph>Halleio<emph.end type="italics"></emph.end>) erat in 23<emph type="sup"></emph>gr<emph.end type="sup"></emph.end> 23′; Inclinatio <lb></lb>Orbitæ ad Eclipticam 83<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 11′; Perihelium in II 25<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 29′. </s> <s>30″; <lb></lb>Diſtantia perihelia a Sole 56020, exiſtente radio Orbis magni <lb></lb>100000, & tempore Perihelii <emph type="italics"></emph>Julii<emph.end type="italics"></emph.end>2<emph type="sup"></emph>d<emph.end type="sup"></emph.end>. </s> <s>3<emph type="sup"></emph>h<emph.end type="sup"></emph.end>. </s> <s>50′. </s> <s>Loca autem Co<lb></lb>metæ in hoc Orbe ab <emph type="italics"></emph>Halleio<emph.end type="italics"></emph.end>computata, & cum locis a <emph type="italics"></emph>Flam<lb></lb>ſtedio<emph.end type="italics"></emph.end>obſervatis collata, exhibentur in Tabula ſequente. <pb xlink:href="039/01/510.jpg" pagenum="479"></pb><arrow.to.target n="table17"></arrow.to.target> <lb></lb><arrow.to.target n="note508"></arrow.to.target></s></p> <p type="margin"> <s><margin.target id="note508"></margin.target>LIBER <lb></lb>TERTIUS.</s></p><table><table.target id="table17"></table.target><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><cell>Long. Comp.</cell><cell>Comp.</cell><cell>Long. Obſ.</cell><cell>Obſer.</cell><cell>Long.</cell><cell>Lat.</cell></row><row><cell></cell><cell>d.</cell><cell>h.</cell><cell>′</cell><cell></cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell></cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><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"></emph>Jul.<emph.end type="italics"></emph.end></cell><cell>13.</cell><cell>12.</cell><cell>55</cell><cell></cell><cell>1.</cell><cell>2.</cell><cell>30</cell><cell></cell><cell>13.</cell><cell>5.</cell><cell>42</cell><cell>29.</cell><cell>28.</cell><cell>13</cell><cell></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"></emph>Aug.<emph.end type="italics"></emph.end></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></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></cell><cell>24.</cell><cell>52.</cell><cell>53</cell><cell>5.</cell><cell>11.</cell><cell>15</cell><cell></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><cell></cell><cell></cell><cell></cell><cell></cell><cell>Auſtr.</cell><cell></cell><cell></cell><cell></cell><cell>Auſtr.</cell><cell></cell><cell></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></cell><cell>24.</cell><cell>45.</cell><cell>31</cell><cell>16.</cell><cell>38.</cell><cell>0</cell><cell></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></lb>apparuit anno 1682. Hujus Nodus aſcendens (computante <emph type="italics"></emph>Hal<lb></lb>leio<emph.end type="italics"></emph.end>) erat in 8 21<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 16′. </s> <s>30″. </s> <s>Inclinatio Orbitæ ad planum Eclip<lb></lb>ticæ 17<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 56′. </s> <s>0″. </s> <s>Perihelium in = 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 52′. </s> <s>50″. </s> <s>Diſtantia peri<lb></lb>helia a Sole 58328. Et tempus æquatum Perihelii <emph type="italics"></emph>Sept.<emph.end type="italics"></emph.end>4<emph type="sup"></emph>d<emph.end type="sup"></emph.end>. </s> <s>7<emph type="sup"></emph>h<emph.end type="sup"></emph.end>. </s> <s>39′. </s> <s><lb></lb>Loca vero ex obſervationibus <emph type="italics"></emph>Flamſtedii<emph.end type="italics"></emph.end>computata, & cum locis <lb></lb>per Theoriam computatis collata, exhibentur in Tabula ſe<lb></lb>quente. <lb></lb><arrow.to.target n="table18"></arrow.to.target> </s></p><table><table.target id="table18"></table.target><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><cell>Long. Comp.</cell><cell>Comp.</cell><cell>Long. Obſ.</cell><cell>Obſer.</cell><cell>Long.</cell><cell>Lat.</cell></row><row><cell></cell><cell>d.</cell><cell>h.</cell><cell>′</cell><cell></cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell></cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><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"></emph>Aug.<emph.end type="italics"></emph.end></cell><cell>19.</cell><cell>16.</cell><cell>38</cell><cell></cell><cell>7.</cell><cell>0.</cell><cell>7</cell><cell></cell><cell>18.</cell><cell>14.</cell><cell>28</cell><cell>25.</cell><cell>50</cell><cell>7</cell><cell></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></cell><cell>6.</cell><cell>29.</cell><cell>53</cell><cell>26.</cell><cell>8.</cell><cell>42</cell><cell></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></cell><cell>12.</cell><cell>37</cell><cell>54</cell><cell>18.</cell><cell>37.</cell><cell>47</cell><cell></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"></emph>Sept.<emph.end type="italics"></emph.end></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></cell><cell>0.</cell><cell>44.</cell><cell>10</cell><cell>8.</cell><cell>49.</cell><cell>10</cell><cell></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 ſatis manifeſtum eſt, quod motus Come<lb></lb>tarum per Theoriam a nobis expoſitam non minus accurate ex-<pb xlink:href="039/01/511.jpg" pagenum="480"></pb><arrow.to.target n="note509"></arrow.to.target>hibentur, quam ſolent motus Planetarum per eorum Theovias. </s> <s>Et <lb></lb>propterea Orbes Cometarum per hanc Theoriam enumerari poſ<lb></lb>ſunt, & tempus periodicum Cometæ in quolibet Orbe revolventis <lb></lb>tandem ſciri, & tum demum Orbium Elliptieorum latera tranſ<lb></lb>verſa & Apheliorum altitudines innoteſcent. </s></p> <p type="margin"> <s><margin.target id="note509"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Cometa retrogradus qui apparuit anno 1607, deſcripſit Orbem <lb></lb>cujus Nodus aſcendens (computante <emph type="italics"></emph>Halleio<emph.end type="italics"></emph.end>) erat in 8 20<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 21′. </s> <s><lb></lb>Inclinatio plani Orbis ad planum Eclipticæ erat 17<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 2′. </s> <s>Peri<lb></lb>helium erat in = 2<emph type="sup"></emph>gr.<emph.end type="sup"></emph.end> 16′, & diſtantia perihelia a Sole erat 58680, <lb></lb>exiſtente radio Orbis magni 100000. Et Cometa erat in Peri<lb></lb>helio <emph type="italics"></emph>Octob.<emph.end type="italics"></emph.end>16<emph type="sup"></emph>d<emph.end type="sup"></emph.end>. </s> <s>3<emph type="sup"></emph>h<emph.end type="sup"></emph.end>. </s> <s>50′. </s> <s>Congruit hic Orbis quamproxime cum <lb></lb>Orbe Cometæ qui apparuit anno 1682. Si Cometæ hi duo fue<lb></lb>rint unus & idem, revolvetur hic Cometa ſpatio annorum 75, & <lb></lb>axis major Orbis ejus erit ad axem majorem Orbis magni, ut <lb></lb>√<emph type="italics"></emph>c<emph.end type="italics"></emph.end>:75X75 ad 1, ſeu 1778 ad 100 circiter. </s> <s>Et diſtantia aphe<lb></lb>lia Cometæ hujus a Sole, erit ad diſtantiam mediocrem Terræ a <lb></lb>Sole, ut 35 ad 1 circiter. </s> <s>Quibus cognitis, haud difficile fuerit <lb></lb>Orbem Ellipticum Cometæ hujus determinare. </s> <s>Atque hæc ita <lb></lb>ſe habebunt ſi Cometa, ſpatio annorum ſeptuaginta quinque, in <lb></lb>hoc Orbe poſthac redierit. </s> <s>Cometæ reliqui majori tempore re<lb></lb>volvi videntur & altius aſcendere. </s></p> <p type="main"> <s>Cæterum Cometæ, ob magnum eorum numerum, & magnam <lb></lb>Apheliorum a Sole diſtantiam, & longam moram in Apheliis, per <lb></lb>gravitates in ſe mutuo nonnihil turbari debent, & eorum eccen<lb></lb>tricitates & revolutionum tempora nunc augeri aliquantulum, <lb></lb>nunc diminui. </s> <s>Proinde non eſt expectandum ut Cometa idem, <lb></lb>in eodem Orbe & iiſdem temporibus periodicis, accurate redeat. </s> <s><lb></lb>Sufficit ſi mutationes non majores obvenerint, quam quæ a cauſis <lb></lb>prædictis oriantur. </s></p> <p type="main"> <s>Et hinc ratio redditur cur Cometæ non comprehendantur Zo<lb></lb>diaco (more Planetarum) ſed inde migrent & motibus variis in <lb></lb>omnes cœlorum regiones ferantur. </s> <s>Scilicet eo fine, ut in Apheliis <lb></lb>ſuis ubi tardiſſime moventur, quam longiſſime diſtent ab invicem <lb></lb>& ſe mutuo quam minime trahant. </s> <s>Qua de cauſa Cometæ qui <lb></lb>altius deſcendunt, adeoque tardiſſime moventur in Apheliis, de<lb></lb>bent altius aſcendere. </s></p> <p type="main"> <s>Cometa qui anno 1680 apparuit, minus diſtabat a Sole in Peri<lb></lb>helio. </s> <s>ſuo quam parte ſexta diametri Solis; & propter ſummam <lb></lb>velocitatem in vicinia illa, & denſitatem aliquam Atmoſphæræ So<lb></lb>lis, reſiſtentiam nonnullam ſentire debuit, & aliquantulum retar-<pb xlink:href="039/01/512.jpg" pagenum="481"></pb>dari & propius ad Solem accedere: & ſingulis revolutionibus ac<lb></lb><arrow.to.target n="note510"></arrow.to.target>cedendo ad Solem, incidet is tandem in corpus Solis. </s> <s>Sed & in <lb></lb>Aphelio ubi tardiſſime movetur, aliquando per attractionem alio<lb></lb>rum Cometarum retardari poteſt & ſubinde in Solem incidere. </s> <s><lb></lb>Sic etiam Stellæ fixæ quæ paulatim expirant in lucern & vapores, <lb></lb>Cometis in ipſas incidentibus refici poſſunt, & novo alimento <lb></lb>accenſæ pro Stellis Novis haberi. </s> <s>Vapores autem qui ex Sole & <lb></lb>Stellis fixis & caudis Cometarum oriuntur, incidere poſſunt per <lb></lb>gravitatem ſuam in Atmoſphæras Planetarum, & ibi condenſari <lb></lb>& converti in aquam & ſpiritus humidos, & ſubinde per lentum <lb></lb>calorem in ſales, & ſulphura, & tincturas, & limum, & lutum, & <lb></lb>argillam, & arenam, & lapides, & coralla, & ſubſtantias alias <lb></lb>terreſtres paulatim migrare. </s> <s>Decreſcente autem corpore Solis <lb></lb>motus medii Planetarum circum Solem paulatim tardeſcent, & <lb></lb>creſcente Terra motus medius Lunæ circum Terram paulatim au<lb></lb>gebitur. </s> <s>Et collatis quidem obſervationibus Eclipſium <emph type="italics"></emph>BabyloNI<lb></lb>cis<emph.end type="italics"></emph.end>cum iis <emph type="italics"></emph>Albategnii<emph.end type="italics"></emph.end>& cum hodiernis, <emph type="italics"></emph>Halleius<emph.end type="italics"></emph.end>noſter motum <lb></lb>medium Lunæ cum motu diurno Terræ collatum, paulatim acce<lb></lb>lerari, primus omnium quod ſciam deprehendit. </s></p> <p type="margin"> <s><margin.target id="note510"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>SCHOLIUM GENERALE.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hypotheſis Vorticum multis premitur difficultatibus. </s> <s>Ut Pla<lb></lb>neta unuſquiſque radio ad Solem ducto areas deſcribat tempori <lb></lb>proportionales, tempora periodica partium Vorticis deberent eſſe <lb></lb>in duplicata ratione diſtantiarum a Sole. </s> <s>Ut periodica Plane<lb></lb>tarum tempora ſint in proportione ſeſquiplicata diſtantiarum a <lb></lb>Sole, tempora periodica partium Vorticis deberent eſſe in eadem <lb></lb>diſtantiarum proportione. </s> <s>Ut Vortices minores circum Satur<lb></lb>num, Jovem & alios Planetas gyrati conſerventur & tranquille <lb></lb>natent in Vortice Solis, tempora periodica partium Vorticis So<lb></lb>laris deberent eſſe æqualia. </s> <s>Revolutiones Solis & Planetarum cir<lb></lb>cum axes ſuos ab omnibus hiſce proportionibus diſcrepant. </s> <s>Mo<lb></lb>tus Cometarum ſunt ſumme regulares, & eaſdem leges cum Pla<lb></lb>netarum motibus obſervant, & per Vortices explicari nequeunt. </s> <s><lb></lb>Feruntur Cometæ motibus valde eccentricis in omnes cælorum <lb></lb>partes, quod fieri non poteſt niſi Vortices tollantur. </s></p> <p type="main"> <s>Projectilia, in aere noſtro, ſolam aeris reſiſtentiam ſentiunt. </s> <s><lb></lb>Sublato aere, ut fit in Vacuo <emph type="italics"></emph>Boyliano,<emph.end type="italics"></emph.end>reſiſtentia ceſſat, ſiqui<lb></lb>dem pluma tenuis & aurum ſolidum æquali cum velocitate in hoc <pb xlink:href="039/01/513.jpg" pagenum="482"></pb><arrow.to.target n="note511"></arrow.to.target>Vacuo cadunt. </s> <s>Et par eſt ratio ſpatiorum cæleſtium quæ ſunt <lb></lb>ſupra atmoſphæram Terræ. </s> <s>Corpora omnia in iſtis ſpatiis liber<lb></lb>rime moveri debent; & propterea Planetæ & Cometæ in orbi<lb></lb>bus ſpecie & poſitione datis, ſecundum leges ſupra expoſitas, per<lb></lb>petuo revolvi. </s> <s>Perſeverabunt quidem in orbibus ſuis per leges <lb></lb>gravitatis, ſed regularem orbium ſitum primitus acquirere per <lb></lb>leges haſce minime potuerunt. </s></p> <p type="margin"> <s><margin.target id="note511"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Planetæ ſex principales revolvuntur circum Solem in circulis <lb></lb>Soli concentricis, eadem motus directione, in eodem plano quam<lb></lb>proxime. </s> <s>Lunæ decem revolvuntur circum Terram, Jovem & Sa<lb></lb>turnum in circulis concentricis, eadem motus directione, in planis <lb></lb>orbium Planetarum quamproxime. </s> <s>Et hi omnes motus regulares <lb></lb>originem non habent ex cauſis Mechanicis; ſiquidem Cometæ in <lb></lb>Orbibus valde eccentricis, & in omnes cælorum partes libere <lb></lb>feruntur. </s> <s>Quo motus genere Cometæ per Orbes Planetarum ce<lb></lb>lerrime & facillime tranſeunt, & in Apheliis ſuis ubi tardiores <lb></lb>ſunt & diutius morantur, quam longiſſime diſtant ab invicem, <lb></lb>& ſe mutuo quam minime trahunt. </s> <s>Elegantiſſima hæcce Solis, <lb></lb>Planetarum & Cometarum compages non niſi conſilio & dominio <lb></lb>Entis intelligentis & potentis oriri potuit. </s> <s>Et ſi Stellæ fixæ ſint <lb></lb>centra ſimilium ſyſtematum; hæc omnia ſimili conſilio conſtructa, <lb></lb>ſuberunt <emph type="italics"></emph>Unius<emph.end type="italics"></emph.end>dominio: præſertim cum lux Fixarum ſit eiuſdem <lb></lb>naturæ ac lux Solis, & ſyſtemata omnia lucem in omnia invicem <lb></lb>immittant. </s></p> <p type="main"> <s>Hic omnia regit, non ut Anima mundi, ſed ut univerſorum Do<lb></lb>minus; & propter dominium ſuum Dominus Deus <lb></lb> <foreign lang="grc">παντικ<gap></gap>τ<gap></gap>ρ</foreign> dici ſolet. </s> <s>Nam <emph type="italics"></emph>Deus<emph.end type="italics"></emph.end>eſt vox relativa <lb></lb>& ad ſervos refertur: & <emph type="italics"></emph>Deitas<emph.end type="italics"></emph.end>eſt dominatio Dei <lb></lb>non in corpus proprium, ſed in ſervos. <emph type="italics"></emph>Deus ſummus<emph.end type="italics"></emph.end>eſt Ens <lb></lb>æternum, infinitum, abſolute perfectum; ſed Ens utcunque per<lb></lb>fectum ſine dominio, non eſt <emph type="italics"></emph>Dominus Deus.<emph.end type="italics"></emph.end>Dicimus enim <emph type="italics"></emph>Deus <lb></lb>meus, Deus veſter, Deus Iſraelis:<emph.end type="italics"></emph.end>ſed non dicimus <emph type="italics"></emph>Æternus meus, <lb></lb>Æternus veſter, Æternus Iſraelis<emph.end type="italics"></emph.end>; non dicimus <emph type="italics"></emph>Infinitus meus, <lb></lb>Infinitus veſter, Infinitus Iſraelis<emph.end type="italics"></emph.end>; non dicimus <emph type="italics"></emph>Perfectus meus, Per<lb></lb>fectus veſter, Perfectus Iſraelis.<emph.end type="italics"></emph.end>Hæ appellationes relationem non <lb></lb>habent ad ſervos. </s> <s>Vox <emph type="italics"></emph>Deus<emph.end type="italics"></emph.end>paſſim ſignificat <emph type="italics"></emph>Dominum,<emph.end type="italics"></emph.end>ſed <lb></lb>omnis Dominus non eſt Deus. </s> <s>Dominatio Entis ſpiritualis <emph type="italics"></emph>Deum<emph.end type="italics"></emph.end><lb></lb>conſtituit, vera verum, ſumma ſummum, ficta fictum. </s> <s>Et ex do<lb></lb>minatione vera ſequitur, Deum verum eſſe vivum, intelligentem & <lb></lb>potentem; ex reliquis perfectionibus ſummum eſſe vel ſumme per-<pb xlink:href="039/01/514.jpg" pagenum="483"></pb>fectum. <emph type="italics"></emph>Æternus<emph.end type="italics"></emph.end>eſt & <emph type="italics"></emph>Infinitus, Omnipotens<emph.end type="italics"></emph.end>& <emph type="italics"></emph>Omniſciens,<emph.end type="italics"></emph.end>id <lb></lb><arrow.to.target n="note512"></arrow.to.target>eſt, durat ab æterno in æternum & adeſt ab infinito in infinitum, <lb></lb>omnia regit & omnia cognoſcit quæ fiunt aut ſciri poſſunt. </s> <s>Non <lb></lb>eſt æternitas vel infinitas, ſed æternus & infinitus; non eſt duratio <lb></lb>vel ſpatium, ſed durat & adeſt. </s> <s>Durat ſemper & adeſt ubique, & <lb></lb>exiſtendo ſemper & ubiQ.E.D.rationem & ſpatium, æternitatem <lb></lb>& infinitatem conſtituit. </s> <s>Cum unaquæque ſpatii particula ſit <lb></lb><emph type="italics"></emph>ſemper,<emph.end type="italics"></emph.end>& unumquodQ.E.D.rationis indiviſibile momentum <emph type="italics"></emph>ubique<emph.end type="italics"></emph.end>; <lb></lb>certe rerum omnium Fabricator ac Dominus non erit <emph type="italics"></emph>nunquam <lb></lb>nuſquam.<emph.end type="italics"></emph.end>Omnipræſens eſt nen per <emph type="italics"></emph>virtutem<emph.end type="italics"></emph.end>ſolam, ſed etiam <lb></lb>per <emph type="italics"></emph>ſubſtantiam<emph.end type="italics"></emph.end>: nam virtus ſine ſubſtantia <lb></lb>ſubſiſtere non poteſt. </s> <s>In ipſo continentur <lb></lb>& moventur univerſa, ſed abſque mutua <emph type="italics"></emph>paſ<lb></lb>ſione.<emph.end type="italics"></emph.end>Deus nihil patitur ex corporum moti<lb></lb>bus: illa nullam ſentiunt reſiſtentiam ex om<lb></lb>nipræſentia Dei. </s> <s>Deum ſummum neceſſario <lb></lb>exiſtere in conſeſſo eſt: Et eadem neceſſitate <lb></lb><emph type="italics"></emph>ſemper<emph.end type="italics"></emph.end>eſt & <emph type="italics"></emph>ubique.<emph.end type="italics"></emph.end>Unde etiam totus eſt ſui ſimilis, totus oculus, <lb></lb>totus auris, totus cerebrum, totus brachium, totus vis ſentiendi, <lb></lb>intelligendi & agendi; ſed more minime humano, more minime <lb></lb>corporeo, more nobis prorſus incognito. </s> <s>Ut cæcus ideam non <lb></lb>habet colorum, ſic nos ideam non habemus modorum quibus <lb></lb>Deus ſapientiſſimus ſentit & intelligit omnia. </s> <s>Corpore omni & <lb></lb>figura corporea prorſus deſtituitur, ideoque videri non poteſt, <lb></lb>nec audiri, nec tangi, nec ſub ſpecie rei alicujus corporei coli de<lb></lb>bet. </s> <s>Ideas habemus attributorum ejus, ſed quid ſit rei alicujus <lb></lb>Subſtantia minime cognoſcimus. </s> <s>Videmus tantum corporum figu<lb></lb>ras & colores, audimus tantum ſonos, tangimus tantum ſuper<lb></lb>ficies externas, olfacimus odores ſolos, & guſtamus ſapores; In<lb></lb>timas ſubſtantias nullo ſenſu, nulla actione reflexa cognoſcimus, & <lb></lb>multo minus ideam habemus ſubſtantiæ Dei. </s> <s>Hunc cognoſcimus <lb></lb>ſolummodo per proprietates ſuas & attributa, & per ſapientiſſi<lb></lb>mas & optimas rerum ſtructuras, & cauſas finales; veneramur au<lb></lb>tem & colimus ob dominium. </s> <s>Deus enim ſine dominio, provi<lb></lb>dentia, & cauſis finalibus, nihil aliud eſt quam Fatum & Na<lb></lb>tura. </s> <s>Et hæc de Deo; de quo utique ex Phænomenis diſſerere, <lb></lb>ad <emph type="italics"></emph>Philoſophiem Experimentalem<emph.end type="italics"></emph.end>pertinet. </s></p> <p type="margin"> <s><margin.target id="note512"></margin.target>LIBER <lb></lb>TERTIUS.</s></p> <p type="main"> <s>Hactenus Phænomena cælorum & maris noſtri per Vim gravi<lb></lb>tatis expoſui, ſed cauſam Gravitatis nondum aſſignavi. </s> <s>Oritur <lb></lb>utique hæc Vis a cauſa aliqua quæ penetrat ad uſque centra Solis <pb xlink:href="039/01/515.jpg" pagenum="484"></pb><arrow.to.target n="note513"></arrow.to.target>& Planetarum, ſine virtutis diminutione; quæque agit non pro <lb></lb>quantitate <emph type="italics"></emph>ſuperficierum<emph.end type="italics"></emph.end>particularum in quas agit (ut ſolent cauſæ <lb></lb>Mechanicæ,) ſed pro quantitate materiæ <emph type="italics"></emph>ſolidæ<emph.end type="italics"></emph.end>; & cujus actio in <lb></lb>immenſas diſtantias undique extenditur, decreſcendo ſemper in <lb></lb>duplicata ratione diſtantiarum. </s> <s>Gravitas in Solem componitur <lb></lb>ex gravitatibus in ſingulas Solis particulas, & recedendo a Sole <lb></lb>decreſcit accurate in duplicata ratione diſtantiarum ad uſque or<lb></lb>bem Saturni, ut ex quiete Apheliorum Planetarum manifeſtum eſt, <lb></lb>& ad uſque ultima Cometarum Aphelia, ſi modo Aphelia illa <lb></lb>quieſcant. </s> <s>Rationem vero harum Gravitatis proprietatum ex <lb></lb>Phænomenis nondum potui deducere, & Hypotheſes non ſingo. </s> <s><lb></lb>Quicquid enim ex Phænomenis non deducitur, <emph type="italics"></emph>Hypotheſis<emph.end type="italics"></emph.end>vo<lb></lb>canda eſt; & Hypotheſes ſeu Metaphyſicæ, ſeu Phyſicæ, ſeu Qua<lb></lb>litatum occultarum, ſeu Mechanicæ, in <emph type="italics"></emph>Philoſophia Experimentali<emph.end type="italics"></emph.end><lb></lb>locum non habent. </s> <s>In hac Philoſophia Propoſitiones deducun<lb></lb>tur ex Phænomenis, & redduntur generales per Inductionem. </s> <s>Sie <lb></lb>impenetrabilitas, mobilitas, & impetus corporum & leges motuum <lb></lb>& gravitatis innotuerunt. </s> <s>Et ſatis eſt quod Gravitas revera ex<lb></lb>iſtat, & agat ſecundum leges a nobis expoſitas, & ad corporum <lb></lb>cæleſtium & maris noſtri motus omnes ſufficiat. </s></p> <p type="margin"> <s><margin.target id="note513"></margin.target>DE MUNDI <lb></lb>SYSTEMATE</s></p> <p type="main"> <s>Adjicere jam liceret nonnulla de Spiritu quodam ſubtiliſſimo cor<lb></lb>pora craſſa pervadente, & in iiſdem latente; cujus vi & actionibus <lb></lb>particulæ corporum ad minimas diſtantias ſe mutuo attrahunt, <lb></lb>& contiguæ factæ cohærent; & corpora Electrica agunt ad di<lb></lb>ſtantias majores, tam repellendo quam attrahendo corpuſcula vi<lb></lb>cina; & Lux emittitur, reflectitur, refringitur, inflectitur, & cor<lb></lb>pora calefacit; & Senſatio omnis excitatur, & membra Anima<lb></lb>lium ad voluntatem moventur, vibrationibus ſcilicet hujus Spiri<lb></lb>tus per ſolida nervorum capillamenta ab externis ſenſuum orga<lb></lb>nis ad cerebrum & a cerebro in muſculos propagatis. </s> <s>Sed hæc <lb></lb>paucis exponi non poſſunt; neque adeſt ſufficiens copia Experi<lb></lb>mentorum, quibus leges actionum hujus Spiritus accurate deter<lb></lb>minari & monſtrari debent. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p><pb xlink:href="039/01/516.jpg"></pb></subchap2></subchap1></chap><chap> <p type="main"> <s><emph type="center"></emph>INDEX RERUM <lb></lb>ALPHABETICUS.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>N.B. <emph type="italics"></emph>Citationes factæ ſunt ad normam ſequentis Exempli.<emph.end type="italics"></emph.end>III, 10: 444, 20: <lb></lb>471, 28 <emph type="italics"></emph>deſignant Libri tertii Propoſitionem decimam: Paginæ<emph.end type="italics"></emph.end>444<emph type="italics"></emph><emph type="sup"></emph>ta<emph.end type="sup"></emph.end> <emph type="italics"></emph>lineam<emph.end type="italics"></emph.end><lb></lb>20<emph type="italics"></emph><emph type="sup"></emph>æm<emph.end type="sup"></emph.end>: Paginæ<emph.end type="italics"></emph.end>471<emph type="italics"></emph><emph type="sup"></emph>æm<emph.end type="sup"></emph.end> lineam<emph.end type="italics"></emph.end>28<emph type="italics"></emph><emph type="sup"></emph>æm<emph.end type="sup"></emph.end><emph.end type="italics"></emph.end>. <lb></lb></s></p> <p type="main"> <s><emph type="center"></emph>A.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>ÆQuinoctiorum præceſſio </s></p> <p type="main"> <s>cauſæ hujus motus indicantur III, <lb></lb>21 </s></p> <p type="main"> <s>quantitas motus ex cauſis computatur III, 39 <lb></lb>Aeris </s></p> <p type="main"> <s>denſitas ad quamlibet altitudinem colligitur <lb></lb>ex Prop. </s> <s>22. Lib. </s> <s>II. quanta ſit ad altitu<lb></lb>dinem unius ſemidiametri Terreſtris oſten<lb></lb>ditur 470, 11 </s></p> <p type="main"> <s>elaſtica vis quali cauſæ tribui poſſit II, 23 </s></p> <p type="main"> <s>gravitas cum Aquæ gravitate collata 470, 3 </s></p> <p type="main"> <s>reſiſtentia quanta ſit, per Experimenta Pen<lb></lb>dulorum colligitur 286, 28; per Experi<lb></lb>menta corporum cadentium & Theoriam <lb></lb>accuratius invenitur 327, 13 </s></p> <p type="main"> <s>Anguli contactus non ſunt omne; ejaſdem gene<lb></lb>ris, ſed alii aliis inſinite minores p. </s> <s>32 </s></p> <p type="main"> <s>Apſidum motus expendltur I, Sect. </s> <s>9 </s></p> <p type="main"> <s>Areæ quas corpora in gyros acta, radiis ad con<lb></lb>trum virium ductis, deſcribunt, conferuntur <lb></lb>cum temporibus deſcriptionum I, 1, 2, 3, <lb></lb>58, 65 </s></p> <p type="main"> <s>Attractio corporum univerſorum demonſtratur <lb></lb>III, 7; qualis ſit hujus demonſtrationis certi<lb></lb>tudo oſtenditur 358, 28: 484, 11 </s></p> <p type="main"> <s>Attractionis cauſam vel modum nullibi definit <lb></lb>Author 5, 17: 147, 32: 172, 31: 483, 34. </s></p> <p type="main"> <s><emph type="center"></emph>C.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Cali </s></p> <p type="main"> <s>reſiſtentia deſtituuntur III, 10: 444, 20: <lb></lb>471, 28; & propterea Fluido omni corpo<lb></lb>rco 328, 18 </s></p> <p type="main"> <s>tranſitum Luci præbent abſque ulla refracti<lb></lb>one 467, 33 </s></p> <p type="main"> <s>Calore virga ferrea comperta eſt augeri longi<lb></lb>tudine 386, 4 </s></p> <p type="main"> <s>Calor Solis quantus ſit in diverſis a Sole diſtantiis <lb></lb>466, 20 </s></p> <p type="main"> <s>quantus apud Mercurium 372, 12 <lb></lb></s></p> <p type="main"> <s>quantus apud Cometam anni 1680 in Peri<lb></lb>helio verſantem 466, 22 </s></p> <p type="main"> <s>Centrum commune gravitatis corporum plu<lb></lb>rium, ab actionibus corporum inter ſe, non <lb></lb>mutat ſtatum ſuum vel motus vel quietis <lb></lb>p. </s> <s>17 </s></p> <p type="main"> <s>Centrum commune gravitatis Terræ, Solis & <lb></lb>Planctarum omnium quicſcere III, 11; con<lb></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></lb>motu annuo percurrit Orbem magnum 376, 6 <lb></lb>quibur intervallis diſtata Terra & Luna 430, 22 </s></p> <p type="main"> <s>Centrun Virium quibus corpora revolventia in <lb></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></lb>bus invenitur I, 5 </s></p> <p type="main"> <s>Circuli circumſerentia, qua lege vis centripetæ <lb></lb>tendentis ad punctum quodcunQ.E.D.tum de<lb></lb>ſcribi poteſt a corpore revolvente I, 4, 7, 8 </s></p> <p type="main"> <s>Cometæ </s></p> <p type="main"> <s>Genus ſunt Planetarum, non Meteororum <lb></lb>444, 24: 466, 15 </s></p> <p type="main"> <s>Luna ſuperiores ſunt, & in regione Planeta<lb></lb>rum verſantur p. </s> <s>439 </s></p> <p type="main"> <s>Diſtantia eorum qua ratione per Obſervatio<lb></lb>nes colligi poteſt quamproxime 439, 21 </s></p> <p type="main"> <s>Plures obſervati ſunt in hemiſphærio Solem <lb></lb>verſus, quam in hemiſphærio oppoſito; & <lb></lb>unde hoc fiat 444, 5 </s></p> <p type="main"> <s>Splendent luce Solis a ſe reflexa 444, 4; Lux <lb></lb>illa quanta eſſet ſolet 441, 12 </s></p> <p type="main"> <s>Cinguntur Atmoſphæris ingentibus 442, 12: <lb></lb>444, 27 </s></p> <p type="main"> <s>Qui ad Solem propius accedunt ut plurimum <lb></lb>minores eſſe exiſtimantur 475, 7 </s></p> <p type="main"> <s>Quo fine non comprehenduntur Zodiaco <lb></lb>(more Planetarum) ſed in omnes tælorum <lb></lb>regiones varie feruntur 480, 30 </s></p> <p type="main"> <s>Poſſunt aliquando in Solem incidere & no<lb></lb>vum illi alimentum ignis præbere 480, 37 </s></p> <p type="main"> <s>Uſus eorum ſuggeritur 473, 1: 481, 7 </s></p><pb xlink:href="039/01/517.jpg"></pb> <p type="main"> <s>Cometaram caudr </s></p> <p type="main"> <s>avertuntur a Sole 408, 39 </s></p> <p type="main"> <s>maximæ ſunt & ſulgentiſſimæ ſtatim poſt <lb></lb>tranſitum per vicinam Solis 467, 8 </s></p> <p type="main"> <s>inſignis 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; ſpatio a capite aſcendunt 471, 1 </s></p> <p type="main"> <s>Cometæ </s></p> <p type="main"> <s>Moventur in Sectionibus Conicis umbilicos <lb></lb>in centro Solis habentibus, & radiis ad So<lb></lb>lem ductis deſcribunt areas temporibus pro<lb></lb>portionales. </s> <s>Et quidem in Ellipſibus mo<lb></lb>ventur ſi in Orbem redeunt, hæ tamen <lb></lb>Parabolis erunt maximæ ſinitimæ III, 40 </s></p> <p type="main"> <s>Trajectoria Paral olica ex datis tribus Obſer<lb></lb>vationibus invenitur III, 41; Inventa cor<lb></lb>rigitur III, 42 </s></p> <p type="main"> <s>Locus in Parabola invenitur ad tempus da<lb></lb>tum 445, 30: I, 30 </s></p> <p type="main"> <s>Velocitas cum velocitate Planetarum conſer<lb></lb>tur 445, 17 </s></p> <p type="main"> <s>Cometa annorum 1664 & 1665 </s></p> <p type="main"> <s>Huius motus obſervatus expenditur, & cum <lb></lb>Theoria accurate congruere deprehenditur <lb></lb>p. </s> <s>477 </s></p> <p type="main"> <s>Cometa annorum 1680 & 1681 </s></p> <p type="main"> <s>Hujus motus obſervatus cum Theoria accu<lb></lb>rate congruere invenitur p. </s> <s>455 & <expan abbr="ſeqq.">ſeqque</expan> </s></p> <p type="main"> <s>Videbatur in Ellipſi revolvi ſpatio annorum <lb></lb>pluſquam quingentorum 464, 37 </s></p> <p type="main"> <s>Trajectoria illius & Cauda ſingulis in locis <lb></lb>delineantur p. </s> <s>465 </s></p> <p type="main"> <s>Cometa anni 1682 </s></p> <p type="main"> <s>Hajus motus accurate teſpondet Theoriæ <lb></lb>p. </s> <s>479 </s></p> <p type="main"> <s>Comparuiſſe viſus eſt anno 1607, iterumque re<lb></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 reſpondet Theoriæ <lb></lb>p. </s> <s>478 </s></p> <p type="main"> <s>Curvæ diſtinguuntur in Geometrice rationales & <lb></lb>Geometrice irrationales 100, 5 </s></p> <p type="main"> <s>Curvatura figurarum qua ratione æſtimanda ſit <lb></lb>235, 28: 398, 33 </s></p> <p type="main"> <s>Cycloidis ſeu Epicycloidis <lb></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></lb>poſiti quarum vires ſunt reciproce ut qua<lb></lb>drata diſtantiarum 198, 1 </s></p> <p type="main"> <s><emph type="center"></emph>D.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Dei Natura p. </s> <s>482 & 483 </s></p> <p type="main"> <s>Deſcenſus graviuni in vacuo quantus ſit, ex lon<lb></lb>gitudine Penduii colligitur 379, 1 </s></p> <p type="main"> <s>Deſcenſus vel Aſcenſus rectilinci ſpatia deſcri<lb></lb>pta, tempora deſcriptionum & velocitates ac<lb></lb><lb></lb>quiſitæ conferuntur, poſita cujuſcunque ge<lb></lb>neris vi centripeta I, Sect. </s> <s>7 </s></p> <p type="main"> <s>Deſcenſus & Aſcenſus corporum in Mediis re<lb></lb>ſiſtentibus II, 3, 8, 9, 40, 13, 14 </s></p> <p type="main"> <s><emph type="center"></emph>E.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Ellipſis </s></p> <p type="main"> <s>qua lege vis contripetæ tendentis ad centrum <lb></lb>figuræ deſcribitur a corpore revolvente <lb></lb>I, 10, 64 </s></p> <p type="main"> <s>qua lege vis centripetæ tendentis ad umbili<lb></lb>cum figuræ deſcribitur a corpore revol<lb></lb>vente I, 11 </s></p> <p type="main"> <s><emph type="center"></emph>F.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Fleidi definitio p. </s> <s>260 </s></p> <p type="main"> <s>Flaidorum denſitas & compreſſio quas leges ha<lb></lb>bent, oſtenditur II, Sect. </s> <s>5 </s></p> <p type="main"> <s>Fluidorum per foramen in vaſe factum effluen<lb></lb>tium determinatur motus II, 36 </s></p> <p type="main"> <s>Fumi in camino aſcenſus obiter explicatur 472, 4 </s></p> <p type="main"> <s><emph type="center"></emph>G.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Graduum in Meridiano Terreſtri menſura exhi<lb></lb>betur, & quam ſit exigua inæqualitas oſten<lb></lb>ditur ex Theoria III, 20 </s></p> <p type="main"> <s>Gravitas </s></p> <p type="main"> <s>diverſi eſt generis a vi Magnetica 368, 29 </s></p> <p type="main"> <s>mutua eſt inter Terram & ejus partes 22, 18 </s></p> <p type="main"> <s>ejus cauſa non aſſignatur 483, 34 </s></p> <p type="main"> <s>datur in Planetas univerſos 365, 15; & per<lb></lb>gendo a ſuperficiebus Planetarum ſurſum <lb></lb>decreſcit in duplicata ratione diſtantiarum <lb></lb>a centro III, 8, deorſum decreſcit in ſim<lb></lb>plici ratione quamproxime III, 9 </s></p> <p type="main"> <s>datur in corpora omnia, & proportionalis eſt <lb></lb>quantitati materiæ in ſingulis III, 7 </s></p> <p type="main"> <s>Gravitatem eſſe vim illam qua Luna retinetur <lb></lb>in Orbe III, 4, computo accuratiori com<lb></lb>probatur 430, 25 </s></p> <p type="main"> <s>Gravitatem eſſe vim illam qua Planetæ primarii <lb></lb>& Satellites Jovis & Saturni retinentur in <lb></lb>Orbibus III, 5 </s></p> <p type="main"> <s><emph type="center"></emph>H.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Hydroſtaticæ 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></lb>tro deſcribitur a corpore revolvente 47, 26 </s></p> <p type="main"> <s>qua lege vis centrifugæ tendentis ab umbilico <lb></lb>figuræ deſcribitur a corpore revolvente 51, 6 </s></p> <p type="main"> <s>qua lege vis centripetæ tendentis ad umbilicum <lb></lb>figurædeſcribitur a corpore revolvente I, 12 </s></p> <p type="main"> <s>Hypotheſes cujuſcunque generis rejiciuntur ab <lb></lb>hac Philoſophia 484, 8. </s></p><pb xlink:href="039/01/518.jpg"></pb> <p type="main"> <s><emph type="center"></emph>I.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Inertiæ vis deſinitur p. </s> <s>2 </s></p> <p type="main"> <s>Jovis </s></p> <p type="main"> <s>diſtantia a Sole 361, </s></p> <p type="main"> <s>ſemidiameter apparens 371, 3 </s></p> <p type="main"> <s>ſemidiameter vera 371, 14 </s></p> <p type="main"> <s>attractiva vis quanta ſit 370, 33 </s></p> <p type="main"> <s>pondus corporum in ejus ſuperficie 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 ſit 375, 33 </s></p> <p type="main"> <s>diametrorum proportio computo exhibetur <lb></lb>381, 27 </s></p> <p type="main"> <s>converſio citcum axem quo tempore abſolvi<lb></lb>tur 381, 25 </s></p> <p type="main"> <s>cingulæ cauſa ſubindicatur 444 32. </s></p> <p type="main"> <s><emph type="center"></emph>L.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Locus definitur, & diſtinguitur in abſolutum & <lb></lb>relativum 6, 12 </s></p> <p type="main"> <s>Loca corporum in Sectionibus conicis moto<lb></lb>rum inveniuntur ad tempus aſſignatum I, <lb></lb>Sect. </s> <s>6 </s></p> <p type="main"> <s>Lucis </s></p> <p type="main"> <s>propagatio non eſt inſtantanea 207, 5; non <lb></lb>fit per agitationem Medii alicujus Ætherci <lb></lb>342, 36 </s></p> <p type="main"> <s>velocitas in diverſis Mediis diverſa I, 95 </s></p> <p type="main"> <s>reflexio quædam explicatur I, 96 </s></p> <p type="main"> <s>refractio explicatur I, 94; non ſit in puncto <lb></lb>ſolum incidentiæ 207, 29 </s></p> <p type="main"> <s>incurvatio prope corporum terminos Expe<lb></lb>rimentis obſervata 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 cauſa patefacta, cur candem ſemper fa<lb></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 ſuperficie 430, 20 </s></p> <p type="main"> <s>denſitas 430, 15 </s></p> <p type="main"> <s>quantitas materiæ 430, 19 </s></p> <p type="main"> <s>diſtantia mediocris a Terra quot continet <lb></lb>maximas Terræ ſemidiametros 430, 25, <lb></lb>quot mediocres 431, 18 </s></p> <p type="main"> <s>parallaxis maxima in longitudinem paulo ma<lb></lb>jor eſt quam paraliaxis maxima in latitu<lb></lb>dinem 387, 8 </s></p> <p type="main"> <s>vis ad Mare movendum quanta ſit III, 37; <lb></lb>non ſentiri poteſt in Experimentis pendu<lb></lb>lorum, vel in Staticis aut Hydroſtaticis <lb></lb>quibuſcunque 430, 1 </s></p> <p type="main"> <s>tempus periodicum 430, 32 </s></p> <p type="main"> <s>tempus revolutionis ſynodicæ 398, 1 </s></p> <p type="main"> <s>motus medius cum diurno motu Terræ col<lb></lb><lb></lb>latus paulatim accelerari deprehenditur ab <lb></lb><emph type="italics"></emph>Helleio<emph.end type="italics"></emph.end>481, 16 </s></p> <p type="main"> <s>Lunæ motus & motuum inæqualitates a cauſis <lb></lb>ſuis derivantur III, 22: p. </s> <s>421 & <expan abbr="ſeqq.">ſeqque</expan> </s></p> <p type="main"> <s>tardius revolvitur Luna dilatato Orbe, in pe<lb></lb>rihelio Terræ, citius in ophelio, contracto <lb></lb>Orbe III, 22: 421, 6 </s></p> <p type="main"> <s>tardius revolvitur, dilatato Orbe, in Apogæi <lb></lb>Syzygiis cum Sole; citius in Quadraturis <lb></lb>Apogæi, contracto Orbe 422, 1 </s></p> <p type="main"> <s>tardius revolvitur, dilatato Orbe, in Syzygiis <lb></lb>Nodi cum Sole; citius in Quadraturis No<lb></lb>di, contracto Orbe 422, 21 </s></p> <p type="main"> <s>tardius movetur in Quadraturis ſuis cum Sole, <lb></lb>citius in Syzygiis; & radio ad Terram <lb></lb>ducto deſeribit aream pro tempere mino<lb></lb>rem in priore caſu, majorem in poſteriore <lb></lb>III, 22: Inæqualitas harum Arearum com<lb></lb>putatur III, 26. Orbem inſuper habet ma<lb></lb>gis curvum & longius a Terra recedit in <lb></lb>priore caſu, minus curvum habet Orbem <lb></lb>& propius ad Terram accedit in poſteriore <lb></lb>III, 22. Orbis hujus figura & proportio <lb></lb>diametrorum ejus computo colligitur III, <lb></lb>28. Et ſabinde proponitur methodus in<lb></lb>veniendi diſtantiam Lunæ a Terra ex motu <lb></lb>ejus horario III, 27 </s></p> <p type="main"> <s>Apogæum tardius movetur in Aphelio Terræ, <lb></lb>velocius in Perihclio III, 22: 421, 21 </s></p> <p type="main"> <s>Apogæum ubi eſt in Solis Syzygiis, maxime <lb></lb>progreditur; in Quadraturis regreditur III, <lb></lb>22: 422, 37 </s></p> <p type="main"> <s>Eccentricitas maxima eſt in Apogæi Syzygiis <lb></lb>cum Sole, minima in Quadraturis III, 22: <lb></lb>422, 39 </s></p> <p type="main"> <s>Nodi tardius moventur in Aphelio Terræ, ve<lb></lb>locius in Perihelio III, 22: 421, 21 </s></p> <p type="main"> <s>Nodi quieſcunt in Syzygiis ſuis cum Sole, & <lb></lb>velociſſime regrediuntur in Quadraturis <lb></lb>III, 22. Nodorum motus & inæqualitates <lb></lb>motuum computantur ex Theoria Gravi<lb></lb>tatis III, 30, 31, 32, 33 </s></p> <p type="main"> <s>Inclinatio Oibis ad Ecſipticam maxima eſt in <lb></lb>Syzygiis Nodorum cum Sole, minima in <lb></lb>Quadraturis I, 66 Cor. </s> <s>10. Inclinationis va<lb></lb>riationes computantur ex Theoria Gravita<lb></lb>tis III, 34, 35 </s></p> <p type="main"> <s>Lunarium motuum Æquationes ad uſus Aſtro<lb></lb>nomicos p. </s> <s>421 & <expan abbr="ſeqq.">ſeqque</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 ſemeſtris prima 412, 1 </s></p> <p type="main"> <s>Æquatio ſemeſtris ſecunda 422, 21 </s></p> <p type="main"> <s>Æquatio centri prima 423, 20: p. </s> <s>101 & <lb></lb><expan abbr="ſeqq.">ſeqque</expan> </s></p> <p type="main"> <s>Æquatio centri ſecunda 424, 15 </s></p> <p type="main"> <s>Variatio prima III, 29 </s></p> <p type="main"> <s>Variatio ſecunda 425, 5 </s></p><pb xlink:href="039/01/519.jpg"></pb> <p type="main"> <s>Motus medii Apogæi </s></p> <p type="main"> <s>Æquatio annua 421, 21 </s></p> <p type="main"> <s>Æquatio ſemeſtris 422, 37 </s></p> <p type="main"> <s>Eccentricitatis </s></p> <p type="main"> <s>Æquatio ſemeſtris 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 ſemeſtris III, 33 </s></p> <p type="main"> <s>Inclinationis Orbitæ ad Eclipticam </s></p> <p type="main"> <s>Æquatio ſemeſtris 420, 22 </s></p> <p type="main"> <s>Lunarium motuum Theoria, qua Methodo ſta<lb></lb>bilienda ſit per Obſervationes 425, 33. </s></p> <p type="main"> <s><emph type="center"></emph>M.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Magnetica vis 22, 13: 271, 25: 368, 29: <lb></lb>431, 23 </s></p> <p type="main"> <s>Maris æſtus a cauſis ſuis derivatur III, 24, 36, 37 </s></p> <p type="main"> <s>Martis </s></p> <p type="main"> <s>diſtantia 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 inſita ſeu vis inertiæ definitur p. </s> <s>2 </s></p> <p type="main"> <s>vis impreſſa definitur p. </s> <s>2 </s></p> <p type="main"> <s>extenſio, durities, impenetrabilitas, mobilitas, <lb></lb>vis inertiæ, gravitas, qua ratione innoteſ<lb></lb>cunt 357, 16: 484, 10 </s></p> <p type="main"> <s>diviſibilitas nondum conſtat 358, 18 </s></p> <p type="main"> <s>Materia ſubtilis <emph type="italics"></emph>Carteſianorum<emph.end type="italics"></emph.end>ad examen quod<lb></lb>dam revocatur 292, 12 </s></p> <p type="main"> <s>Materia vel ſubtiiiſſima Gravitate non deſtitui<lb></lb>tur 368, 1 </s></p> <p type="main"> <s>Mechanicæ, quæ dicuntur, Potentiæ explicantur <lb></lb>& demonſtrantur p. </s> <s>14 & 15: p. </s> <s>23 </s></p> <p type="main"> <s>Mercurii </s></p> <p type="main"> <s>diſtantia 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>Tranſmutandi figuras in alias quæ ſunt ejuſ<lb></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></lb>& 447 </s></p> <p type="main"> <s>Inveniendi Curvarum omnium quadraturas <lb></lb>proxime veras 447, 8 </s></p> <p type="main"> <s>Serierum convergentium adhibetur ad ſolu<lb></lb>tionem Problematum difficiliorum p. </s> <s>127: <lb></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 abſolutus & relativus p. </s> <s>6: 7: 8: 9 2b <lb></lb>invicem ſecerni poſſunt, exemplo demonſtra<lb></lb>tur p. </s> <s>10 </s></p> <p type="main"> <s>Motus Leges p. </s> <s>12 & <expan abbr="ſeqq.">ſeqque</expan> </s></p> <p type="main"> <s>Motuum compoſitio & reſolutio p. </s> <s>14 </s></p> <p type="main"> <s>Motus corporum congredientium poſt reflexio<lb></lb>nem, quali Experimento recte colligi poſſunt, <lb></lb><lb></lb>oſtenditur 19, 21 </s></p> <p type="main"> <s>Motus corporum </s></p> <p type="main"> <s>in Conicis ſectionibus 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 Superſiciebus datis & Funependulorum <lb></lb>motus reciprocus I, Sect. </s> <s>10 </s></p> <p type="main"> <s>Motus corporum viribus centripetis ſe mutuo <lb></lb>petentium I, Sect. </s> <s>11 </s></p> <p type="main"> <s>Motus corporum Minimorum, quæ viribus cen<lb></lb>tripetis ad ſingulas Magni alicujus corporis <lb></lb>partes tendentibus agitantur I, Sect. </s> <s>14 </s></p> <p type="main"> <s>Motus corporum quibus reſiſtitur </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 ejuſ<lb></lb>dem ratione duplicata II, Sect. </s> <s>3 </s></p> <p type="main"> <s>Motus </s></p> <p type="main"> <s>corporum ſola vi inſita progredientium in <lb></lb>Mediis reſiſtentibus II, 1, 2, 5, 6, 7, 11, <lb></lb>12: 302, 1 </s></p> <p type="main"> <s>corporum recta aſcendentium vel deſcenden<lb></lb>tium in Mediis reſiſtentibus, agente vi Gra<lb></lb>vitatis uniformi II, 3, 8, 9, 40, 13, 14 </s></p> <p type="main"> <s>corporum projectorum in Mediis reſiftenti<lb></lb>bus, agente vi Gravitatis unifor mi II, 4, 10 </s></p> <p type="main"> <s>corporum circumgyrantium in Mediis reſi<lb></lb>ſtentibus II, Sect. </s> <s>4 </s></p> <p type="main"> <s>corporum Funependulorum in Mediis reſi<lb></lb>ſtentibus II, Sect. </s> <s>6 </s></p> <p type="main"> <s>Motus & reſiſtentia 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 ſeu Vorticoſus Fluidorum II, <lb></lb>Sect. </s> <s>9 </s></p> <p type="main"> <s>Mundus originem non habet ex cauſis Mecha<lb></lb>nicis p. </s> <s>482, 12. </s></p> <p type="main"> <s><emph type="center"></emph>N.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Navium conſtructioni Propoſitio non inutilis <lb></lb>300, 4. </s></p> <p type="main"> <s><emph type="center"></emph>O.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Opticarum ovalium inventio quam <emph type="italics"></emph>Carteſius<emph.end type="italics"></emph.end>ce<lb></lb>laverat I, 97. <emph type="italics"></emph>Carteſiani<emph.end type="italics"></emph.end>Problematis genera<lb></lb>lior ſolutio I, 98 </s></p> <p type="main"> <s>Orbitarum inventio </s></p> <p type="main"> <s>quas corpora deſcribunt, de loco dato data <lb></lb>cum velocitate, ſecundum datum rectam <lb></lb>egreſſa; ubi vis centripeta eſt reciproce ut <lb></lb>quadratum diſtantiæ & vis illius quantitas <lb></lb>abſoluta cognoſcitur I, 17 </s></p> <p type="main"> <s>quas corpora deſcribunt ubi vires centripetæ <lb></lb>ſunt reciproce ut cubi diſtantiarum 45, 18: <lb></lb>118, 27: 125, 25 </s></p> <p type="main"> <s>quas corpora viribus quibuſcunque centripetis <lb></lb>agitata deſcribunt I, Sect. </s> <s>8. </s></p><pb xlink:href="039/01/520.jpg"></pb> <p type="main"> <s><emph type="center"></emph>P.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Parabola, qua lege vis centripetæ tendentis ad <lb></lb>umbilicum figuræ, deſcribitur a corpore revol<lb></lb>vente I, 13 </s></p> <p type="main"> <s>Pendulorum affectiones explicantur I, 50, 51, <lb></lb>52, 53: II, Sect. </s> <s>6. </s></p> <p type="main"> <s>Pendulotum iſochronorum longitudines diverſæ <lb></lb>in diverſis loeorum Latitudinibus inter ſe <lb></lb>conſeruntur, tum per Obſervatienes, tum per <lb></lb>Theoriam Gravitatis III, 20 </s></p> <p type="main"> <s>Philoſophandi 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></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 Ellipſibus umbilicum habenti<lb></lb>bus in centro Solis III, 13 </s></p> <p type="main"> <s>radiis ad Solem ductis deſcribunt areas tem<lb></lb>poribus proportionales 361, 15: III, 13 </s></p> <p type="main"> <s>temporibus periodicis revolvuntur quæ ſunt <lb></lb>in ſeſquiplicata ratione diſtantiarum a <lb></lb>Sole 360, 17: III, 13 & I, 15 </s></p> <p type="main"> <s>retinentur in Orbibus ſuis a vi Gravitatis <lb></lb>quæ reſpicit Solem, & eſt reciproce ut <lb></lb>quadratum diſtantiæ ab ipſius centro <lb></lb>III, 2, 5 </s></p> <p type="main"> <s>Secundarii </s></p> <p type="main"> <s>moventur in Ellipſibus umbilicum habenti<lb></lb>bus in centro Primariorum III, 22 </s></p> <p type="main"> <s>radiis ad Primarios ſuos ductis deſcribunt <lb></lb>areas temporibus proportionales 359, 3, <lb></lb>22: 361, 27: III, 22 </s></p> <p type="main"> <s>temporibus periodicis revolvuntur quæ ſunt <lb></lb>in ſeſquiplicata ratione diſtantiarum a <lb></lb>Primariis ſuis 359, 3, 22: III, 22 & I, 15 </s></p> <p type="main"> <s>retinentur in Orbibus ſuis a vi Gravitatis <lb></lb>quæ reſpicit Primarios, & eſt reciproce <lb></lb>ut quadratum diſtantiæ ab eorum centris <lb></lb>III, 1, 3, 4, 5 </s></p> <p type="main"> <s>Planetarum </s></p> <p type="main"> <s>diſtantiæ a Sole 361, 1 </s></p> <p type="main"> <s>Orbium Aphelia & Nodi prope quieſcunt <lb></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>denſitas calori quem a Sole recipiunt, ac<lb></lb>commodatur 372, 7 </s></p> <p type="main"> <s>converſiones diurnæ ſunt æquabiles III, 17</s></p> <p type="main"> <s>axes ſunt minores diametris quæ ad eoſdem<lb></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></lb>paribus diſtantiis ab eorum centris, ſunt ut <lb></lb>quantitates materiæ in corporibus III, 6 </s></p> <p type="main"> <s>non pendent ab eorum formis & texturis <lb></lb>367, 35 <lb></lb></s></p> <p type="main"> <s>in diverſis Terræ regionibus inveniuntur & <lb></lb>inter ſe comparantur III, 20 </s></p> <p type="main"> <s>Problematis </s></p> <p type="main"> <s><emph type="italics"></emph>Kepleriani<emph.end type="italics"></emph.end>ſolutio per Trochoidem & per <lb></lb>Approximationes I, 31 </s></p> <p type="main"> <s><emph type="italics"></emph>Veterum<emph.end type="italics"></emph.end>de quatuor lineis, a <emph type="italics"></emph>Pappo<emph.end type="italics"></emph.end>memorati, <lb></lb>a <emph type="italics"></emph>Carteſio<emph.end type="italics"></emph.end>par calculum Analyticum tentati, <lb></lb>compoſitio Geometrica 70, 19 </s></p> <p type="main"> <s>Projectilia, ſepoſita Medii reſiſtentia, moveri in <lb></lb>Parabola colligitur 47, 23: 202, 23: 236, 29 </s></p> <p type="main"> <s>Projectilium motus in Mediis reſiſtentibus II, <lb></lb>4, 10 </s></p> <p type="main"> <s>Pulſuam Aeris, quibes Soni propagantur, deter<lb></lb>minantur intervalla ſeu latitudines II, 50: 344, <lb></lb>18. Hæc intervalla in apertarum Fiſtularum <lb></lb>ſonis æquari duplis longitudinibus Fiſtularum <lb></lb>veroſimile eſt 344, 26 </s></p> <p type="main"> <s><emph type="center"></emph><expan abbr="q.">que</expan><emph.end type="center"></emph.end></s></p> <p type="main"> <s>Quadratura generalis Ovalium dari non poteſt <lb></lb>per finitos terminos I, Lem, 28. p. </s> <s>98 </s></p> <p type="main"> <s>Qualitates corporum qua ratione innoteſcunt & <lb></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"></emph>R.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Reſiſtentiæ 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 cujuſcunque generis 302, 32 </s></p> <p type="main"> <s>Reſiſtentiarum Theoria confirmatur </s></p> <p type="main"> <s>per Experimenta Pendulorum II, 30, 31, Sch. </s> <s><lb></lb>Gen. </s> <s>p. </s> <s>284 </s></p> <p type="main"> <s>per Experimenta corporum cadentium II, 40, <lb></lb>Sch. </s> <s>p. </s> <s>319 </s></p> <p type="main"> <s>Reſiſtentia Mediorum </s></p> <p type="main"> <s>eſt ut eorundem denſitas, cæteris paribu, <lb></lb>290, 29: 291, 35: II, 33, 35, 38: 327, 14 </s></p> <p type="main"> <s>eſt in duplicata ratione velocitatis corporum <lb></lb>quibus reſiſtitur, cæteris paribus 219, 24: <lb></lb>284, 33; II, 33, 35, 38: 324, 23 </s></p> <p type="main"> <s>eſt in duplicata ratione diametri corporum <lb></lb>Sphærieorum quibus reſiſtitur, cæteris pa<lb></lb>ribus 288, 4: 289, 11: II, 33, 35, 38: <lb></lb>Sch. </s> <s>p. </s> <s>319 </s></p> <p type="main"> <s>non minuitur ab actione Fluidi in partes po<lb></lb>ſticas corporis moti 312, 23 </s></p> <p type="main"> <s>Reſiſtentia Fluidorum duplex eſt; oriturque vel <lb></lb>ab Inertia materiæ fluidæ, vel ab Elaſticitate, <lb></lb>Tenacitate & Frictione partium ejus 318, 1. <lb></lb>Reſiſtentia quæ ſentitur in Fluidis fere tota <lb></lb>eſt prioris generis 326, 32, & minui non po<lb></lb>teſt per ſubtilitatem partium Fluidi, manente <lb></lb>denſitate 328, 7 </s></p> <p type="main"> <s>Reſiſtentiæ Globi ad reſiſtentiam Cylindri pro<lb></lb>portio, in Mediis non continuis II, 34 </s></p><pb xlink:href="039/01/521.jpg"></pb> <p type="main"> <s>Reſiſtentia quam patitur a Fluido ſruſtum Co<lb></lb>nicum, qua ratione fiat minima 299, 30 </s></p> <p type="main"> <s>Reſiſtentiæ minimæ Solidum 300, 15. </s></p> <p type="main"> <s><emph type="center"></emph>S.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Satellitis</s></p> <p type="main"> <s>Jovialis extimi elongatio maxima heliocentrica <lb></lb>a centro Jovis 370, 35 </s></p> <p type="main"> <s><emph type="italics"></emph>Hugeniani<emph.end type="italics"></emph.end>elongatio maxima heliocentrica a <lb></lb>centro Saturni 371, 5 </s></p> <p type="main"> <s>Satellitum </s></p> <p type="main"> <s>Jovialium tempora periodica & diſtantiæ a <lb></lb>centro Jovis 359, 12 </s></p> <p type="main"> <s>Saturniorum tempora periodica & diſtantiæ a <lb></lb>centro Saturni 360, 1 </s></p> <p type="main"> <s>Jorialium & Saturniorum inæquales motus <lb></lb>a motibus Lanæ derivari poſſe oſſenditur <lb></lb>III, 23 </s></p> <p type="main"> <s>Saturni </s></p> <p type="main"> <s>diſtantia a Sole 361, 1 </s></p> <p type="main"> <s>ſemidiameter apparens 371, 9 </s></p> <p type="main"> <s>ſemidiameter vera 371, 14 </s></p> <p type="main"> <s>vis attractiva quanta ſit 370, 33 </s></p> <p type="main"> <s>pondus corporum in ejus ſuperficie 371, 19 </s></p> <p type="main"> <s>denſitas 371, 37 </s></p> <p type="main"> <s>quantitas materiæ 371, 27 </s></p> <p type="main"> <s>perturbatio a Jove quanta ſit 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></lb>dentis ad punctum quodcunQ.E.D.tum, deſcri<lb></lb>buntur a corporibus revolventibus 58, 20 </s></p> <p type="main"> <s>Sectionum Conicarum deſcriptio 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></lb>tur Centra vel Aſymptoti 87, 9 </s></p> <p type="main"> <s>Seſquiplicata ratio definitur 31, 40 </s></p> <p type="main"> <s>Sol </s></p> <p type="main"> <s>circum Planetarum omnium commune gravi<lb></lb>tatis centrum movetur III, 12 </s></p> <p type="main"> <s>ſemidiameter ejus mediocris apparens 371, 12 </s></p> <p type="main"> <s>ſemidiameter vera 371, 14 </s></p> <p type="main"> <s>parallaxis ejus horizontalis 370, 33 </s></p> <p type="main"> <s>parallaxis menſtrua 376, 4 </s></p> <p type="main"> <s>vis ejus attractiva quanta ſit 370, 33 </s></p> <p type="main"> <s>pondus corporum in ejus ſuperficie 371, 19 </s></p> <p type="main"> <s>denſitas 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></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></lb>fit per agitationem Aeris 343, 1 </s></p> <p type="main"> <s>velocitas computo colligitur 343. 8, paulu<lb></lb>lum major eſſe debet Æſtivo quam Hyber<lb></lb>no tempore, per Thecriam 344, 11 </s></p> <p type="main"> <s>ceſſatio fit ſtatim ubi ceſſat motus corporis <lb></lb>ſonori 344, 29 <lb></lb></s></p> <p type="main"> <s>augmentatio per tubos ſtenterophonicos <lb></lb>344, 32 </s></p> <p type="main"> <s>Spatium </s></p> <p type="main"> <s>abſolutum & relativum p. </s> <s>6, 7 </s></p> <p type="main"> <s>non eſt æqualiter plenum 368, 16 </s></p> <p type="main"> <s>Sphæroidis attractio, cujus particularum vires <lb></lb>ſunt reciproce ut quadrata diſtantiarum <lb></lb>198, 21 </s></p> <p type="main"> <s>Spiralis quæ ſecat radios ſuos omnes in angulo <lb></lb>dato, qua lege vis centripetæ tendenti ad <lb></lb>centrum Spiralis deſcribi poteſt a corpore <lb></lb>revolvente, oſtenditur I, 9: II, 15, 16 </s></p> <p type="main"> <s>Spiritum Q.E.D.m corpora pervadentem & in <lb></lb>corporibus latentem, ad plurima naturæ phæ<lb></lb>nomena ſolvenda, requiri ſuggeritur 484, 17 </s></p> <p type="main"> <s>Stellarum fixarum </s></p> <p type="main"> <s>quies demonſtratur 376, 18 </s></p> <p type="main"> <s>radiatio & ſcintillatio quibus cauſis referendæ <lb></lb>ſint 467, 38 </s></p> <p type="main"> <s>Stellæ Novæ unde oriri poſſint 481, 5 </s></p> <p type="main"> <s>Subſtantiæ rerum omnium occultæ ſunt 483, 22 </s></p> <p type="main"> <s><emph type="center"></emph>T.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Tempus abſolutum & relativum p. </s> <s>5, 7 </s></p> <p type="main"> <s>Temporis Æquatio Aſtronomica per Horolo<lb></lb>gium oſcillatorium & Eclipſes Satellitum Jo<lb></lb>vis comprobatur 7, 15 </s></p> <p type="main"> <s>Tempora periodica corporum revolventium in <lb></lb>Ellipſibus, ubi vires centripetæ ad umbilicum <lb></lb>tendunt I, 15 </s></p> <p type="main"> <s>Terræ </s></p> <p type="main"> <s>dimenſio per <emph type="italics"></emph>Picartum<emph.end type="italics"></emph.end>378, 11, per <emph type="italics"></emph>Caſſinum<emph.end type="italics"></emph.end><lb></lb>378, 21, per <emph type="italics"></emph>Norwoodum<emph.end type="italics"></emph.end>378, 28 </s></p> <p type="main"> <s>figura invenitur, & proportio diametrorum, <lb></lb>& menſura graduum in Meridiano III, <lb></lb>19, 20 </s></p> <p type="main"> <s>altitudinis ad Æquatorem ſupra altitudinem ad <lb></lb>Polos quantus ſit exceſſus 381, 7: 387, 1 </s></p> <p type="main"> <s>ſemidiameter maxima, minima & mediocris <lb></lb>387, 10 </s></p> <p type="main"> <s>globus denſior eſt quam ſi totus ex Aqua con<lb></lb>ſtaret 372, 31 </s></p> <p type="main"> <s>globus denſior eſt ad centrum quam ad ſuper<lb></lb>ficiem 386, 1 </s></p> <p type="main"> <s>molem indies augeri veroſimile eſt 473, 18 <lb></lb>481, 13 </s></p> <p type="main"> <s>axis nutatio III, 21 </s></p> <p type="main"> <s>motus annuus in Orbe magno demonſtratur <lb></lb>III, 12, 13: 478, 26 </s></p> <p type="main"> <s>Eccentricitas quanta ſit 421, 15 </s></p> <p type="main"> <s>Aphelii motus quantus ſit 376, 33. </s></p> <p type="main"> <s><emph type="center"></emph>V.<emph.end type="center"></emph.end></s></p> <p type="main"> <s>Vacuum datur, vel ſpatia omnia (ſi dicantur <lb></lb>eſſe plena) non ſunt æqualiter plena 328, 18: <lb></lb>368, 25 </s></p><pb xlink:href="039/01/522.jpg"></pb> <p type="main"> <s>Velocitas maxima quam Globus, in Medio re<lb></lb>ſiſtente cadendo, poteſt acquirere II, 38, <lb></lb>Cor. </s> <s>2 </s></p> <p type="main"> <s>Velocitates corporum in Sectionibus conicis mo<lb></lb>torum, ubi vires centripetæ ad umbilicum <lb></lb>tendunt I, 16 </s></p> <p type="main"> <s>Veneris </s></p> <p type="main"> <s>diſtantia 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 compoſitio & reſolutio p. </s> <s>14 </s></p> <p type="main"> <s>Vires attractivæ corporum </s></p> <p type="main"> <s>ſphærieorum ex particulis quacunque lege <lb></lb>trahentibus compoſitorum, expenduntur <lb></lb>I, Sect. </s> <s>12 </s></p> <p type="main"> <s>non ſphærieorum ex particulis quacunque <lb></lb>lege trahentibus compoſitorum, expendun<lb></lb>tur I, Sect. </s> <s>13 </s></p> <p type="main"> <s>Vis centrifuga corporum in Æquatore Terræ <lb></lb>quanta ſit 379. 22 </s></p> <p type="main"> <s>Vis centripeta deſinitur p. </s> <s>2 </s></p> <p type="main"> <s>quantitas ejus abſoluta 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></lb>ratione colligenda ſit, oſtenditur 40, 1 </s></p> <p type="main"> <s>Virium centripetarum inventio, ubi corpus in <lb></lb>ſpatio non reſiſtente, circa centrum immo<lb></lb>bile, in Orbe quocunque revolvitur I, 6: I, <lb></lb>Sect. </s> <s>2 & 3 </s></p> <p type="main"> <s>Viribus centripetis datis ad quodcunque pun<lb></lb>ctum tendentibus, quibus Figura quævis a <lb></lb><lb></lb>corpore revolvente deſcribi poteſt; dantur <lb></lb>vires centripetæ ad aliud quodvis punctum <lb></lb>tendentes, quibus eadem Figura eodem tem<lb></lb>pore periodico deſcribi poteſt 44, 3 </s></p> <p type="main"> <s>Viribus centripetis datis quibus Figura qurvis <lb></lb>deſcribitur a corpore revolvente; dantur vires <lb></lb>quibus Figura nova deſcribi poteſt, ſi Ordi<lb></lb>natæ augeantur vel minuantur in ratione qua<lb></lb>cunQ.E.D.ta, vel angulus Ordinationis utcun<lb></lb>que mutetur, manente tempore periodico <lb></lb>47, 28 </s></p> <p type="main"> <s>Viribus centripetis in duplicata ratione diſtantia<lb></lb>rum decreſcentibus, quænam Figura deſcribi <lb></lb>poſſunt, oſtenditur 53, 1: 150, 8 </s></p> <p type="main"> <s>Vicentripeta </s></p> <p type="main"> <s>quæ ſit reciproce ut cubus ordinatim applica<lb></lb>tæ tendentis ad centrum virium maxime <lb></lb>longinquum, corpus movebitur in data <lb></lb>quavis coni ſectione 45, 1 </s></p> <p type="main"> <s>quæ ſit ut cubus ordinatim applicatæ tenden<lb></lb>tis ad centrum virium maxime longinquum, <lb></lb>corpus movebitur in Hyperbola 202, 26 </s></p> <p type="main"> <s>Umbra Terreſtris in Eclipſibus Lunæ augenda eſt, <lb></lb>propter Atmoſphæræ refractionem 425, 27 </s></p> <p type="main"> <s>Umbræ Terreſtris dian etri non ſunt æquales; <lb></lb>quanta ſit differentia oſtenditur 387, 8 </s></p> <p type="main"> <s>Undarum in aquæ ſtagtantis ſuperficie propa<lb></lb>gatarum velocitas invenitur II, 46 </s></p> <p type="main"> <s>Vorticum natura & conſtitutio ad examen re<lb></lb>vocatur II, Sect. </s> <s>9: 481, 21 </s></p> <p type="main"> <s><emph type="italics"></emph>Ut.<emph.end type="italics"></emph.end>Hujus voculæ fignificatio Mathematica de<lb></lb>fiuitur 30, 19. </s></p> <p type="main"> <s><emph type="center"></emph><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s></p> </chap> <pb xlink:href="039/01/523.jpg"></pb> </body> <back></back> </text></archimedes>