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It appears here for continuity's sake. --> <!ELEMENT info (author, title, date, place, editor, publisher, translator, lang, chunk, locator?) > <!-- how many of these should be required? --> <!-- what about bringing in line with dublin core? --> <!ATTLIST info id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT lang (#PCDATA) > <!ATTLIST lang id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT locator (#PCDATA) > <!ATTLIST locator id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT lb EMPTY > <!ATTLIST lb ed CDATA #IMPLIED id ID #IMPLIED n CDATA #IMPLIED> <!-- <lb> occurs at <s> level and at <p> level --> <!-- <lb> at end of <s> must be placed after </s> --> <!-- unrecognized symbols will appear inline acc. to special conventions. --> <!-- (as in DE specs) --> <!-- emph and and emph.end are elements that are quer to the xml structure --> <!ELEMENT p (s | pb|lb|emph|emph.end|gap )+ > <!ATTLIST p id ID #IMPLIED n CDATA #IMPLIED type (main|marked|foot|margin|list|head|caption|table) "main" > <!-- captions go in paragr's immediately after figure, with arrow.to.target --> <!-- pb can occur within p but NOT at the start or end --> <!ELEMENT pb EMPTY > <!ATTLIST pb ed CDATA #IMPLIED id ID #IMPLIED n CDATA #IMPLIED pagenum CDATA #IMPLIED> <!-- the "ed" attribute gives the edition in which the page break occurs. If no --> <!-- edition is given, it is assumed to be that given in info --> <!-- pagenum gives the page number actually printed on the page. --> <!-- n gives the page number numbered consecutively from page 1 of text. --> <!-- pb can occur inside <s> etc. and inside <p> and inside <chap> --> <!-- <pb> at end of <p> must be placed after </p> --> <!ELEMENT place (#PCDATA) > <!ATTLIST place id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT publisher (#PCDATA)* > <!ATTLIST publisher id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT s (#PCDATA| foreign | expan | foot.target|margin.target|arrow.to.target|pb|lb|emph|emph.end|gap)* > <!ATTLIST s id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT section ( p | pb | figure)+ > <!ATTLIST section id ID #IMPLIED n CDATA #IMPLIED type CDATA #IMPLIED> <!ELEMENT table ( row+ ) > <!ATTLIST table id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT row ( cell+ ) > <!ATTLIST row id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT cell (#PCDATA| foreign | expan | foot.target|margin.target|arrow.to.target|pb|lb|emph|emph.end|gap)* > <!ATTLIST cell id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT text (pb?, front, pb?, body, pb?, back) > <!-- if front and back are going to be optional, then pb can be allowed to --> <!-- occur only within front body back. --> <!ATTLIST text type CDATA #IMPLIED id ID #IMPLIED n CDATA #IMPLIED> <!ELEMENT title (#PCDATA) > <!ATTLIST title id ID #IMPLIED n CDATA #IMPLIED type CDATA #IMPLIED > <!ELEMENT translator (#PCDATA)* > <!ATTLIST translator id ID #IMPLIED n CDATA #IMPLIED > <!ENTITY shy "[-]"> <!ENTITY AElig "[AElig]"> <!ENTITY Prime "[Prime]"> <!ENTITY acirc "[acirc]"> <!ENTITY aelig "[aelig]"> <!ENTITY agrave "[agrave]"> <!ENTITY amp "[amp]"> <!ENTITY atilde "[atilde]"> <!ENTITY deg "[deg]"> <!ENTITY egrave "[egrave]"> <!ENTITY etilde "[etilde]"> <!ENTITY euml "[euml]"> <!ENTITY horbar "[horbar]"> <!ENTITY igrave "[igrave]"> <!ENTITY lbrk "[lbrk]"> <!ENTITY longs "[longs]"> <!ENTITY nbsp "[nbsp]"> <!ENTITY oelig "[oelig]"> <!ENTITY ograve "[ograve]"> <!ENTITY otilde "[otilde]"> <!ENTITY prime "[prime]"> <!ENTITY radic "[radic]"> <!ENTITY sdot "[sdot]"> <!ENTITY shy "[shy]"> <!ENTITY tprime "[tprime]"> <!ENTITY ugrave "[ugrave]"> <!ENTITY utilde "[utilde]"> <!ENTITY uuml "[uuml]"> <!ENTITY verbar "[verbar]"> ]> <archimedes> <info> <author>Isaac Newton</author> <title>Philosophiae Naturalis Principia Mathmatica</title> <date>1713</date> <place>Cambridge</place> <editor></editor> <publisher></publisher> <translator></translator> <lang>la</lang> <chunk unit="page*">page</chunk> <locator>000000031.xml</locator> </info> <text> <front> </front> <body> <chap> <pb/> <p type="main"> <s><emph type="center"/>PHILOSOPHIÆ <lb/>NATURALIS <lb/>PRINCIPIA <lb/>MATHEMATICA.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>AUCTORE <lb/>ISAACO NEWTONO, <lb/>EQUITE A RATO.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>EDITIO SECUNDA AUCTIOR ET EMBNDATIOR.<emph.end type="center"/></s> </p> <figure></figure> <p type="main"> <s><emph type="center"/>CANTABRIGIÆ, MDCCXIII.<emph.end type="center"/></s> </p> <pb/> <p type="main"> <s><emph type="center"/>ILLUSTRISSIMÆ <lb/>SOCIETATI REGALI, <lb/>A <lb/>SERENISSIMO REGE <lb/>CAROLO II <lb/>AD PHILOSOPHIAM PROMOVENDAM <lb/>FUNDATÆ, <lb/>ET <lb/>AUSPICIIS <lb/>AUGUSTISSIMÆ REGINÆ <lb/>ANNÆ <lb/>FLORENTI,<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>TRACTATUM HUNC D.D.D.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>JS. NEWTONUS.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="center"/>IN <lb/>VIRI PRÆSTANTISSIMI <lb/>ISAACI NEWTONI <lb/>OPUS HOCCE <lb/>MATHEMATICO PHYSICUM<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Sæculi Genti&longs;que no&longs;træ Decus egregium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>EN tibi norma Poli, & divæ libramina Molis, <lb/>Computus en Jovis; & quas, dum primordia rerum. </s> <s><lb/>Conderet, omnipotens &longs;ibi Leges ip&longs;e Creator <lb/>Dixerit, atque operum quæ fundamenta locarit. </s> <s><lb/>Intima panduntur victi penetralia Cæli, <lb/>Nec latet, extremos quæ Vis circumrotet Orbes. </s> <s><lb/>Sol &longs;olio re&longs;idens ad &longs;e jubet omnia prono <lb/>Tendere de&longs;cen&longs;u, nec recto tramite currus <lb/>Sidereos patitur va&longs;tum per inane moveri; <lb/>Sed rapit immotis, &longs;e centro, &longs;ingula gyris. </s> <s><lb/>Hinc patet, horrificis qua &longs;it via flexa Cometis: <lb/>Di&longs;cimus hinc tandem, qua cau&longs;a argentea Phœbe <lb/>Pa&longs;&longs;ibus haud æquis eat, & cur &longs;ubdita nulli <lb/>Hactenus A&longs;tronomo numerorum fræna recu&longs;et: <lb/>Cur remeent Nodi, curque Auges progrediantur. </s> <s><lb/>Di&longs;cimus, & quantis refluum vaga Cynthia Pontum <lb/>Viribus impellat; fe&longs;&longs;is dum fluctibus ulvam <lb/>De&longs;erit, ac nautis &longs;u&longs;pectas nudat arenas; <lb/>Alterni&longs;ve ruens &longs;pumantia littora pul&longs;at. <pb/>Quæ toties animos veterum tor&longs;ere Sophorum, <lb/>Quæque Scholas hodie rauco certamine vexant, <lb/>Obvia con&longs;picimus; nubem pellente Mathe&longs;i: <lb/>Quæ &longs;uperas penetrare domos, atque ardua Cæli, <lb/>NEWTONI au&longs;picils, jam dat contingere Templa. </s> <s><lb/>Surgite Mortales, terrenas mittite curas; <lb/>Atque hinc cæligenæ vites cogno&longs;cite Mentis, <lb/>A pecudum vita longe longeque remotæ. </s> <s><lb/>Qui &longs;criptis primus Tabulis compe&longs;cere Cædes, <lb/>Furta & Adulteria, & perjuræ crimina Fraudis; <lb/>Quive vagis populis circumdare mœnibus Urbes <lb/>Auctor erat; Cereri&longs;ve beavit munere gentes; <lb/>Vel qui curarum lenimen pre&longs;&longs;it ab Uva; <lb/>Vel qui Niliaca mon&longs;travit arundine pictos <lb/>Con&longs;ociare &longs;onos, oculi&longs;que exponere Voces; <lb/>Humanam &longs;ortem minus extulit; utpote pauca <lb/>In commune ferens mi&longs;eræ &longs;olatia vitæ. </s> <s><lb/>Jam vero Superis convivæ admittimur, alti <lb/>Jura poli tractare licet, jamque abdita diæ <lb/>Clau&longs;tra patent Naturæ, & rerum immobilis ordo; <lb/>Et quæ præteritis latuere incognita &longs;æclis. </s> <s><lb/>Talia mon&longs;trantem ju&longs;tis celebrate Camænis, <lb/>Vos qui cæle&longs;ti gaudetis nectare ve&longs;ci, <lb/>NEWTONUM clau&longs;i re&longs;erantem &longs;crinia Veri, <lb/>NEWTONUM Mu&longs;is carum, cui pectore puro <lb/>Phœbus ade&longs;t, totoque ince&longs;&longs;it Numine mentem: <lb/>Nec fas e&longs;t propius Mortali attingere Divos. <lb/><emph type="italics"/>EDM. HALLET.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="center"/>AUCTORIS <lb/>PRÆFATIO <lb/>AD <lb/>LECTOREM.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>CUM Veteres<emph.end type="italics"/> Mechanicam (<emph type="italics"/>uti Auctor e&longs;t<emph.end type="italics"/> Pappus) <emph type="italics"/>in rerum <lb/>Naturalium inve&longs;tigatione maximi fecerint; & Recentiores, <lb/>mi&longs;&longs;is formis &longs;ub&longs;tautialibus & qualitatibus occultis, Phænomena <lb/>Naturæ ad leges Mathematicas revocare aggre&longs;&longs;i fint: Vi&longs;um e&longs;t <lb/>in hoc Tractatu<emph.end type="italics"/> Mathe&longs;in <emph type="italics"/>excolere, quatenus ea ad<emph.end type="italics"/> Philo&longs;ophiam <lb/><emph type="italics"/>&longs;pectat.<emph.end type="italics"/> Mechanicam <emph type="italics"/>vero duplicem Veteres con&longs;tituerunt<emph.end type="italics"/>: Ra­<lb/>tionalem <emph type="italics"/>quæ per Demon&longs;trationes accurate procedit, &<emph.end type="italics"/> Practi­<lb/>cam. <emph type="italics"/>Ad <gap/>acticam &longs;pectant Artes omnes Manuales, a quibus <lb/>utique<emph.end type="italics"/> Mechanica <emph type="italics"/>nomen mutuata e&longs;t. </s> <s>Cum autem Artifices pa­<lb/>rum accurate operari &longs;oleant, fit ut<emph.end type="italics"/> Mechanica <emph type="italics"/>omnis a<emph.end type="italics"/> Geome­<lb/>tria <emph type="italics"/>ita di&longs;tinguatur, ut quicquid accuratum &longs;it ad<emph.end type="italics"/> Geometriam <lb/><emph type="italics"/>referatur, quicquid minus accuratum ad<emph.end type="italics"/> Mechanicam. <emph type="italics"/>Attamen <lb/>errores non &longs;unt Artis &longs;ed Artificum. </s> <s>Qui minus accurate ope­<lb/>ratur, imperfectior e&longs;t Mechanicus, & &longs;i quis accurati&longs;&longs;ime ope­<lb/>rari po&longs;&longs;et, hic foret Mechanicus omnium perfecti&longs;&longs;imus. </s> <s>Nam & <lb/>Linearum rectarum & Circulorum de&longs;criptiones in quibus<emph.end type="italics"/> Geo­<lb/>metria <emph type="italics"/>fundatur, ad<emph.end type="italics"/> Mechanicam <emph type="italics"/>pertinent. </s> <s>Has lineas de&longs;cri­<lb/>bere<emph.end type="italics"/> Geometria <emph type="italics"/>non docet &longs;ed po&longs;tulat. </s> <s>Po&longs;tulat enim ut Tyro <lb/>ea&longs;dem accurate de&longs;cribere prius didicerit quam linen atting at<emph.end type="italics"/><lb/>Geometriæ; <emph type="italics"/>deia, quomodo per has operationes Problemata &longs;ol­<lb/>uantur, docet. </s> <s>Rectas & Circulos de&longs;cribere Problemata &longs;unt,<emph.end type="italics"/> <pb/><emph type="italics"/>&longs;ed non Geometrica. </s> <s>Ex<emph.end type="italics"/> Mechanica <emph type="italics"/>po&longs;lulatur horum &longs;olutio, in<emph.end type="italics"/><lb/>Geometria <emph type="italics"/>docetur &longs;olutorum u&longs;us. </s> <s>Ac gloriatur<emph.end type="italics"/> Geometria <lb/><emph type="italics"/>quod tam paucis principiis aliunde petitis tam multa præ&longs;tet. </s> <s>Fun­<lb/>datur igitur<emph.end type="italics"/> Geometria <emph type="italics"/>in praxi Mechanica, & nihil aliud e&longs;t <lb/>quam<emph.end type="italics"/> Mechanicæ univer&longs;alis <emph type="italics"/>pars illa quæ artem men&longs;urandi ac­<lb/>curate proponit ac demon&longs;trat. </s> <s>Cum autem artes Manuales in <lb/>corporibus movendis præcipue ver&longs;entur, fit ut<emph.end type="italics"/> Geometria <emph type="italics"/>ad mag­<lb/>nitudinem,<emph.end type="italics"/> Mechanica <emph type="italics"/>ad motum vulgo referatur. </s> <s>Quo &longs;en&longs;u<emph.end type="italics"/> Me­<lb/>chanica rationalis <emph type="italics"/>erit Scientia Motuum qui ex viribus quibu&longs;­<lb/>cunque re&longs;ultant, & Virium quæ ad motus quo&longs;cunque requirun­<lb/>tur, accurate propo&longs;ita ac demon&longs;trata. </s> <s>Pars hæc<emph.end type="italics"/> Mechanicæ <emph type="italics"/>a <lb/>Veteribus in<emph.end type="italics"/> Potentiis quinque <emph type="italics"/>ad artes manuales &longs;pectantibus <lb/>exculta fuit, qui Gravitatem (cum potentia manualis non &longs;it) vix <lb/>aliter quam in ponderibus per potentias illas movendis con&longs;iderarunt. </s> <s><lb/>Nos autem non Artibus &longs;ed Philo&longs;ophiæ con&longs;ulentes, deque poten­<lb/>tiis non manualibus &longs;ed naturalibus &longs;cribentes, ea maxime tracta­<lb/>mus quæ ad Gravitatem, Levitatem, vim Ela&longs;ticam, re&longs;i&longs;tentiam <lb/>Fluidorum & eju&longs;modi vires &longs;eu attractivas &longs;eu impul&longs;ivas &longs;pe­<lb/>ctant: Et ea propter, hæc no&longs;tra tanquam Philo&longs;ophiæ principia <lb/>Mathematica proponimus. </s> <s>Omnis enim Philo&longs;ophiæ difficultas in <lb/>eo ver&longs;ari videtur, ut a Phænomenis motuum inve&longs;tigemus vires <lb/>Naturæ, deinde ab his viribus demon&longs;tremus phænomena <gap/>liquæ. </s> <s><lb/>Et huc &longs;pectant Propo&longs;itiones generales quas Libro primo & &longs;ecundo <lb/>pertractavimus. </s> <s>In Libro autem tertio Exemplum hujus rei propo­<lb/>&longs;uimus per explicationem Sy&longs;tematis mundaui. </s> <s>Ibi enim, ex phæ­<lb/>nomenis cæleflibus, per Propo&longs;itiones in Libris prioribus Mathe­<lb/>matice aemon&longs;tratas, derivantur vires Gravitatis quibus corpora <lb/>ad Solem & Planetas &longs;ingulos tendunt. </s> <s>Deinde ex his viribus <lb/>per Propo&longs;itiones etiam Mathematicas, deducuntur motus Planeta­<lb/>rum, Cometarum, Lunæ & Maris. </s> <s>Utinam cætera Naturæ phæ­<lb/>nomena ex principiis Mechanicis eodem argumentandi genere deri­<lb/>vare liceret. </s> <s>Nam multa me movent ut nonnihil &longs;u&longs;picer e<gap/> om-<emph.end type="italics"/> <pb/><emph type="italics"/>nia ex viribus quibu&longs;dam pendere po&longs;&longs;e, quibus corporum particulæ <lb/>per cau&longs;as nondum cognitas vel in &longs;e mutuo impelluntur & &longs;e­<lb/>cundum figuras regulares cohærent, vel ab invicem fugantur & <lb/>recedunt: quibus viribus ignotis, Philo&longs;ophi hactenus Naturam fru­<lb/>&longs;tra tentarunt. </s> <s>Spero autem quod vel huic Philo&longs;ophandi modo, <lb/>vel veriori alicui, Principia hic po&longs;ita lucem aliquam præbebunt.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>In his edendis, Vir acuti&longs;&longs;imus & in omni literarum genere <lb/>eruditi&longs;&longs;imus<emph.end type="italics"/> Edmundus Halleius <emph type="italics"/>operam navavit, nec &longs;olum <lb/>Typothetarum Sphalmata correxit & Schemata incidi curavit, &longs;ed <lb/>etiam Auctor fuit ut horum editionem aggrederer. </s> <s>Quippe cum <lb/>demon&longs;tratam a me Figuram Orbium cæle&longs;tium impetraverat, ro­<lb/>gare non de&longs;titit ut eandem cum<emph.end type="italics"/> Societate Regali <emph type="italics"/>communicarem, <lb/>Quæ deinde hortatibus & benignis &longs;uis au&longs;piciis effecit ut de ea­<lb/>dem in lucem emittenda cogitare inciperem. </s> <s>At po&longs;tquam Mo­<lb/>tuum Lunarium inæqualitates aggre&longs;&longs;us e&longs;&longs;em, deinde etiam ælia <lb/>tentare cæpi&longs;&longs;em quæ ad leges & men&longs;uras Gravitatis & aliarum <lb/>virium, & Figuras a corporibus &longs;ecundum datas qua&longs;cunque leges <lb/>attractis de&longs;cribendas, ad motus corporum plurium inter &longs;e, ad <lb/>motus corporum in Mediis re&longs;i&longs;tentibus, ad vires, den&longs;itates & <lb/>motus Mediorum, ad Orbes Cometarum & &longs;imilia &longs;pectant, edi­<lb/>tionem in aliud tempus differendam e&longs;&longs;e putavi, ut cætera rima­<lb/>rer & una in publicum darem. </s> <s>Quæ ad motus Lunares &longs;pectant, <lb/>(imperfecta cum &longs;int,) in Corollariis Propo&longs;itionis<emph.end type="italics"/> LXVI. <emph type="italics"/>&longs;imul <lb/>complexus &longs;um, ne &longs;ingula methodo prolixiore quam pro rei digni­<lb/>tate proponere, & &longs;igillatim demon&longs;trare tenerer, & &longs;eriem reli­<lb/>quarum Propo&longs;itionum interrumpere. </s> <s>Nonnulla &longs;ero inventa lo­<lb/>cis minus idoneis in&longs;erere malui, quam numerum Propo&longs;itionum <lb/>& citationes mutare. </s> <s>Ut omnia candide legantur, & defectus, <lb/>in materia tam difficili non tam reprehendantur, quam novis Le­<lb/>ctorum conatibus inve&longs;tigentur, & benigne &longs;uppleantur, enixe rogo.<emph.end type="italics"/></s> </p> <p type="main"> <s>Dabam <emph type="italics"/>Cantabrigiæ,<emph.end type="italics"/> e Collegio <lb/><emph type="italics"/>S. Trinitatis,<emph.end type="italics"/> Maii 8. 1686. </s> </p> <p type="main"> <s><emph type="italics"/>IS. NEWTON.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="italics"/>IN hac Secunda Principiorum Editione, multa &longs;par&longs;im emen­<lb/>dantur & nonnulla adjiciuntur. </s> <s>In Libri primi Sectione<emph.end type="italics"/> II, <lb/><emph type="italics"/>Inventio virium quibus corpora in Orbibus datis revolvi po&longs;&longs;int, <lb/>facilior redditur & amplior. </s> <s>In Libri &longs;ecundi Sectione<emph.end type="italics"/> VII, <lb/><emph type="italics"/>Theoria re&longs;i&longs;tentiæ Fluidorum accuratius inve&longs;tigatur & novis <lb/>Experimentis confirmatur. </s> <s>In Libro tertio Theoria Lunæ & Præ­<lb/>ce&longs;&longs;io Æquinoctiorum ex Principiis &longs;uis plenius deducuntur, & <lb/>Theoria Cometarum pluribus & accuratius computatis Orbium <lb/>exemplis confirmatur.<emph.end type="italics"/></s> </p> <p type="main"> <s>Dabam <emph type="italics"/>Londini,<emph.end type="italics"/><lb/>Mar. </s> <s>28. 1713. </s> </p> <p type="main"> <s><emph type="italics"/>IS. NEWTON.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="center"/>EDITORIS <lb/>PRÆFATIO.<emph.end type="center"/></s> </p> <p type="main"> <s>NEWTONIANÆ Philo&longs;ophiæ novam tibi, Lector benevole, <lb/>diuque de&longs;ideratam Editionem, plurimum nunc emenda­<lb/>tam atque auctiorem exhibemus. </s> <s>Quæ poti&longs;&longs;imum conti­<lb/>neantur in hoc Opere celeberrimo, intelligere potes ex Indicibus <lb/>adjectis: quæ vel addantur vel immutentur, ip&longs;a te fere docebit <lb/>Auctoris Præfatio. </s> <s>Reliquum e&longs;t, ut adjiciantur nonnulla de Me­<lb/>thodo hujus Philo&longs;ophiæ. </s> </p> <p type="main"> <s>Qui Phy&longs;icam tractandam &longs;u&longs;ceperunt, ad tres fere cla&longs;&longs;es re­<lb/>vocari po&longs;&longs;unt. </s> <s>Extiterunt enim, qui &longs;ingulis rerum &longs;peciebus Quali­<lb/>tates &longs;pecificas & occultas tribuerint; ex quibus deinde corporum <lb/>&longs;ingulorum operationes, ignota quadam ratione, pendere volue­<lb/>runt. </s> <s>In hoc po&longs;ita e&longs;t &longs;umma doctrinæ Schola&longs;ticæ, ab <emph type="italics"/>Ari&longs;totele<emph.end type="italics"/><lb/>& Peripateticis derivatæ: Affirmant utique &longs;ingulos Effectus ex <lb/>corporum &longs;ingularibus Naturis oriri; at unde &longs;int illæ Naturæ <lb/>non docent; nihil itaque docent. </s> <s>Cumque toti &longs;int in rerum no­<lb/>minibus, non in ip&longs;is rebus; Sermonem quendam Philo&longs;ophicum <lb/>cen&longs;endi &longs;unt adinveni&longs;&longs;e, Philo&longs;ophiam tradidi&longs;&longs;e non &longs;unt cen­<lb/>&longs;endi. </s> </p> <p type="main"> <s>Alii ergo melioris diligentiæ laudem con&longs;equi &longs;perarunt, rejecta <lb/>Vocabulorum inutili farragine. </s> <s>Statuerunt itaque Materiam uni­<lb/>ver&longs;am homogeneam e&longs;&longs;e, omnem vero Formarum varietatem, quæ <lb/>in corporibus cernitur, ex particularum componentium &longs;implici&longs;&longs;i­<lb/>mis quibu&longs;dam & intellectu facillimis affectionibus oriri. </s> <s>Et recte <lb/>quidem progre&longs;&longs;io in&longs;tituitur a &longs;implicioribus ad magis compo&longs;ita, <lb/>&longs;i particularum primariis illis affectionibus non alios tribuunt mo­<lb/>dos, quam quos ip&longs;a tribuit Natura. </s> <s>Verum ubi licentiam &longs;ibi <lb/>a&longs;&longs;umunt, ponendi qua&longs;cunque libet ignotas partium figuras & <lb/>magnitudines, incerto&longs;que &longs;itus & motus; quin & fingendi Fluida <lb/>quædam occulta, quæ corporum poros liberrime permeent, omni­<lb/>potente prædita &longs;ubtilitate, motibu&longs;que occultis agitata; jam ad <lb/>&longs;omnia delabuntur, neglecta rerum con&longs;titutione vera: quæ fane <lb/>fru&longs;tra petenda e&longs;t ex fallacibus conjecturis, cum vix etiam per <lb/>certi&longs;&longs;imas Ob&longs;ervationes inve&longs;tigari po&longs;&longs;it. </s> <s>Qui &longs;peculationum <pb/>&longs;uarum fundamentum de&longs;umunt ab Hypothe&longs;ibus, etiam&longs;i deinde <lb/>&longs;ecundum leges Mechanicas accurati&longs;&longs;ime procedant; Fabulam qui­<lb/>dem elegantem forte & venu&longs;tam, Fabulam tamen concinnare di­<lb/>cendi &longs;unt. </s> </p> <p type="main"> <s>Relinquitur adeo tertium genus, qui Philo&longs;ophiam &longs;cilicet Ex­<lb/>perimentalem profitentur. </s> <s>Hi quidem ex &longs;implici&longs;&longs;imis quibus <lb/>po&longs;&longs;unt principiis rerum omnium cau&longs;as derivandas e&longs;&longs;e volunt: <lb/>nihil autem Principii loco a&longs;&longs;umunt, quod nondum ex Phænome­<lb/>nis comprobatum fuerit. </s> <s>Hypothe&longs;es non commini&longs;cuntur, neque <lb/>in Phy&longs;icam recipiunt, ni&longs;i ut Quæ&longs;tiones de quarum veritare di&longs;­<lb/>putetur. </s> <s>Duplici itaque Methodo incedunt, Analytica & Syn­<lb/>thetica. </s> <s>Naturæ vires lege&longs;que virium &longs;impliciores ex &longs;electis <lb/>quibu&longs;dam Phænomenis per Analy&longs;in deducunt, ex quibus deinde <lb/>per Synthe&longs;in reliquorum con&longs;titutionem tradunt. </s> <s>Hæc illa e&longs;t <lb/>Philo&longs;ophandi ratio longe optima, quam præ ceteris merito am­<lb/>plectendam cen&longs;uit Celeberrimus Auctor no&longs;ter. </s> <s>Hanc &longs;olam uti­<lb/>que dignam judicavit, in qua excolenda atque adornanda operam <lb/>&longs;uam collocaret. </s> <s>Hujus igitur illu&longs;tri&longs;&longs;imum dedit Exemplum, <lb/>Mundani nempe Sy&longs;tematis explicationem e Theoria Gravitatis <lb/>felici&longs;&longs;ime deductam. </s> <s>Gravitatis virtutem univer&longs;is corporibus in­<lb/>e&longs;&longs;e, &longs;u&longs;picati &longs;unt vel finxerunt alii: primus Ille & folus ex Ap­<lb/>parentiis demon&longs;trare potuit, & &longs;peculationibus egregiis firmi&longs;&longs;i­<lb/>mum ponere fundamentum. </s> </p> <p type="main"> <s>Scio equidem nonnullos magni etiam nominis Viros, præjudiciis <lb/>quibu&longs;dam plus æquo occupatos, huic novo Principio ægre a&longs;&longs;en­<lb/>tiri potui&longs;&longs;e, & certis incerta identidem prætuli&longs;&longs;e. </s> <s>Horum famam vel­<lb/>licare non e&longs;t animus: Tibi potius, Benevole Lector, illa paucis ex­<lb/>ponere lubet, ex quibus Tute ip&longs;e judicium non iniquum feras. </s> </p> <p type="main"> <s>Igitur ut Argumenti &longs;umatur exordium a &longs;implici&longs;&longs;imis & pro­<lb/>ximis; de&longs;piciamus pauli&longs;per qualis &longs;it in Terre&longs;tribus natura Gra­<lb/>vitatis, ut deinde tutius progrediamur ubi ad corpora Cæle&longs;tia, lon­<lb/>gi&longs;&longs;ime a &longs;edibus no&longs;tris remota, perventum fuerit. </s> <s>Convenit jam <lb/>inter omnes Philo&longs;ophos, corpora univer&longs;a circumterre&longs;tria gra­<lb/>vitare in Terram. </s> <s>Nulla dari corpora vere levia, jamdudum <lb/>confirmavit Experientia multiplex. </s> <s>Quæ dicitur Levitas relativa, <lb/>non e&longs;t vera Levitas, &longs;ed apparens &longs;olummodo; & oritur a præ­<lb/>pollente Gravitate corporum contiguorum. </s> </p> <p type="main"> <s>Porro, ut corpora univer&longs;a gravitant in Terram, ita Terra vici&longs;­<lb/>&longs;im in corpora æqualiter gravitat; Gravitatis enim actionem e&longs;&longs;e <lb/>mutuam & utrinque æqualem, &longs;ic o&longs;tenditur. </s> <s>Di&longs;tinguatur Terræ <pb/>totius moles in binas qua&longs;cunque partes, vel æquales vel utcunque <lb/>inæquales: jam &longs;i pondera partium non e&longs;&longs;ent in &longs;e mutuo æqua­<lb/>lia; cederet pondus minus majori, & partes conjunctæ pergerent <lb/>recta moveri ad in&longs;initum, ver&longs;us plagam in quam tendit pondus <lb/>majus: omnino contra Experientiam. </s> <s>Itaque dicendum erit, pon­<lb/>dera partium in æquilibrio e&longs;&longs;e con&longs;tituta: hoc e&longs;t, Gravitatis <lb/>actionem e&longs;&longs;e mutuam & utrinque æqualem. </s> </p> <p type="main"> <s>Pondera corporum, æqualiter a centro Terræ di&longs;tantium, &longs;unt ut <lb/>quantitates materiæ in corporibus. </s> <s>Hoc utique colligitur ex <lb/>æquali acceleratione corporum omnium, e quiete per ponderum <lb/>vires cadentium: nam vires quibus inæqualia corpora æqualiter <lb/>accelerantur, debent e&longs;&longs;e proportionales quantitatibus materiæ <lb/>movendæ. </s> <s>Jam vero corpora univer&longs;a cadentia æqualiter acce­<lb/>lerari, ex eo patet, quod in Vacuo <emph type="italics"/>Boyliano<emph.end type="italics"/> temporibus æqualibus <lb/>æqualia &longs;patia cadendo de&longs;cribunt, &longs;ublata &longs;cilicet Aeris re&longs;i&longs;tentia: <lb/>accuratius autem comprobatur per Experimenta Pendulorum. </s> </p> <p type="main"> <s>Vires attractivæ corporum, in æqualibus di&longs;tantiis, &longs;unt ut <lb/>quantitates materiæ in corporibus. </s> <s>Nam cum corpora in Ter­<lb/>ram & Terra vici&longs;&longs;im in corpora momentis æqualibus gravitent; <lb/>Terræ pondus in unumquodque corpus, &longs;eu vis qua corpus Ter­<lb/>ram attrahit, æquabitur ponderi corporis eju&longs;dem in Terram. </s> <s><lb/>Hoc autem pondus erat ut quantitas materiæ in corpore: itaque <lb/>vis qua corpus unumquodque Terram attrahit, &longs;ive corporis vis <lb/>ab&longs;oluta, erit ut eadem quantitas materiæ. </s> </p> <p type="main"> <s>Oritur ergo & componitur vis attractiva corporum integrorum <lb/>ex viribus attractivis partium: &longs;iquidem aucta vel diminuta mole <lb/>materiæ, o&longs;ten&longs;um e&longs;t, proportionaliter augeri vel diminui ejus vir­<lb/>tutem. </s> <s>Actio itaque Telluris ex conjunctis partium Actionibus <lb/>conflari cen&longs;enda erit; atque adeo corpora omnia Terre&longs;tria &longs;e <lb/>mutuo trahere oportet viribus ab&longs;olutis, quæ &longs;int in ratione ma­<lb/>teriæ trahentis. </s> <s>Hæc e&longs;t natura Gravitatis apud Terram: videa­<lb/>mus jam qualis &longs;it in Cælis. </s> </p> <p type="main"> <s>Corpus omne per&longs;everare in &longs;tatu &longs;uo vel quie&longs;cendi vel movendi <lb/>uniformiter in directum, ni&longs;i quatenus a viribus impre&longs;&longs;is cogitur <lb/>&longs;tatum illum mutare; Naturæ lex e&longs;t ab omnibus recepta Philo&longs;o­<lb/>phis. </s> <s>Inde vero &longs;equitur, corpora quæ in Curvis moventur, atque <lb/>adeo de lineis rectis Orbitas &longs;uas tangentibus jugiter abeunt, Vi <lb/>aliqua perpetuo agente retineri in itinere curvilineo. </s> <s>Planetis <lb/>igitur in Orbibus curvis revolventibus nece&longs;&longs;ario aderit Vis aliqua, <lb/>per cujus actiones repetitas inde&longs;inenter a Tangentibus deflectantur. </s> </p> <pb/> <p type="main"> <s>Jam illud concedi æquum e&longs;t, quod Mathematicis rationibus <lb/>colligitur & certi&longs;&longs;ime demon&longs;tratur; Corpora nempe omnia, quæ <lb/>moventur in linea aliqua curva in plano de&longs;cripta, quæque radio <lb/>ducto ad punctum vel quie&longs;cens vel utcunque motum de&longs;cribunt <lb/>areas circa punctum illud temporibus proportionales, urgeri a <lb/>Viribus quæ ad idem punctum tendunt. </s> <s>Cum igitur in confe&longs;&longs;o <lb/>&longs;it apud A&longs;tronomos, Planetas primarios circum Solem, &longs;ecunda­<lb/>rios vero circum &longs;uos primarios, areas de&longs;cribere temporibus pro­<lb/>portionales; con&longs;equens e&longs;t ut Vis illa, qua perpetuo detorquen­<lb/>tur a Tangentibus rectilineis, & in Orbitis curvilineis revolvi ce­<lb/>guntur, ver&longs;us corpora dirigatur quæ &longs;ita &longs;unt in Orbitarum cen­<lb/>tris. </s> <s>Hæc itaque Vis non inepte vocari pote&longs;t, re&longs;pectu quidem <lb/>corporis revolventis, Centripeta; re&longs;pectu autem corporis cen­<lb/>tralis, Attractiva; a quacunque demum cau&longs;a oriri fingatur. </s> </p> <p type="main"> <s>Quin & hæc quoque concedenda &longs;unt, & Mathematice demon­<lb/>&longs;trantur: Si corpora plura motu æquabili revolvantur in Circulis <lb/>concentricis, & quadrata temporum periodicorum &longs;int ut cubi di­<lb/>&longs;tantiarum a centro communi; Vires centripetas revolventium <lb/>fore reciproce ut quadrata di&longs;tantiarum. </s> <s>Vel, &longs;i corpora revol­<lb/>vantur in Orbitis quæ &longs;unt Circulis finitimæ, & quie&longs;cant Orbita­<lb/>rum Ap&longs;ides; Vires centripetas revolventium fore reciproce ut <lb/>quadrata di&longs;tantiarum. </s> <s>Obtinere ca&longs;um alterutrum in Planetis <lb/>univer&longs;is con&longs;entiunt A&longs;tronomi. </s> <s>Itaque Vires centripetæ Plane­<lb/>tarum omnium &longs;unt reciproce ut quadrata di&longs;tantiarum ab Or­<lb/>bium centris. </s> <s>Si quis objiciat Planetarum, & Lunæ præ&longs;ertim, <lb/>Ap&longs;ides non penitus quie&longs;cere; &longs;ed motu quodam lento ferri in <lb/>con&longs;equentia: re&longs;ponderi pote&longs;t, etiam&longs;i concedamus hunc mo­<lb/>tum tardi&longs;&longs;imum exinde profectum e&longs;&longs;e quod Vis centripetæ pro­<lb/>portio aberret aliquantum a duplicata, aberrationem illam per <lb/>computum Mathematicum inveniri po&longs;&longs;e & plane in&longs;en&longs;ibilem <lb/>e&longs;&longs;e. </s> <s>Ip&longs;a enim ratio Vis centripetæ Lunaris, quæ omnium ma­<lb/>xime turbari debet, paululum quidem duplicatam &longs;uperabit; ad <lb/>hanc vero &longs;exaginta fere vicibus propius accedet quam ad tripli­<lb/>catam. </s> <s>Sed verior erit re&longs;pon&longs;io, &longs;i dicamus hanc Ap&longs;idum progre&longs;­<lb/>&longs;ionem, non ex aberratione a duplicata proportione, &longs;ed ex alia <lb/>pror&longs;us diver&longs;a cau&longs;a oriri, quemadmodum egregie common&longs;tratur <lb/>in hac Philo&longs;ophia. </s> <s>Re&longs;tat ergo ut Vires centripetæ, quibus Pla­<lb/>netæ primarii tendunt ver&longs;us Solem & &longs;ecundarii ver&longs;us primarios <lb/>&longs;uos, &longs;int accurate ut quadrata di&longs;tantisrum reciproce. </s> </p> <pb/> <p type="main"> <s>Ex iis quæ hactenus dicta &longs;unt, con&longs;tat Planetas in Orbitis &longs;uis <lb/>retineri per Vim aliquam in ip&longs;os perpetuo agentem: con&longs;tat <lb/>Vim illam dirigi &longs;emper ver&longs;us Orbitarum centra: con&longs;tat hujus <lb/>efficaciam augeri in acce&longs;&longs;u ad centrum, diminui in rece&longs;&longs;n ab eo­<lb/>dem: & augeri quidem in eadem proportione qua diminuitur qua­<lb/>dratum di&longs;tantiæ, diminui in eadem proportione qua di&longs;tantiæ <lb/>quadratum augetur. </s> <s>Videamus jam, comparatione in&longs;tituta inter <lb/>Planetarum Vires centripetas & Vim Gravitatis, annon eju&longs;dem <lb/>forte &longs;int generis. </s> <s>Eju&longs;dem vero generis erunt, &longs;i deprehendan­<lb/>tur hinc & inde leges eædem eædemque affectiones. </s> <s>Primo ita­<lb/>que Lunæ, quæ nobis proxima e&longs;t, Vim centripetam expendamus. </s> </p> <p type="main"> <s>Spatia rectilinea, quæ a corporibus e quiete demi&longs;&longs;is dato tem­<lb/>pore &longs;ub ip&longs;o motus initio de&longs;eribuntur, ubi a viribus quibu&longs;cun­<lb/>que urgentur, proportionalia &longs;unt ip&longs;is viribus: Hoc utique con­<lb/>&longs;equitur ex ratiociniis Mathematicis. </s> <s>Erit igitur Vis centripeta <lb/>Lunæ in Orbita &longs;ua revolventis, ad Vim Gravitatis in &longs;uperficie <lb/>Terræ, ut &longs;patium quod tempore quam minimo de&longs;criberet Luna <lb/>de&longs;cendendo per Vim centripetam ver&longs;us Terram, &longs;i circulari om­<lb/>ni motu privari fingeretur, ad &longs;patium quod eodem tempore quam <lb/>minimo de&longs;cribit grave corpus in vicinia Terræ, per Vim gravita­<lb/>tis &longs;uæ cadendo. </s> <s>Horum &longs;patiorum prius æquale e&longs;t arcus a Luna <lb/>per idem tempus de&longs;cripti &longs;inui ver&longs;o, quippe qui Lunæ tran&longs;la­<lb/>tionem de Tangente, factam a Vi centripeta, metitur; atque adeo <lb/>computari pote&longs;t ex datis tum Lunæ tempore periodico tum di­<lb/>&longs;tantia ejus a centro Terræ. </s> <s>Spatium po&longs;terius invenitur per Ex­<lb/>perimenta Pendulorum, quemadmodum docuit <emph type="italics"/>Hugenius.<emph.end type="italics"/> Inito <lb/>itaque calculo, &longs;patium prius ad &longs;patium pofterius, &longs;eu vis cen­<lb/>tripeta Lunæ in Orbita &longs;ua revolventis ad vim Gravitatis in &longs;u­<lb/>perficie Terræ, erit ut quadratum &longs;emidiametri Terræ ad Orbitæ <lb/>&longs;emidiametri quadratum. </s> <s>Eandem habet rationem, per ea quæ <lb/>&longs;uperius o&longs;tenduntur, vis centripeta Lunæ in Orbita &longs;ua revol­<lb/>venris ad vim Lunæ centripetam prope Terræ &longs;uperficiem. </s> <s>Vis <lb/>itaque centripeta prope Terræ &longs;uperficiem æqualis e&longs;t vi Gravita­<lb/>tis. </s> <s>Non ergo diver&longs;æ &longs;unt vires, &longs;ed una atque eadem: &longs;i enim <lb/>diver&longs;æ e&longs;&longs;ent, corpora viribus conjunctis duplo celerius in Ter­<lb/>ram caderent quam ex vi &longs;ola Gravitatis. </s> <s>Con&longs;tat igitur Vim <lb/>illam centripetam, qua Luna perpetuo de Tangente vel trahitur <lb/>vel impellitur & in Orbita retinetur, ip&longs;am e&longs;fe vim Gravitatis <lb/>terre&longs;tris ad Lunam u&longs;que pertingentem. </s> <s>Et rationi quidem con­<lb/>&longs;entaneum e&longs;t ut ad ingentes diftantias illa &longs;e&longs;e Virtus extendat, <pb/>cum nullam ejus &longs;en&longs;ibilem imminutionem, vel in alti&longs;&longs;imis montium <lb/>cacuminibus, ob&longs;ervare licet. </s> <s>Gravitat itaque Luna in Terram: <lb/>quin & actione mutua, Terra vici&longs;&longs;im in Lunam æqualiter gravitat: <lb/>id quod abunde quidem confirmatur in hac Philo&longs;ophia, ubi agi­<lb/>tur de Maris æ&longs;tu & Æquinoctiorum præce&longs;&longs;ione, ab actione tum <lb/>Lunæ tum Solis in Terram oriundis. </s> <s>Hinc & illud tandem edo­<lb/>cemur, qua nimirum lege vis Gravitatis decre&longs;cat in majoribus a <lb/>Tellure di&longs;tantiis. </s> <s>Nam cum Gravitas non diver&longs;a &longs;it a Vi cen­<lb/>tripeta Lunari, hæc vero &longs;it reciproce proportionalis quadrato <lb/>di&longs;tantiæ; diminuetur & Gravitas in eadem ratione. </s> </p> <p type="main"> <s>Progrediamur jam ad Planetas reliquos. </s> <s>Quoniam revolu­<lb/>tiones primariorum circa Solem & &longs;ecundariorum circa Jovem & <lb/>Saturnum &longs;unt Phænomena generis eju&longs;dem ac revolutio Lunæ <lb/>circa Terram, quoniam porro demon&longs;tratum e&longs;t vires centripetas <lb/>primariorum dirigi ver&longs;us centrum Solis, &longs;ecundariorum ver&longs;us <lb/>centra Jovis & Saturni, quemadmodum Lunæ vis centripeta ver&longs;us <lb/>Terræ centrum dirigitur; adhæc, quoniam omnes illæ vires &longs;unt <lb/>reciproce ut quadrata di&longs;tantiarum a centris, quemadmodum vis <lb/>Lunæ e&longs;t ut quadratum di&longs;tantiæ a Terra: concludendum erit <lb/>eandem e&longs;&longs;e naturam univer&longs;is. </s> <s>Itaque ut Luna gravitat in Ter­<lb/>ram, & Terra vici&longs;&longs;im in Lunam; &longs;ic etiam gravitabunt omnes <lb/>&longs;ecundarii in primarios &longs;uos, & primarii vici&longs;&longs;im in &longs;ecundarios; <lb/>&longs;ic & omnes primarii in Solem, & Sol vici&longs;&longs;im in primarios. </s> </p> <p type="main"> <s>Igitur Sol in Planetas univer&longs;os gravitat & univer&longs;i in Solem. </s> <s><lb/>Nam &longs;ecundarii dum primarios &longs;uos comitantur, revolvuntur in­<lb/>terea circum Solem una cum primariis. </s> <s>Eodem itaque argumento, <lb/>utriu&longs;que generis Planetæ gravitant in Solem, & Sol in ip&longs;os. </s> <s><lb/>Secundarios vero Planetas in Solem gravitare abunde in&longs;uper <lb/>con&longs;tat ex inæqualitatibus Lunaribus; quarum accurati&longs;&longs;imam <lb/>Theoriam, admiranda &longs;agacitate patefactam, in tertio hujus Operis <lb/>libro expo&longs;itam habemus. </s> </p> <p type="main"> <s>Solis virtutem attractivam quoquover&longs;um propagari ad ingen­<lb/>tes u&longs;que di&longs;tantias, & &longs;e&longs;e diffundere ad &longs;ingulas circumjecti &longs;pa­<lb/>tii partes, aperti&longs;&longs;ime colligi pote&longs;t ex motu Cometarum; qui ab <lb/>immen&longs;is intervallis profecti feruntur in viciniam Solis, & non­<lb/>nunquam adeo ad ip&longs;um proxime accedunt ut Globum ejus, in <lb/>Periheliis &longs;uis ver&longs;antes, tantum non contingere videantur. </s> <s>Ho­<lb/>rum Theoriam ab A&longs;tronomis antehac fru&longs;tra quæ&longs;itam, no&longs;tro <lb/>tandem &longs;æculo feliciter inventam & per Ob&longs;ervationes certi&longs;­<lb/>&longs;ime demon&longs;tratam, Præ&longs;tanti&longs;&longs;imo no&longs;tro Auctori debemus. </s> <s>Patet <pb/>igitur Cometas in Sectionibus Conicis umbilicos in centro Solis <lb/>habentibus moveri, & radiis ad Solem ductis areas temporibus <lb/>proportionales de&longs;cribere. </s> <s>Ex hi&longs;ce vero Phænomenis manife­<lb/>&longs;tum e&longs;t & Mathematice comprobatur, vires illas, quibus Cometæ <lb/>retinentur in orbitis &longs;uis, re&longs;picere Solem & e&longs;&longs;e reciproce ut qua­<lb/>drata di&longs;tantiarum ab ip&longs;ius centro. </s> <s>Gravitant itaque Cometæ <lb/>in Solem: atque adeo Solis vis attractiva non tantum ad corpora <lb/>Planetarum in datis di&longs;tantiis & in eodem fere plano collocata, <lb/>&longs;ed etiam ad Cometas in diver&longs;i&longs;&longs;imis Cælorum regionibus & in <lb/>diver&longs;i&longs;&longs;imis di&longs;tantiis po&longs;itos pertingit. </s> <s>Hæc igitur e&longs;t natura <lb/>corporum gravitantium, ut vires &longs;uas edant ad omnes di&longs;tantias in <lb/>omnia corpora gravitantia. </s> <s>Inde vero &longs;equitur, Planetas & Co­<lb/>metas univer&longs;os &longs;e mutuo trahere, & in &longs;e mutuo graves e&longs;&longs;e: <lb/>quod etiam confirmatur ex perturbatione Jovis & Saturni, A&longs;tro­<lb/>nomis non incognita, & ab actionibus horum Planetarum in &longs;e in­<lb/>vicem oriunda; quin & ex motu illo lenti&longs;&longs;imo Ap&longs;idum, qui &longs;u­<lb/>pra memoratus e&longs;t, quique a cau&longs;a con&longs;imili profici&longs;citur. </s> </p> <p type="main"> <s>Eo demum pervenimus ut dicendum &longs;it, & Terram & Solem & <lb/>corpora omnia cæle&longs;tia, quæ Solem comitantur, &longs;e mutuo attrahere. </s> <s><lb/>Singulorum ergo particulæ quæque minimæ vires &longs;uas attractivas <lb/>habebunt, pro quantitate materiæ pollentes; quemadmodum &longs;u­<lb/>pra de Terre&longs;tribus o&longs;ten&longs;um e&longs;t. </s> <s>In diver&longs;is autem di&longs;tantiis, <lb/>erunt & harum vires in duplicata ratione di&longs;tantiarum reciproce: <lb/>nam ex particulis hac lege trahentibus componi debere Globos <lb/>eadem lege trahentes, Mathematice demon&longs;tratur. </s> </p> <p type="main"> <s>Conclu&longs;iones præcedentes huic innituntur Axiomati, quod a <lb/>nullis non recipitur Philo&longs;ophis; Effectuum &longs;cilicet eju&longs;dem ge­<lb/>neris, quorum nempe quæ cogno&longs;cuntur proprietates eædem &longs;unt, <lb/>ea&longs;dem e&longs;&longs;e cau&longs;as & ea&longs;dem e&longs;&longs;e proprietates quæ nondum cog­<lb/>no&longs;cuntur. </s> <s>Quis enim dubitat, &longs;i Gravitas &longs;it cau&longs;a de&longs;cen&longs;us <lb/>Lapidis in <emph type="italics"/>Europa,<emph.end type="italics"/> quin eadem &longs;it cau&longs;a de&longs;cen&longs;us in <emph type="italics"/>America?<emph.end type="italics"/><lb/>Si Gravitas mutua fuerit inter Lapidem & Terram in <emph type="italics"/>Europa<emph.end type="italics"/>; <lb/>quis negabit mutuam e&longs;&longs;e in <emph type="italics"/>America?<emph.end type="italics"/> Si vis attractiva Lapidis <lb/>& Terræ componatur, in <emph type="italics"/>Europa,<emph.end type="italics"/> ex viribus attractivis partium; <lb/>quis negabit &longs;imilem e&longs;&longs;e compo&longs;itionem in <emph type="italics"/>America?<emph.end type="italics"/> Si attractio <lb/>Terræ ad omnia corporum genera & ad omnes di&longs;tantias propa­<lb/>getur in <emph type="italics"/>Europa<emph.end type="italics"/>; quidni pariter propagari dicamus in <emph type="italics"/>America?<emph.end type="italics"/><lb/>In hac Regula fundatur omnis Philo&longs;ophia: quippe qua &longs;ublata <lb/>nihil affirmare po&longs;&longs;imus de Univer&longs;is. </s> <s>Con&longs;titutio rerum &longs;ingula­<lb/>rum innote&longs;cit per Ob&longs;ervationes & Experimenta: inde vero non <pb/>ni&longs;i per hanc Regulam de rerum univer&longs;arum natura judica­<lb/>mus. </s> </p> <p type="main"> <s>Jam cum Gravia &longs;int omnia corpora, quæ apud Terram vel in <lb/>Cælis reperiuntur, de quibus Experimenta vel Ob&longs;ervationes in­<lb/>&longs;tituere licet; omnino dicendum erit, Gravitatem corporibus uni­<lb/>ver&longs;is competere. </s> <s>Et quemadmodum nulla concipi debent cor­<lb/>pora, quæ non &longs;int Exten&longs;a, Mobilia, & Impenetrabilia; ita nulla <lb/>concipi debere, quæ non &longs;int Gravia. </s> <s>Corporum Exten&longs;io, Mobi­<lb/>litas, & Impenetrabilitas non ni&longs;i per Experimenta innote&longs;cunt: <lb/>eodem plane modo Gravitas innote&longs;cit. </s> <s>Corpora omnia de qui­<lb/>bus Ob&longs;ervationes habemus, Exten&longs;a &longs;unt & Mobilia & Impene­<lb/>trabilia: & inde concludimus corpora univer&longs;a, etiam illa de qui­<lb/>bus Ob&longs;ervationes non habemus, Exten&longs;a e&longs;&longs;e & Mobilia & Im­<lb/>penetrabilia. </s> <s>Ita corpora omnia &longs;unt Gravia, de quibus Ob&longs;er­<lb/>vationes habemus: & inde concludimus corpora univer&longs;a, etiam <lb/>illa de quibus Ob&longs;ervationes non habemus, Gravia e&longs;&longs;e. </s> <s>Si quis <lb/>dicat corpora Stellarum inerrantium non e&longs;&longs;e Gravia, quandoqui­<lb/>dem eorum Gravitas nondum e&longs;t ob&longs;ervata; eodem argumento <lb/>dicere licebit neque Exten&longs;a e&longs;&longs;e, nec Mobilia, nec Impenetrabilia, <lb/>cum hæ Fixarum affectiones nondum &longs;int ob&longs;ervatæ. </s> <s>Quid opus <lb/>e&longs;t verbis? </s> <s>Inter primarias qualitates corporum univer&longs;orum vel <lb/>Gravitas habebit locum; vel Exten&longs;io, Mobilitas, & Impenetra­<lb/>bilitas non habebunt. </s> <s>Et natura rerum vel recte explicabitur <lb/>per corporum Gravitatem, vel non recte explicabitur per corpo­<lb/>rum Exten&longs;ionem, Mobilitatem, & Impenetrabilitatem. </s> </p> <p type="main"> <s>Audio nonnullos hanc improbare conclu&longs;ionem, & de occultis <lb/>qualitatibus ne&longs;cio quid mu&longs;&longs;itare. </s> <s>Gravitatem &longs;cilicet Occultum <lb/>e&longs;&longs;e quid, perpetuo argutari &longs;olent; occultas vero cau&longs;as pro­<lb/>cul e&longs;&longs;e ablegandas a Philo&longs;ophia. </s> <s>His autem facile re&longs;pon­<lb/>detur; occultas e&longs;&longs;e cau&longs;as, non illas quidem quarum exi&longs;tentia <lb/>per Ob&longs;ervationes clari&longs;&longs;ime demon&longs;tratur, &longs;ed has &longs;olum quarum <lb/>occulta e&longs;t & ficta exi&longs;tentia nondum vero comprobata. </s> <s>Gravitas <lb/>ergo non erit occulta cau&longs;a motuum cæle&longs;tium; &longs;iquidem ex Phæ­<lb/>nomenis o&longs;ten&longs;um e&longs;t, hanc virtutem revera exi&longs;tere. </s> <s>Hi potius <lb/>ad occultas confugiunt cau&longs;as; qui ne&longs;cio quos Vortices, materiæ <lb/>cuju&longs;dam pror&longs;us fictitiæ & &longs;en&longs;ibus omnino ignotæ, motibus <lb/>ii&longs;dem regendis præficiunt. </s> </p> <p type="main"> <s>Ideone autem Gravitas occulta cau&longs;a dicetur, eoque nomine <lb/>rejicietur e Philo&longs;ophia, quod cau&longs;a ip&longs;ius Gravitatis occulta e&longs;t <lb/>& nondum inventa? </s> <s>Qui &longs;ic &longs;tatuunt, videant nequid &longs;tatu- <pb/>ant ab&longs;urdi, unde totius tandem Philo&longs;ophiæ fundamenta convel­<lb/>lantur. </s> <s>Etenim cau&longs;æ continuo nexu procedere &longs;olent a compo­<lb/>&longs;itis ad &longs;impliciora: ubi ad cau&longs;am &longs;implici&longs;&longs;imam perveneris, jam <lb/>non licebit ulterius progredi. </s> <s>Cau&longs;æ igitur &longs;implici&longs;&longs;imæ nulla <lb/>dari pote&longs;t mechanica explicatio: &longs;i daretur enim, cau&longs;a non­<lb/>dum e&longs;&longs;et &longs;implici&longs;&longs;ima. </s> <s>Has tu proinde cau&longs;as &longs;implici&longs;&longs;imas <lb/>appellabis occultas, & exulare jubebis? </s> <s>&longs;imul vero exulabunt <lb/>& ab his proxime pendentes & quæ ab illis porro pendent, <lb/>u&longs;que dum a cau&longs;is omnibus vacua fuerit & probe purgata Phi­<lb/>lo&longs;ophia. </s> </p> <p type="main"> <s>Sunt qui Gravitatem præter naturam e&longs;&longs;e dicunt, & Miraculum <lb/>perpetuum vocant. </s> <s>Itaque rejiciendam e&longs;&longs;e volunt, cum in Phy­<lb/>&longs;ica præternaturales cau&longs;æ locum non habeant. </s> <s>Huic ineptæ <lb/>pror&longs;us objectioni diluendæ, quæ & ip&longs;a Philo&longs;ophiam &longs;ubruit <lb/>univer&longs;am, vix operæ pretium e&longs;t immorari. </s> <s>Vel enim Gravita­<lb/>tem corporibus omnibus inditam e&longs;&longs;e negabunt, quod tamen dici <lb/>non pote&longs;t: vel eo nomine præter naturam e&longs;&longs;e affirmabunt, quod <lb/>ex aliis corporum affectionibus atque adeo ex cau&longs;is Mechanicis <lb/>originem non habeat. </s> <s>Dantur certe primariæ corporum affecti­<lb/>ones; quæ, quoniam &longs;unt primariæ, non pendent ab aliis. </s> <s>Vide­<lb/>rint igitur annon & hæ omnes &longs;int pariter præter naturam, eo­<lb/>que pariter rejiciendæ: viderint vero qualis &longs;it deinde futura <lb/>Philo&longs;ophia. </s> </p> <p type="main"> <s>Nonnulli &longs;unt quibus hæc tota Phy&longs;ica cæle&longs;tis vel ideo minus <lb/>placet, quod cum <emph type="italics"/>Carte&longs;ii<emph.end type="italics"/> dogmatibus pugnare & vix conciliari <lb/>po&longs;&longs;e videatur. </s> <s>His &longs;ua licebit opinione frui; ex æquo autem <lb/>agant oportet: non ergo denegabunt aliis eandem libertatem <lb/>quam &longs;ibi concedi po&longs;tulant. </s> <s>NEWTONIANAM itaque Philo&longs;ophi­<lb/>am, quæ nobis verior habetur, retinere & amplecti licebit, & cau&longs;as <lb/>&longs;equi per Phænomena comprobatas, potius quam fictas & nondum <lb/>comprobatas. </s> <s>Ad veram Philo&longs;ophiam pertinet, rerum naturas <lb/>ex cau&longs;is vere exi&longs;tentibus derivare: eas vero leges quærere, qui­<lb/>bus voluit Summus opifex hunc Mundi pulcherrimum ordinem <lb/>&longs;tabilire; non eas quibus potuit, &longs;i ita vi&longs;um fui&longs;&longs;et. </s> <s>Rationi enim <lb/>con&longs;onum e&longs;t, ut a pluribus cau&longs;is, ab invicem nonnihil diver&longs;is, <lb/>idem po&longs;&longs;it Effectus profici&longs;ci: hæc autem vera erit cau&longs;a, ex qua <lb/>vere atque actu profici&longs;citur; reliquæ locum non habent in Philo­<lb/>&longs;ophia vera. </s> <s>In Horologiis automatis idem Indicis horarii mo­<lb/>tus vel ab appen&longs;o Pondere vel ab intus conclu&longs;o Elatere oriri po­<lb/>te&longs;t. </s> <s>Quod &longs;i oblatum Horologium revera &longs;it in&longs;tructum Pondere; <pb/>ridebitur qui finget Elaterem, & ex Hypothe&longs;i &longs;ic præpropere con­<lb/>ficta motum Indicis explicare &longs;u&longs;cipiet: oportuit enim internam <lb/>Machinæ fabricam penitius per&longs;crutari, ut ita motus propo&longs;iti prin­<lb/>cipium verum exploratum habere po&longs;&longs;et. </s> <s>Idem vel non ab&longs;imile <lb/>feretur judicium de Philo&longs;ophis illis, qui materia quadam &longs;ubti­<lb/>li&longs;&longs;ima Cælos e&longs;&longs;e repletos, hanc autem in Vortices inde&longs;inenter <lb/>agi voluerunt. </s> <s>Nam &longs;i Phænomenis vel accurati&longs;&longs;ime &longs;atisfacere <lb/>po&longs;&longs;ent ex Hypothe&longs;ibus &longs;uis; veram tamen Philo&longs;ophiam tradi­<lb/>di&longs;&longs;e, & veras cau&longs;as motuum cæle&longs;tium inveni&longs;&longs;e nondum di­<lb/>cendi &longs;unt; ni&longs;i vel has revera exi&longs;tere, vel &longs;altem alias non ex­<lb/>i&longs;tere demon&longs;traverint. </s> <s>Igitur &longs;i o&longs;ten&longs;um fuerit, univer&longs;orum <lb/>corporum Attractionem habere verum locum in rerum natura; <lb/>quinetiam o&longs;ten&longs;um fuerit, qua ratione motus omnes cæle&longs;tes ab­<lb/>inde &longs;olutionem recipiant; vana fuerit & merito deridenda objectio, <lb/>&longs;i quis dixerit eo&longs;dem motus per Vortices explicari debere, etiam&longs;i <lb/>id fieri po&longs;&longs;e vel maxime conce&longs;&longs;erimus. </s> <s>Non autem concedimus: <lb/>Nequeunt enim ullo pacto Phænomena per Vortices explicari; <lb/>quod ab Auctore no&longs;tro abunde quidem & clari&longs;&longs;imis rationibus <lb/>evincitur; ut &longs;omniis plus æquo indulgeant oporteat, qui inep­<lb/>ti&longs;&longs;imo figmento re&longs;arciendo, novi&longs;que porro commentis ornando <lb/>infelicem operam addicunt. </s> </p> <p type="main"> <s>Si corpora Planetarum & Cometarum circa Solem deferantur <lb/>a Vorticibus; oportet corpora delata & Vorticum partes proxime <lb/>ambientes eadem velocitate eademque cur&longs;us determinatione mo­<lb/>veri, & eandem habere den&longs;itatem vel eandem Vim inertiæ pro <lb/>mole materiæ. </s> <s>Con&longs;tat vero Planetas & Cometas, dum ver&longs;an­<lb/>tur in ii&longs;dem regionibus Cælorum, velocitatibus variis variaque <lb/>cur&longs;us determinatione moveri. </s> <s>Nece&longs;&longs;ario itaque &longs;equitur, ut <lb/>Fluidi cæle&longs;tis partes illæ, quæ &longs;unt ad ea&longs;dem di&longs;tantias a Sole, <lb/>revolvantur eodem tempore in plagas diver&longs;as cum diver&longs;is ve­<lb/>locitatibus: etenim alia opus erit directione & velocitate, ut tran­<lb/>&longs;ire po&longs;&longs;int Planetæ; alia, ut tran&longs;ire po&longs;&longs;int Cometæ. </s> <s>Quod cum <lb/>explicari nequeat; vel fatendum erit, univer&longs;a corpora cæle&longs;tia <lb/>non deferri a materia Vorticis; vel dicendum erit, eorundem mo­<lb/>tus repetendos e&longs;le non ab uno eodemque Vortice, &longs;ed a pluribus <lb/>qui ab invicem diver&longs;i &longs;int, idemque &longs;patium Soli circumjectum <lb/>pervadant. </s> </p> <p type="main"> <s>Si plures Vortices in eodem &longs;patio contineri, & &longs;e&longs;e mutuo pe­<lb/>netrare, motibu&longs;que diver&longs;is revolvi ponantur; quoniam hi mo­<lb/>tus debent e&longs;&longs;e conformes delatorum corporum motibus, qui <pb/>&longs;unt &longs;umme regulares, & peraguntur in Sectionibus Conicis, nunc <lb/>valde eccentricis, nunc ad Circulorum proxime formam acceden­<lb/>tibus; jure quærendum erit, qui fieri po&longs;&longs;it, ut iidem integri con­<lb/>&longs;erventur, nec ab actionibus materiæ occur&longs;antis per tot &longs;æcula <lb/>quicquam perturbentur. </s> <s>Sane &longs;i motus hi fictitii &longs;unt magis com­<lb/>po&longs;iti & difficilius explicantur, quam veri illi motus Planetarum <lb/>& Cometarum; fru&longs;tra mihi videntur in Philo&longs;ophiam recipi: <lb/>omnis enim Cau&longs;a debet e&longs;&longs;e E&longs;fectu &longs;uo &longs;implicior. </s> <s>Conce&longs;&longs;a <lb/>Fabularum licentia, affirmaverit aliquis Planetas omnes & Cometas <lb/>circumcingi Atmo&longs;phæris, adin&longs;tar Telluris no&longs;træ; quæ quidem <lb/>Hypothe&longs;is rationi magis con&longs;entanea videbitur quam Hypothe­<lb/>&longs;is Vorticum. </s> <s>Affirmaverit deinde has Atmo&longs;phæras, ex natura <lb/>&longs;ua, circa Solem moveri & Sectiones Conicas de&longs;cribere; qui <lb/>&longs;ane motus multo facilius concipi pote&longs;t, quam con&longs;imilis motus <lb/>Vorticum &longs;e invicem permeantium. </s> <s>Denique Planetas ip&longs;os & <lb/>Cometas circa Solem deferri ab Atmo&longs;phæris &longs;uis credendum e&longs;&longs;e <lb/>&longs;tatuat, & ob repertas motuum cæle&longs;tium cau&longs;as triumphum agat. </s> <s><lb/>Qui&longs;quis autem hanc Fabulam rejiciendam e&longs;&longs;e putet, idem & alte­<lb/>ram Fabulam rejiciet: nam ovum non e&longs;t ovo &longs;imilius, quam Hy­<lb/>pothe&longs;is Atmo&longs;phærarum Hypothe&longs;i Vorticum. </s> </p> <p type="main"> <s>Docuit <emph type="italics"/>Galilæus,<emph.end type="italics"/> lapidis projecti & in Parabola moti deflexio­<lb/>nem a cur&longs;u rectilineo oriri a Gravitate lapidis in Terram, ab oc­<lb/>culta &longs;cilicet qualitate. </s> <s>Fieri tamen pote&longs;t ut alius aliquis, na&longs;i <lb/>acutioris, Philo&longs;ophus cau&longs;am aliam commini&longs;catur. </s> <s>Finget igi­<lb/>tur ille materiam quandam &longs;ubtilem, quæ nec vi&longs;u, nec tactu, <lb/>neque ullo &longs;en&longs;u percipitur, ver&longs;ari in regionibus quæ proxime <lb/>contingunt Telluris &longs;uperficiem. </s> <s>Hanc autem materiam, in di­<lb/>ver&longs;as plagas, variis & plerumque contrariis motibus ferri, & li­<lb/>neas Parabolicas de&longs;cribere contendet. </s> <s>Deinde vero lapidis de­<lb/>flexionem pulchre &longs;ic expediet, & vulgi plau&longs;um merebitur. </s> <s>La­<lb/>pis, inquiet, in Fluido illo &longs;ubtili natat; & cur&longs;ui ejus ob&longs;equen­<lb/>do, non pote&longs;t non eandem una &longs;emitam de&longs;cribere. </s> <s>Fluidum <lb/>vero movetur in lineis Parabolicis; ergo lapidem in Parabola <lb/>moveri nece&longs;&longs;e e&longs;t. </s> <s>Quis nunc non mirabitur acuti&longs;&longs;imum huju&longs;ce <lb/>Philo&longs;ophi ingenium, ex cau&longs;is Mechanicis, materia &longs;cilicet & <lb/>motu, phænomena Naturæ ad vulgi etiam captum præclare de­<lb/>ducentis? </s> <s>Quis vero non &longs;ub&longs;annabit bonum illum <emph type="italics"/>Galilæum,<emph.end type="italics"/> qui <lb/>magno molimine Mathematico qualitates occultas, e Philo&longs;ophia <lb/>feliciter exclu&longs;as, denuo revocare &longs;u&longs;tinuerit? </s> <s>Sed pudet nugis <lb/>diutius immorari. </s> </p> <pb/> <p type="main"> <s>Summa rei huc tandem redìt: Cometarum ingens e&longs;t numerus; <lb/>motus eorum &longs;unt &longs;umme regulares, & ea&longs;dem leges cum Plane­<lb/>tarum motibus ob&longs;ervant. </s> <s>Moventur in Orbibus Conicis, hi or­<lb/>bes &longs;unt valde admodum eccentrici. </s> <s>Feruntur undique in omnes <lb/>Cælorum partes, & Planetarum regiones liberrime pertran&longs;eunt, <lb/>& &longs;æpe contra Signorum ordinem incedunt. </s> <s>Hæc Phænomena <lb/>certi&longs;&longs;ime confirmantur ex Ob&longs;ervationibus A&longs;tronomicis: & per <lb/>Vortices nequeunt explicari: Imo, ne quidem cum Vorticibus <lb/>Planetarum con&longs;i&longs;tere po&longs;&longs;unt. </s> <s>Cometarum motibus omnino lo­<lb/>cus non erit; ni&longs;i materia illa fictitia penitus e Cælis amo­<lb/>veatur. </s> </p> <p type="main"> <s>Si enim Planetæ circum Solem a Vorticibus devehuntur; Vor­<lb/>ticum partes, quæ proxime ambiunt unumquemque Planetam, eju&longs;­<lb/>dem den&longs;itatis erunt ac Planeta; uti &longs;upra dictum e&longs;t. </s> <s>Itaque <lb/>materia illa omnis quæ contigua e&longs;t Orbis magni perimetro, pa­<lb/>rem habebit ac Tellus den&longs;itatem: quæ vero jacet intra Orbem <lb/>magnum atque Orbem Saturni, vel parem vel majorem habebit. </s> <s><lb/>Nam ut con&longs;titutio Vorticis permanere po&longs;&longs;it, debent partes mi­<lb/>nus den&longs;æ centrum occupare, magis den&longs;æ longius a centro abire. </s> <s><lb/>Cum enim Planetarum tempora periodica &longs;int in ratione &longs;e&longs;qui­<lb/>plicata di&longs;tantiarum a Sole, oportet partium Vorticis periodos <lb/>eandem rationem &longs;ervare. </s> <s>Inde vero &longs;equitur, vires centrifugas <lb/>harum partium fore reciproce ut quadrata di&longs;tantiarum. </s> <s>Quæ <lb/>igitur majore intervallo di&longs;tant a centro, nituntur ab eodem re­<lb/>cedere minore vi: unde &longs;i minus den&longs;æ fuerint, nece&longs;&longs;e e&longs;t ut ce­<lb/>dant vi majori, qua partes centro propiores a&longs;cendere conantur. </s> <s><lb/>A&longs;cendent ergo den&longs;iores, de&longs;cendent minus den&longs;æ, & locorum <lb/>fiet invicem permutatio; donec ita fuerit di&longs;po&longs;ita atque ordinata <lb/>materia fluida totius Vorticis, ut conquie&longs;cere jam po&longs;&longs;it in æqui­<lb/>librio con&longs;tituta. </s> <s>Si bina Fluida, quorum diver&longs;a e&longs;t den&longs;itas, <lb/>in eodem va&longs;e continentur; utique futurum e&longs;t ut Fluidum, cu­<lb/>jus major e&longs;t den&longs;itas, majore vi Gravitatis infimum petat locum: <lb/>& ratione non ab&longs;imili omnino dicendum e&longs;t, den&longs;iores Vorticis <lb/>partes majore vi centrifuga petere &longs;upremum locum. </s> <s>Tota igi­<lb/>tur illa & multo maxima pars Vorticis, quæ jacet extra Telluris <lb/>orbem, den&longs;itatem habebit atque adeo vim inertiæ pro mole ma­<lb/>teriæ, quæ non minor erit quam den&longs;itas & vis inertiæ Telluris: <lb/>inde vero Cometis trajectis orietur ingens re&longs;i&longs;tentia, & valde ad­<lb/>modum &longs;en&longs;ibilis; ne dicam, quæ motum eorundem penitus &longs;i&longs;tere <lb/>atque ab&longs;orbere po&longs;&longs;e merito videatur. </s> <s>Con&longs;tat autem ex motu Co- <pb/>metarum pror&longs;us regulari, nullam ip&longs;os re&longs;i&longs;tentiam pati quæ vel <lb/>minimum &longs;entiri pote&longs;t; atque adeo neutiquam in materiam ul­<lb/>lam incur&longs;are, cujus aliqua &longs;it vis re&longs;i&longs;tendi, vel proinde cujus ali­<lb/>qua &longs;it den&longs;itas &longs;eu vis Inertiæ. </s> <s>Nam re&longs;i&longs;tentia Mediorum ori­<lb/>tur vel ab inertia materiæ fluidæ, vel a defectu lubricitatis. </s> <s>Quæ <lb/>oritur a defectu lubricitatis, admodum exigua e&longs;t; & &longs;ane vix <lb/>ob&longs;ervari pote&longs;t in Fluidis vulgo notis, ni&longs;i valde tenacia fuerint <lb/>adin&longs;tar Olei & Mellis. </s> <s>Re&longs;i&longs;tentia quæ &longs;entitur in Aere, Aqua, <lb/>Hydrargyro, & huju&longs;modi Fluidis non tenacibus fere tota e&longs;t <lb/>prioris generis; & minui non pote&longs;t per ulteriorem quemcunque <lb/>gradum &longs;ubtilitatis, manente Fluidi den&longs;itate vel vi inertiæ, cui <lb/>&longs;emper proportionalis e&longs;t hæc re&longs;i&longs;tentia; quemadmodum clari&longs;­<lb/>&longs;ime demon&longs;tratum e&longs;t ab Auctore no&longs;tro in peregregia Re&longs;i&longs;ten­<lb/>tiarum Theoria, quæ paulo nunc accuratius exponitur, hac &longs;e­<lb/>cunda vice, & per Experimenta corporum cadentium plenius <lb/>confirmatur. </s> </p> <p type="main"> <s>Corpora progrediendo motum &longs;uum Fluido ambienti paulatim <lb/>communicant, & communicando amittunt, amittendo autem re­<lb/>tardantur. </s> <s>E&longs;t itaque retardatio motui communicato proportio­<lb/>nalis; motus vero communicatus, ubi datur corporis progredientis <lb/>velocitas, e&longs;t ut Fluidi den&longs;itas; ergo retardatio &longs;eu re&longs;i&longs;tentia <lb/>erit ut eadem Fluidi den&longs;itas; neque ullo pacto tolli pote&longs;t, ni&longs;i <lb/>a Fluido ad partes corporis po&longs;ticas recurrente re&longs;tituatur motus <lb/>ami&longs;&longs;us. </s> <s>Hoc autem dici non poterit, ni&longs;i impre&longs;&longs;io Fluidi in cor­<lb/>pus ad partes po&longs;ticas æqualis fuerit impre&longs;&longs;ioni corporis in Flui­<lb/>dum ad partes anticas, hoc e&longs;t, ni&longs;i velocitas relativa qua Flui­<lb/>dum irruit in corpus a tergo, æqualis fuerit velocitati qua cor­<lb/>pus irruit in Fluidum, id e&longs;t, ni&longs;i velocitas ab&longs;oluta Fluidi re­<lb/>currentis duplo major fuerit quam velocitas ab&longs;oluta Fluidi pro­<lb/>pul&longs;i; quod fieri nequit. </s> <s>Nullo igitur modo tolli pote&longs;t Flui­<lb/>dorum re&longs;i&longs;tentia, quæ oritur ab corundem den&longs;itate & vi in­<lb/>ertiæ. </s> <s>Itaque concludendum erit; Fluidi cæle&longs;tis nullam e&longs;&longs;e <lb/>vim inertiæ, cum nulla &longs;it vis re&longs;i&longs;tendi: nullam e&longs;&longs;e vim qua <lb/>motus communicetur, cum nulla &longs;it vis inertiæ: nullam e&longs;&longs;e vim <lb/>qua mutatio quælibet vel corporibus &longs;ingulis vel pluribus indu­<lb/>catur, cum nulla &longs;it vis qua motus communicetur: nullam e&longs;&longs;e <lb/>omnino efficaciam, cum nulla &longs;it facultas mutationem quamlibet <lb/>inducendi. </s> <s>Quidni ergo hanc Hypothe&longs;in, quæ fundamento <lb/>plane de&longs;tituitur, quæque naturæ rerum explicandæ ne minimum <lb/>quidem in&longs;ervit, inepti&longs;&longs;imam vocare liceat & Philo&longs;opho pror- <pb/>&longs;us indignam. </s> <s>Qui Cælos materia fluida repletos e&longs;&longs;e volunt, <lb/>hanc vero non inertem e&longs;&longs;e &longs;tatuunt; Hi verbis tollunt Vacuum, <lb/>re ponunt. </s> <s>Nam cum huju&longs;modi materia fluida ratione nulla <lb/>&longs;ecerni po&longs;&longs;it ab inani Spatio; di&longs;putatio tota fit de rerum no­<lb/>minibus, non de naturis. </s> <s>Quod &longs;i aliqui &longs;int adeo u&longs;que de­<lb/>diti Materiæ, ut Spatium a corporibus vacuui nullo pacto ad­<lb/>mittendum credere velint; videamus quo tandem oporteat illos <lb/>pervenire. </s> </p> <p type="main"> <s>Vel enim dicent hanc, quam confingunt, Mundi per omnia <lb/>pleni con&longs;titutionem ex voluntate Dei profectam e&longs;&longs;e, propter <lb/>eum finem, ut operationibus Naturæ &longs;ub&longs;idium præ&longs;ens haberi <lb/>po&longs;&longs;et ab Æthere &longs;ubtili&longs;&longs;imo cuncta permeante & implente; <lb/>quod tamen dici non pote&longs;t, &longs;iquidem jam o&longs;ten&longs;um e&longs;t ex Co­<lb/>metarum phænomenis, nullam e&longs;&longs;e hujus Ætheris efficaciam: vel <lb/>dicent ex voluntate Dei profectam e&longs;&longs;e, propter finem aliquem <lb/>ignotum; quod neque dici debet, &longs;iquidem diver&longs;a Mundi con­<lb/>&longs;titutio eodem argumento pariter &longs;tabiliri po&longs;&longs;et: vel denique <lb/>non dicent ex voluntate Dei profectam e&longs;&longs;e, &longs;ed ex nece&longs;&longs;itate <lb/>quadam Naturæ. </s> <s>Tandem igitur delabi oportet in &longs;æces &longs;ordi­<lb/>das Gregis impuri&longs;&longs;imi. </s> <s>Hi &longs;unt qui &longs;omniant Fato univer&longs;a <lb/>regi, non Providentia; Materiam ex nece&longs;&longs;itate &longs;ua &longs;emper & ubi­<lb/>que extiti&longs;&longs;e, infinitam e&longs;&longs;e & æternam. </s> <s>Quibus po&longs;itis, erit <lb/>etiam undiquaque uniformis: nam varietas formarum cum nece&longs;­<lb/>&longs;itate omnino pugnat. </s> <s>Erit etiam immota: nam &longs;i nece&longs;&longs;ario <lb/>moveatur in plagam aliquam determinatam, cum determinata ali­<lb/>qua velocitate; pari nece&longs;&longs;itate movebitur in plagam diver&longs;am <lb/>cum diver&longs;a velocitate; in plagas autem diver&longs;as, cum diver&longs;is <lb/>velocitatibus, moveri non pote&longs;t; oportet igitur immotam e&longs;&longs;e. </s> <s><lb/>Neutiquam profecto potuit oriri Mundus, pulcherrima forma­<lb/>rum & motuum varietate di&longs;tinctus, ni&longs;i ex liberrima voluntate <lb/>cuncta providentis & gubernantis Dei. </s> </p> <p type="main"> <s>Ex hoc igitur fonte promanarunt illæ omnes quæ dicuntur <lb/>Naturæ leges: in quibus multa &longs;ane &longs;apienti&longs;&longs;imi con&longs;ilii, nulla <lb/>nece&longs;&longs;itatis apparent ve&longs;tigia. </s> <s>Has proinde non ab incertis con­<lb/>jecturis petere, &longs;ed Ob&longs;ervando atque Experiendo addi&longs;cere de­<lb/>bemus. </s> <s>Qui veræ Phy&longs;icæ principia Lege&longs;que rerum, &longs;ola men­<lb/>tis vi & interno rationis lumine fretum, invenire &longs;e po&longs;&longs;e confi­<lb/>dit; hunc oportet vel &longs;tatuere Mundum ex nece&longs;&longs;itate fui&longs;le, Le­<lb/>ge&longs;que propo&longs;itas ex eadem nece&longs;&longs;itate &longs;equi; vel &longs;i per volun­<lb/>tatem Dei con&longs;titutus &longs;it ordo Naturæ, &longs;e tamen, homuncionem <pb/>mi&longs;ellum, quid optimum factu &longs;it per&longs;pectum habere. </s> <s>Sana om­<lb/>nis & vera Philo&longs;ophia fundatur in Phænomenis rerum: quæ &longs;i <lb/>nos vel invitos & reluctantes ad huju&longs;modi principia deducunt, <lb/>in quibus clari&longs;&longs;ime cernuntur Con&longs;ilium optimum & Dominium <lb/>&longs;ummum &longs;apienti&longs;&longs;imi & potenti&longs;&longs;imi Entis; non erunt hæc ideo <lb/>non admittenda principia, quod quibu&longs;dam for&longs;an hominibus <lb/>minus grata &longs;int futura. </s> <s>His vel Miracula vel Qualitates occultæ <lb/>dicantur, quæ di&longs;plicent: verum nomina malitio&longs;e indita non &longs;unt <lb/>ip&longs;is rebus vitio vertenda; ni&longs;i illud fateri tandem velint, utique <lb/>debere Philo&longs;ophiam in Athei&longs;mo fundari. </s> <s>Horum hominum <lb/>gratia non erit labefactanda Philo&longs;ophia, &longs;iquidem rerum ordo <lb/>non vult immutari. </s> </p> <p type="main"> <s>Obtinebit igitur apud probos & æquos Judices præ&longs;tanti&longs;&longs;ima <lb/>Philo&longs;ophandi ratio, quæ fundatur in Experimentis & Ob&longs;erva­<lb/>tionibus. </s> <s>Huic vero, dici vix poterit, quanta lux accedat, quanta <lb/>dignitas, ab hoc Opere præclaro Illu&longs;tri&longs;&longs;imi no&longs;tri Auctoris; cujus <lb/>eximiam ingenii felicitatem, difficillima quæque Problemata eno­<lb/>dantis, & ad ea porro pertingentis ad quæ nec &longs;pes erat humanam <lb/>mentem a&longs;&longs;urgere potui&longs;&longs;e, merito admirantur & &longs;u&longs;piciunt qui­<lb/>cunque paulo profundius in hi&longs;ce rebus ver&longs;ati &longs;unt. </s> <s>Clau&longs;tris <lb/>ergo re&longs;eratis, aditum Nobis aperuit ad pulcherrima rerum my­<lb/>&longs;teria. </s> <s>Sy&longs;tematis Mundani compagem eleganti&longs;&longs;imam ita tan­<lb/>dem patefecit & penitius per&longs;pectandam dedit; ut nec ip&longs;e, &longs;i <lb/>nunc revivi&longs;ceret, Rex <emph type="italics"/>Alphon&longs;us<emph.end type="italics"/> vel &longs;implicitatem vel harmoniæ <lb/>gratiam in ea de&longs;ideraret. </s> <s>Itaque Naturæ maje&longs;tatem propius jam <lb/>licet intueri, & dulci&longs;&longs;ima contemplatione frui, Conditorem vero <lb/>ac Dominum Univer&longs;orum impen&longs;ius colere & venerari, qui fructus <lb/>e&longs;t Philo&longs;ophiæ multo uberrimus. </s> <s>Cæcum e&longs;&longs;e oportet, qui ex <lb/>optimis & &longs;apienti&longs;&longs;imis rerum &longs;tructuris non &longs;tatim videat Fabri­<lb/>catoris Omnipotentis infinitam &longs;apientiam & bonitatem: in&longs;anum, <lb/>qui profiteri nolit. </s> </p> <p type="main"> <s>Extabit igitur Eximium NEWTONI Opus adver&longs;us Atheorum <lb/>impetus muniti&longs;&longs;imum præ&longs;idium: neque enim alicunde felicius, <lb/>quam ex hac pharetra, contra impiam Catervam tela depromp&longs;eris. </s> <s><lb/>Hoc &longs;en&longs;it pridem, & in pereruditis Concionibus Anglice Latineque <lb/>editis, primus egregie demon&longs;travit Vir in omni Literarum genere <lb/>præclarus idemque bonarum Artium fautor eximius RICHARDUS <lb/>BENTLEIUS, Sæculi &longs;ui & Academiæ no&longs;træ magnum Orna­<lb/>mentum, Collegii no&longs;tri <emph type="italics"/>S. Trinitatis<emph.end type="italics"/> Magi&longs;ter digni&longs;&longs;imus & in­<lb/>tegerrimus. </s> <s>Huic ego me pluribus nominibus ob&longs;trictum fateri <pb/>debeo: Huic & Tuas quæ debentur gratias, Lector benevole, non <lb/>denegabis. </s> <s>Is enim, cum a longo tempore Celeberrimi Auctoris <lb/>amicitia intima frueretur, (qua etiam apud Po&longs;teros cen&longs;eri non <lb/>minoris æ&longs;timat, quam propriis Scriptis quæ literato orbi in de­<lb/>liciis &longs;unt inclare&longs;cere) Amici &longs;imul famæ & &longs;cientiarum incre­<lb/>mento con&longs;uluit. </s> <s>Itaque cum Exemplaria prioris Editionis rari&longs;­<lb/>&longs;ima admodum & immani pretio coemenda &longs;upere&longs;&longs;ent; &longs;ua&longs;it Ille <lb/>crebris efflagitationibus & tantum non objurgando perpulit deni­<lb/>que Virum Præ&longs;tanti&longs;&longs;imum, nec mode&longs;tia minus quam eruditi­<lb/>one &longs;umma In&longs;ignem, ut novam hanc Operis Editionem, per om­<lb/>nia elimatam denuo & egregiis in&longs;uper acce&longs;&longs;ionibus ditatam, &longs;uis <lb/>&longs;umptibus & au&longs;piciis prodire pateretur: Mihi vero, pro jure <lb/>&longs;uo, pen&longs;um non ingratum demandavit, ut quam po&longs;&longs;et emendate <lb/>id fieri curarem. </s> </p> <p type="main"> <s><emph type="italics"/>Cantabrigiæ,<emph.end type="italics"/><lb/>Maii 12. 1713. </s> </p> <p type="main"> <s>ROGERUS COTES Collegii <emph type="italics"/>S. Trinitatis<emph.end type="italics"/> Socius, <lb/>A&longs;tronomiæ & Philo&longs;ophiæ Experimentalis <lb/>Profe&longs;&longs;or <emph type="italics"/>Plumianus.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="center"/>INDEX CAPITUM <lb/>TOTIUS OPERIS.<emph.end type="center"/></s> </p> <p type="main"> <s>PAG. </s> </p> <p type="main"> <s>DEFINITIONES. 1 </s> </p> <p type="main"> <s>AXIOMATA, SIVE LEGES MOTUS. 12 </s> </p> <p type="main"> <s><emph type="center"/>DE MOTU CORPORUM LIBER PRIMUS.<emph.end type="center"/></s> </p> <p type="main"> <s>SECT. I. <emph type="italics"/>DE Methodo rationum primarum & ultima­<lb/>rum.<emph.end type="italics"/> 24 </s> </p> <p type="main"> <s>SECT. II. <emph type="italics"/>De inventione Virium centripetarum.<emph.end type="italics"/> 34 </s> </p> <p type="main"> <s>SECT. III. <emph type="italics"/>De motu corporum in Conicis &longs;ectionibus eccentri­<lb/>cis.<emph.end type="italics"/> 48 </s> </p> <p type="main"> <s>SECT. IV. <emph type="italics"/>De inventione Orbium Ellipticorum, Parabolicorum <lb/>& Hyperbolicorum ex Umbilico dato.<emph.end type="italics"/> 59 </s> </p> <p type="main"> <s>SECT. V. <emph type="italics"/>De inventione Orbium ubi Umbilicus neuter datur.<emph.end type="italics"/> 66 </s> </p> <p type="main"> <s>SECT. VI. <emph type="italics"/>De inventione Motuum in Orbibus datis.<emph.end type="italics"/> 97 </s> </p> <p type="main"> <s>SECT. VII. <emph type="italics"/>De corporum A&longs;cen&longs;u & De&longs;cen&longs;u rectilineo.<emph.end type="italics"/> 105 </s> </p> <p type="main"> <s>SECT. VII. <emph type="italics"/>De inventione Orbium in quibus corpora Viribus <lb/>quibu&longs;cunque centripetis agitata revolvuntur.<emph.end type="italics"/> 114 </s> </p> <p type="main"> <s>SECT. IX. <emph type="italics"/>De Motu corporum in Orbibus mobilibus, deque <lb/>Motu Ap&longs;idum.<emph.end type="italics"/> 121 </s> </p> <p type="main"> <s>SECT. X. <emph type="italics"/>De Motu corporum in Superficiebus datis, deque <lb/>Funependulorum Motu reciproco.<emph.end type="italics"/> 132 </s> </p> <p type="main"> <s>SECT. XI. <emph type="italics"/>De Motu corporum Viribus centripetis &longs;e mutuo pe­<lb/>tentium.<emph.end type="italics"/> 147 </s> </p> <p type="main"> <s>SECT. XII. <emph type="italics"/>De corporum Sphæricorum Viribus attractivis.<emph.end type="italics"/> 173 </s> </p> <pb/> <p type="main"> <s>SECT. XIII. <emph type="italics"/>De corporum non Sphæricorum Viribus attracti­<lb/>vis.<emph.end type="italics"/> 192 </s> </p> <p type="main"> <s>SECT. XIV. <emph type="italics"/>De Motu corporum Minimorum, quæ Veribus cen­<lb/>tripetis ad &longs;ingulas Magni alicujus corporis partes ten­<lb/>dentibus agitantur.<emph.end type="italics"/> 203 </s> </p> <p type="main"> <s><emph type="center"/>DE MOTU CORPORUM LIBER SECUNDUS.<emph.end type="center"/></s> </p> <p type="main"> <s>SECT. I. <emph type="italics"/>DE Motu corporum quibus re&longs;i&longs;titur in ratione <lb/>Velocitatis.<emph.end type="italics"/> 211 </s> </p> <p type="main"> <s>SECT. II. <emph type="italics"/>De Motu corporum quibus re&longs;i&longs;titur in duplicata ra­<lb/>tione Velocitatis.<emph.end type="italics"/> 220 </s> </p> <p type="main"> <s>SECT. III. <emph type="italics"/>De Motu corporum quibus re&longs;i&longs;titur partim in ratione <lb/>Velocitatis, partim in eju&longs;dem ratione duplicata.<emph.end type="italics"/> 245 </s> </p> <p type="main"> <s>SECT. IV. <emph type="italics"/>De corporum Circulari motu in Mediis re&longs;i&longs;tentibus.<emph.end type="italics"/><lb/>253 </s> </p> <p type="main"> <s>SECT. V. <emph type="italics"/>De den&longs;itate & compre&longs;&longs;ione Fluidorum, deque Hy­<lb/>dro&longs;tatica.<emph.end type="italics"/> 260 </s> </p> <p type="main"> <s>SECT. VI. <emph type="italics"/>De Motu & Re&longs;i&longs;tentia corporum Funependulorum.<emph.end type="italics"/><lb/>272 </s> </p> <p type="main"> <s>SECT. VII. <emph type="italics"/>De motu Fluidorum & re&longs;i&longs;tentia Projectilium.<emph.end type="italics"/> 294 </s> </p> <p type="main"> <s>SECT. VIII. <emph type="italics"/>De motu per Fluida propagato.<emph.end type="italics"/> 329 </s> </p> <p type="main"> <s>SECT. IX. <emph type="italics"/>De motu Circulari Fluidorum.<emph.end type="italics"/> 345 </s> </p> <p type="main"> <s><emph type="center"/>DE MUNDI SYSTEMATE LIBER TERTIUS.<emph.end type="center"/></s> </p> <p type="main"> <s>REGULÆ PHILOSOPHANDI 357 </s> </p> <p type="main"> <s>PHÆNOMENA 359 </s> </p> <p type="main"> <s>PROPOSITIONES 362 </s> </p> <p type="main"> <s>SCHOLIUM GENERALE. 481 </s> </p> <pb/> <p type="main"> <s><emph type="center"/>PHILOSOPHIÆ <lb/>NATURALIS <lb/>Principia <lb/>MATHEMATICA.<emph.end type="center"/><lb/><gap desc="hr tag"/></s> </p> <p type="main"> <s><emph type="center"/>DEFINITIONES.<emph.end type="center"/><lb/><gap desc="hr tag"/></s> </p> <p type="main"> <s><emph type="center"/>DEFINITIO I.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Quantitas Materiæ e&longs;t men&longs;ura eju&longs;dem orta ex illius Den&longs;itate & <lb/>Magnitudine conjunctim.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>AER, den&longs;itate duplicata, in &longs;patio etiam duplicato fit qua­<lb/>druplus; in triplicato &longs;extuplus. </s> <s>Idem intellige de Nive & <lb/>Pulveribus per compre&longs;&longs;ionem vel liquefactionem conden­<lb/>&longs;atis. </s> <s>Et par e&longs;t ratio corporum omnium, quæ per cau&longs;as qua&longs;cun­<lb/>que diver&longs;imode conden&longs;antur. </s> <s>Medii interea, &longs;i quod fuerit, in­<lb/>ter&longs;titia partium libere pervadentis, hic nullam rationem habeo. </s> <s><lb/>Hanc autem Quantitatem &longs;ub nomine Corporis vel Ma&longs;&longs;æ in &longs;e­<lb/>quentibus pa&longs;&longs;im intelligo. </s> <s>Innote&longs;cit ea per corporis cuju&longs;que <lb/>Pondus. </s> <s>Nam Ponderi proportionalem e&longs;&longs;e reperi per experi­<lb/>menta Pendulorum accurati&longs;&longs;ime in&longs;tituta, uti po&longs;thac docebitur. </s> </p> <p type="main"> <s><emph type="center"/>DEFINITIO II.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Quantitas Motus e&longs;t men&longs;ura eju&longs;dem orta ex Velocitate & Quan­<lb/>titate Materiæ conjunctim.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Motus totius e&longs;t &longs;umma motuum in partibus &longs;ingulis; adeoque <lb/>in corpore duplo majore æquali cum velocitate duplus e&longs;t, & du­<lb/>pla cum velocitate quadruplus. </s> </p> <pb pagenum="2"/> <p type="main"> <s><emph type="center"/>DEFINITIO III.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Materiæ Vis In&longs;ita e&longs;t potentia re&longs;i&longs;tendi, qua corpus unumquodque, <lb/>quantum in &longs;e e&longs;t, per&longs;everat in &longs;tatu &longs;uo vel quie&longs;cendi vel <lb/>movendi uniformiter in directum.<emph.end type="italics"/></s> </p> <p type="main"> <s>Hæc &longs;emper proportionalis e&longs;t &longs;uo corpori, neque differt quic­<lb/>quam ab Inertia ma&longs;&longs;æ, ni&longs;i in modo concipiendi. </s> <s>Per inertiam <lb/>materiæ, fit ut corpus omne de &longs;tatu &longs;uo vel quie&longs;cendi vel moven­<lb/>di difficulter deturbetur. </s> <s>Unde etiam vis in&longs;ita nomine &longs;ignifican­<lb/>ti&longs;&longs;imo Vis Inertiæ dici po&longs;&longs;it. </s> <s>Exercet vero corpus hanc vim &longs;olum­<lb/>modo in mutatione &longs;tatus &longs;ui per vim aliam in &longs;e impre&longs;&longs;am facta; <lb/><expan abbr="e&longs;tq;">e&longs;tque</expan> exercitium ejus &longs;ub diver&longs;o re&longs;pectu & Re&longs;i&longs;tentia & Impetus: <lb/>re&longs;i&longs;tentia, quatenus corpus ad con&longs;ervandum &longs;tatum &longs;uum relucta­<lb/>tur vi impre&longs;&longs;æ; impetus, quatenus corpus idem, vi re&longs;i&longs;tentis ob­<lb/>&longs;taculi difficulter cedendo, conatur &longs;tatum ejus mutare. </s> <s>Vulgus <lb/>re&longs;i&longs;tentiam quie&longs;centibus & impetum moventibus tribuit: &longs;ed mo­<lb/>tus & quies, uti vulgo concipiuntur, re&longs;pectu &longs;olo di&longs;tinguuntur <lb/>ab invicem; <expan abbr="neq;">neque</expan> &longs;emper vere quie&longs;cunt quæ vulgo tanquam quie­<lb/>&longs;centia &longs;pectantur. </s> </p> <p type="main"> <s><emph type="center"/>DEFINITIO IV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Vis Impre&longs;&longs;a e&longs;t actio in corpus exercita, ad mutandum ejus &longs;tatum <lb/>vel quie&longs;cendi vel movendi uniformiter in directum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Con&longs;i&longs;tit hæc vis in actione &longs;ola, neque po&longs;t actionem permanet <lb/>in corpore. </s> <s>Per&longs;everat enim corpus in &longs;tatu omni novo per &longs;olam <lb/>vim inertiæ. </s> <s>E&longs;t autem vis impre&longs;&longs;a diver&longs;arum originum, ut ex <lb/>Ictu, ex Pre&longs;&longs;ione, ex vi Centripeta. </s> </p> <p type="main"> <s><emph type="center"/>DEFINITIO V.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Vis Centripeta e&longs;t, qua corpora ver&longs;us punctum aliquod tanquam ad <lb/>Centrum undique trahuntur, impelluntur, vel utcunq tendunt.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Hujus generis e&longs;t Gravitas, qua corpora tendunt ad centrum ter­<lb/>ræ; Vis Magnetica, qua ferrum petit magnetem; & Vis illa, <lb/><expan abbr="quæcunq;">quæcunque</expan> &longs;it, qua Planetæ perpetuo retrahuntur a motibus rectili­<lb/>neis, & in lineis curvis revolvi coguntur. </s> <s>Lapis, in funda circum- <pb pagenum="3"/>actus, a circumagente manu abire conatur; & conatu &longs;uo fundam <lb/>di&longs;tendit, <expan abbr="eoq;">eoque</expan> fortius quo celerius revolvitur; &, quamprimum di­<lb/>mittitur, avolat. </s> <s>Vim conatui illi contrariam, qua funda lapidem <lb/>in manum perpetuò retrahit & in orbe retinet, quoniam in manum <lb/>ceu orbis centrum dirigitur, Centripetam appello. </s> <s>Et par e&longs;t ratio <lb/>corporum omnium, quæ in gyrum aguntur. </s> <s>Conantur ea omnia a <lb/>centris orbium recedere; & ni&longs;i ad&longs;it vis aliqua conatui i&longs;ti contra­<lb/>ria, qua cohibeantur & in orbibus retineantur, quamque ideò Centri­<lb/>petam appello, abibunt in rectis lineis uniformi cum motu. </s> <s>Pro­<lb/>jectile, &longs;i vi Gravitatis de&longs;titueretur, non deflecteretur in terram, &longs;ed <lb/>in linea recta abiret in cælos; idque uniformi cum motu, &longs;i modo <lb/>aeris re&longs;i&longs;tentia tolleretur. </s> <s>Per gravitatem &longs;uam retrahitur a cur&longs;u <lb/>rectilineo & in terram perpetuo flectitur, idque magis vel minus <lb/>pro gravitate &longs;ua & velocitate motus. </s> <s>Quo minor erit ejus gravitas pro quantitate materiæ vel major &c. </s> <s><lb/>vel major velocitas quacum projicitur, eo minus deviabit a cur&longs;u <lb/>rectilineo & longius perget. </s> <s>Si Globus plumbeus, data cum velo­<lb/>citate &longs;ecundum lineam horizontalem a montis alicujus vertice vi <lb/>pulveris tormentarii projectus, pergeret in linea curva ad di&longs;tantiam <lb/>duorum milliarium, priu&longs;quam in terram decideret: hic dupla cum <lb/>velocitate qua&longs;i duplo longius pergeret, & decupla cum velocitate <lb/>qua&longs;i decuplo longius: &longs;i modo aeris re&longs;i&longs;tentia tolleretur. </s> <s>Et augendo <lb/>velocitatem augeri po&longs;&longs;et pro lubitu di&longs;tantia in quam projiceretur, <lb/>& minui curvatura lineæ quam de&longs;criberet, ita ut tandem caderet <lb/>ad di&longs;tantiam graduum decem vel triginta vel nonaginta; vel eriam <lb/>ut terram totam circuiret priu&longs;quam caderet; vel denique ut in <lb/>terram nunquam caderet, &longs;ed in cælos abiret & motu abeundi per­<lb/>geret in infinitum. </s> <s>Et eadem ratione, qua Projectile vi gravitatis <lb/>in orbem flecti po&longs;&longs;et & terram totam circuire, pote&longs;t & Luna vel <lb/>vi gravitatis, &longs;i modo gravis &longs;it, vel alia quacunque vi, qua in ter­<lb/>ram urgeatur, retrahi &longs;emper a cur&longs;u rectilineo terram ver&longs;us, & <lb/>in orbem &longs;uum flecti: & ab&longs;que tali vi Luna in orbe &longs;uo retineri <lb/>non pote&longs;t. </s> <s>Hæc vis, &longs;i ju&longs;to minor e&longs;&longs;et, non &longs;atis flecteret Lunam <lb/>de cur&longs;u rectilineo: &longs;i ju&longs;to major, plus &longs;atis flecteret, ac de orbe <lb/>terram ver&longs;us deduceret. </s> <s>Requiritur quippe, ut &longs;it ju&longs;tæ magnitudinis: <lb/>& Mathematicorum e&longs;t invenire Vim, qua corpus in dato quovis <lb/>orbe data cum velocitate accurate retineri po&longs;&longs;it; & vici&longs;&longs;im inve­<lb/>nire Viam curvilineam, in quam corpus e dato quovis loco data <lb/>cum velocitate egre&longs;&longs;um a data vi flectatur. </s> <s>E&longs;t autem vis hujus cen­<lb/>tripetæ Quantitas trium generum, Ab&longs;oluta, Acceleratrix, & Motrix. </s> </p> <pb pagenum="4"/> <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/>ES.</s> </p> <p type="main"> <s><emph type="center"/>DEFINITIO VI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Vis centripetæ Quantitas Ab&longs;oluta e&longs;t men&longs;ura eju&longs;dem major vel minor <lb/>pro Efficacia cau&longs;æ eam propagantis a centro per regiones in circuitu.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Ut vis Magnetica pro mole magnetis vel inten&longs;ione virtutis major <lb/>in uno magnete, minor in alio. </s> </p> <p type="main"> <s><emph type="center"/>DEFINITIO VII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Vis centripetæ Quantitas Acceleratrix e&longs;t ip&longs;ius men&longs;ura Velocitati <lb/>proportionalis, quam dato tempore generat.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Uti Virtus magnetis eju&longs;dem major in minori di&longs;tantia, minor <lb/>in majori: vel vis Gravitans major in vallibus, minor in cacumini­<lb/>bus præaltorum montium, atque adhuc minor (ut po&longs;thac patebit) <lb/>in majoribus di&longs;tantiis a globo terræ; in æqualibus autem di&longs;tan­<lb/>tiis eadem undique, propterea quod corpora omnia cadentia (gra­<lb/>via an levia, magna an parva) &longs;ublata Aeris re&longs;i&longs;tentia, æqualiter <lb/>accelerat. </s> </p> <p type="main"> <s><emph type="center"/>DEFINITIO VIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Vis centripetæ Quantitas Motrix e&longs;t ip&longs;ius men&longs;ura proportionalis. </s> <s><lb/>Motui, quem dato tempore generat.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Uti Pondus majus in majore corpore, minus in minore; inque <lb/>corpore eodem majus prope terram, minus in cælis. </s> <s>Hæc Quantitas <lb/>e&longs;t corporis totius centripetentia &longs;eu propen&longs;io in centrum, & (ut ita <lb/>dicam) Pondus; & innote&longs;cit &longs;emper per vim ip&longs;i contrariam & æ­<lb/>qualem, qua de&longs;cen&longs;us corporis impediri pote&longs;t. </s> </p> <p type="main"> <s>Ha&longs;ce virium quantitates brevitatis gratia nominare licet vires <lb/>motrices, acceleratrices, & ab&longs;olutas; & di&longs;tinctionis gratia referre ad <lb/>Corpora, centrum petentia, ad corporum Loca, & ad Centrum virium: <lb/>nimirum vim motricem ad Corpus, tanquam conatum & propen&longs;io­<lb/>nem totius in centrum ex propen&longs;ionibus omnium partium compo&longs;i­<lb/>tam; & vim acceleratricem ad Locum corporis, tanquam efficaciam <lb/>quandam, de centro per loca &longs;ingula in circuitu diffu&longs;am, ad movenda <lb/>corpora quæ in ip&longs;is &longs;unt; vim autem ab&longs;olutam ad Centrum, tan­<lb/>quam cau&longs;a aliqua præditum, &longs;ine qua vires motrices non propa­<lb/>gantur per regiones in circuitu; &longs;ive cau&longs;a illa &longs;it corpus aliquod <lb/>centrale (quale e&longs;t Magnes in centro vis magneticæ, vel Terra in <pb pagenum="5"/>centro vis gravitantis) &longs;ive alia aliqua quæ non apparet. </s> <s>Mathe­<lb/>maticus duntaxat e&longs;t hic conceptus. </s> <s>Nam virium cau&longs;as & &longs;edes phy­<lb/>&longs;icas jam non expendo. </s> </p> <p type="main"> <s>E&longs;t igitur vis acceleratrix ad vim motricem ut celeritas ad mo­<lb/>tum. </s> <s>Oritur enim quantitas motus ex celeritate ducta in quanti­<lb/>tatem materiæ, & vis motrix ex vi acceleratrice ducta in quantita­<lb/>tem eju&longs;dem materiæ. </s> <s>Nam &longs;umma actionum vis acceleratricis in <lb/>&longs;ingulas corporis particulas e&longs;t vis motrix totius. </s> <s>Unde juxta <lb/>&longs;uperficiem Terræ, ubi gravitas acceleratrix &longs;eu vis gravitans in <lb/>corporibus univer&longs;is eadem e&longs;t, gravitas motrix &longs;eu pondus e&longs;t ut <lb/>corpus: at &longs;i in regiones a&longs;cendatur ubi gravitas acceleratrix fit mi­<lb/>nor, pondus pariter minuetur, eritque &longs;emper ut corpus in <lb/>gravitatem acceleratricem ductum. </s> <s>Sic in regionibus ubi gravitas <lb/>acceleratrix duplo minor e&longs;t, pondus corporis duplo vel triplo <lb/>minoris erit quadruplo vel &longs;extuplo minus. </s> </p> <p type="main"> <s>Porro attractiones & impul&longs;us eodem &longs;en&longs;u acceleratrices & mo­<lb/>trices nomino. </s> <s>Voces autem Attractionis, Impul&longs;us, vel Propen­<lb/>&longs;ionis cuju&longs;cunque in centrum, indifferenter & pro &longs;e mutuo pro­<lb/>mi&longs;cue u&longs;urpo; has vires non Phy&longs;ice &longs;ed Mathematice tantum con­<lb/>&longs;iderando. </s> <s>Unde caveat lector, ne per huju&longs;modi voces cogitet me <lb/>&longs;peciem vel modum actionis cau&longs;amve aut rationem Phy&longs;icam ali­<lb/>cubi definire, vel centris (quæ &longs;unt puncta Mathematica) vires <lb/>vere & Phy&longs;ice tribuere; &longs;i forte aut centra trahere, aut vires cen­<lb/>trorum e&longs;&longs;e dixero. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Hactenus voces minus notas, quo &longs;en&longs;u in &longs;equentibus acci­<lb/>piendæ &longs;int, explicare vi&longs;um e&longs;t. </s> <s>Nam Tempus, Spatium, Locum <lb/>& Motum, ut omnibus noti&longs;&longs;ima, non definio. </s> <s>Notandum tamen, quod <lb/>vulgus quantitates ha&longs;ce non aliter quam ex relatione ad &longs;en&longs;ibilia <lb/>concipiat. </s> <s>Et inde oriuntur præjudicia quædam, quibus tollendis <lb/>convenit ea&longs;dem in ab&longs;olutas & relativas, veras & apparentes, ma­<lb/>thematicas & vulgares di&longs;tingui. </s> </p> <p type="main"> <s>I. </s> <s>Tempus Ab&longs;olutum, verum, & mathematicum, in &longs;e & natura <lb/>&longs;ua <expan abbr="ab&longs;q;">ab&longs;que</expan> relatione ad externum quodvis, æquabiliter fluit, <expan abbr="alioq;">alioque</expan> <lb/>nomine dicitur Duratio: Relativum, apparens, & vulgare e&longs;t &longs;en&longs;ibilis <lb/>& externa quævis Durationis per motum men&longs;ura (&longs;eu accurata <lb/>&longs;eu inæquabilis) qua vulgus vice veri temporis utitur; ut Hora, <lb/>Dies, Men&longs;is, Annus. <pb pagenum="6"/> <arrow.to.target n="note2"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note2"></margin.target>NI­<lb/>ES.</s> </p> <p type="main"> <s>II. </s> <s>Spatium Ab&longs;olutum, natura &longs;ua ab&longs;que relatione ad externum <lb/>quodvis, &longs;emper manet &longs;imilare & immobile: Relativum e&longs;t &longs;patii <lb/>hujus men&longs;ura &longs;eu dimen&longs;io quælibet mobilis, quæ a &longs;en&longs;ibus no&longs;tris <lb/>per &longs;itum &longs;uum ad corpora definitur, & a vulgo pro &longs;patio immo­<lb/>bili u&longs;urpatur: uti dimen&longs;io &longs;patii &longs;ubterranei, aerei vel cæle&longs;tis <lb/>definita per &longs;itum &longs;uum ad Terram. </s> <s>Idem &longs;unt &longs;patium ab&longs;olutum <lb/>& relativum, &longs;pecie & magnitudine; &longs;ed non permanent idem &longs;em­<lb/>per numero. </s> <s>Nam &longs;i Terra, verbi gratia, movetur; &longs;patium Aeris <lb/>no&longs;tri, quod relative & re&longs;pectu Terræ &longs;emper manet idem, nunc <lb/>erit una pars &longs;patii ab&longs;oluti in quam Aer tran&longs;it, nunc alia pars ejus; <lb/>& &longs;ic ab&longs;olute mutabitur perpetuo. </s> </p> <p type="main"> <s>III. </s> <s>Locus e&longs;t pars &longs;patii quam corpus occupat, <expan abbr="e&longs;tq;">e&longs;tque</expan> pro ratione <lb/>&longs;patii vel Ab&longs;olutus vel Relativus. </s> <s>Pars, inquam, &longs;patii; non Situs <lb/>corporis, vel Superficies ambiens. </s> <s>Nam &longs;olidorum æqualium <lb/>æquales &longs;emper &longs;unt loci; Superficies autem ob di&longs;&longs;imilitudinem <lb/>figurarum ut plurimum inæquales &longs;unt; Situs vero proprie loquen­<lb/>do quantitatem non habent, <expan abbr="neq;">neque</expan> tam &longs;unt loca quam affectiones <lb/>locorum. </s> <s>Motus totius idem e&longs;t cum &longs;umma motuum partium, <lb/>hoc e&longs;t, tran&longs;latio totius de &longs;uo loco eadem e&longs;t cum &longs;umma tran&longs;la­<lb/>tionum partium de locis &longs;uis; <expan abbr="adeoq;">adeoque</expan> locus totius idem cum &longs;umma <lb/>locorum partium, & propterea internus & in corpore toto. </s> </p> <p type="main"> <s>IV. </s> <s>Motus Ab&longs;olutus e&longs;t tran&longs;latio corporis de loco ab&longs;oluto in <lb/>locum ab&longs;olutum, Relativus de relativo in relativum. </s> <s>Sic in navi <lb/>quæ velis pa&longs;&longs;is fertur, relativus corporis Locus e&longs;t navigii regio illa <lb/>in qua corpus ver&longs;atur, &longs;eu cavitatis totius pars illa quam corpus <lb/>implet, <expan abbr="quæq;">quæque</expan> adeo movetur una cum navi: & Quies relativa e&longs;t <lb/>perman&longs;io corporis in eadem illa navis regione vel parte cavita­<lb/>tis. </s> <s>At quies Vera e&longs;t perman&longs;io corporis in eadem parte &longs;patii <lb/>illius immoti in qua navis ip&longs;a una cum cavitate &longs;ua & contentis <lb/>univer&longs;is movetur. </s> <s>Unde &longs;i Terra vere quie&longs;cit, corpus quod rela­<lb/>tive quie&longs;cit in navi, movebitur vere & ab&longs;olute ea cum velocitate <lb/>qua navis movetur in Terra. </s> <s>Sin Terra etiam movetur; orietur <lb/>verus & ab&longs;olutus corporis motus, partim ex Terræ motu vero in <lb/>&longs;patio immoto, partim ex navis motu relativo in Terra: & &longs;i cor­<lb/>pus etiam movetur relative in navi; orietur verus ejus motus, par­<lb/>tim ex vero motu Terræ in &longs;patio immoto, partim ex relativis mo­<lb/>tibus tum navis in Terra, tum corporis in navi; & ex his motibus <lb/>relativis orietur corporis motus relativus in Terra. </s> <s>Ut &longs;i Terræ pars <lb/>illa, ubi navis ver&longs;atur, moveatur vere in orientem cum velocitate <lb/>partium 10010; & velis <expan abbr="ventoq;">ventoque</expan> feratur navis in occidentem cum <lb/>velocitate partium decem; Nauta autem ambulet in navi ori- <pb pagenum="7"/>entem ver&longs;us cum velocitatis parte una: movebitur Nauta vere & <lb/>ab&longs;olute in &longs;patio immoto cum velocitatis partibus 10001 in o­<lb/>rientem, & relative in terra occidentem ver&longs;us cum velocitatis <lb/>partibus novem. </s> </p> <p type="main"> <s>Tempus Ab&longs;olutum a relativo di&longs;tinguitur in A&longs;tronomia per Æ­<lb/>quationem temporis vulgi. </s> <s>Inæquales enim &longs;unt dies Naturales, <lb/>qui vulgo tanquam æquales promen&longs;ura temporis habentur. </s> <s>Hanc <lb/>inæqualitatem corrigunt A&longs;tronomi, ut ex veriore tempore men&longs;urent motus &c. </s> <s><lb/>motus cæle&longs;tes. </s> <s>Po&longs;&longs;ibile e&longs;t, ut nullus &longs;it motus æquabilis quo <lb/>Tempus accurate men&longs;uretur. </s> <s>Accelerari & retardari po&longs;&longs;unt motus <lb/>omnes, &longs;ed fluxus temporis Ab&longs;oluti mutari nequit. </s> <s>Eadem e&longs;t du­<lb/>ratio &longs;eu per&longs;everantia exi&longs;tentiæ rerum; &longs;ive motus &longs;int celeres, &longs;ive <lb/>tardi, &longs;ive nulli: proinde hæc a men&longs;uris &longs;uis &longs;en&longs;ibilibus merito <lb/>di&longs;tinguitur, & ex ii&longs;dem colligitur per Æquationem A&longs;tronomi­<lb/>cam. </s> <s>Hujus autem æquationis in determinandis Phænomenis ne­<lb/>ce&longs;&longs;itas, tum per experimentum Horologii O&longs;cillatorii, tum etiam <lb/>per eclip&longs;es Satellitum Jovis evincitur. </s> </p> <p type="main"> <s>Ut partium Temporis ordo e&longs;t immutabilis, &longs;ic etiam ordo par­<lb/>tium Spatii. </s> <s>Moveantur hæ de locis &longs;uis, & movebuntur (ut ita <lb/>dicam) de &longs;eip&longs;is. </s> <s>Nam tempora & &longs;patia &longs;unt &longs;ui ip&longs;orum & <lb/>rerum omnium qua&longs;i Loca. </s> <s>In Tempore quoad ordinem &longs;ucce&longs;&longs;i­<lb/>onis; in Spatio quoad ordinem &longs;itus locantur univer&longs;a. </s> <s>De illo­<lb/>rum e&longs;&longs;entia e&longs;t ut &longs;int Loca: & loca primaria moveri ab&longs;urdum <lb/>e&longs;t. </s> <s>Hæc &longs;unt igitur ab&longs;oluta Loca; & &longs;olæ tran&longs;lationes de his lo­<lb/>cis &longs;unt ab&longs;oluti Motus. </s> </p> <p type="main"> <s>Verum quoniam hæ Spatii partes videri nequeunt, & ab invi­<lb/>cem per &longs;en&longs;us no&longs;tros di&longs;tingui; earum vice adhibemus men&longs;uras <lb/>&longs;en&longs;ibiles. </s> <s>Ex po&longs;itionibus enim & di&longs;tantiis rerum a corpore ali­<lb/>quo, quod &longs;pectamus ut immobile, de&longs;inimus loca univer&longs;a: deinde <lb/>etiam & omnes motus æ&longs;timamus cum re&longs;pectu ad prædicta loca, <lb/>quatenus corpora ab ii&longs;dem transferri concipimus. </s> <s>Sic vice loco­<lb/>rum & motuum ab&longs;olutorum relativis utimur; nec incommode in <lb/>rebus humanis: in Philo&longs;ophicis autem ab&longs;trahendum e&longs;t a &longs;en&longs;ibus. </s> <s><lb/>Fieri etenim pote&longs;t, ut nullum revera quie&longs;cat corpus, ad quod loca <lb/>motu&longs;que referantur. </s> </p> <p type="main"> <s>Di&longs;tinguuntur autem Quies & Motus ab&longs;oluti & relativi ab invi­<lb/>cem per Proprietates &longs;uas & Cau&longs;as & Effectus. </s> <s>Quietis proprietas <lb/>e&longs;t, quod corpora vere quie&longs;centia quie&longs;cunt inter &longs;e. </s> <s>Ideoque <lb/>cum po&longs;&longs;ibile &longs;it, ut corpus aliquod in regionibus Fixarum, aut longe <lb/>ultra, quie&longs;cat ab&longs;olute; &longs;ciri autem non po&longs;&longs;it ex &longs;itu corporum <lb/>ad invicem in regionibus no&longs;tris, horumne aliquod ad longin-</s> </p> <pb pagenum="8"/> <p type="main"> <s> <arrow.to.target n="note3"></arrow.to.target><lb/>quum illud datam po&longs;itionem &longs;ervet necne; quies vera ex horum <lb/>&longs;itu inter &longs;e definiri nequit. </s> </p> <p type="margin"> <s><margin.target id="note3"></margin.target>NI­<lb/>ES.</s> </p> <p type="main"> <s>Motus proprietas e&longs;t, quod partes, quæ datas &longs;ervant po&longs;itiones <lb/>ad tota, participant motus eorundem totorum. </s> <s>Nam Gyrantium <lb/>partes omnes conantur recedere ab axe motus, & Progredientium <lb/>impetus oritur ex conjuncto impetu partium &longs;ingularum. </s> <s>Motis <lb/>igitur corporibus ambientibus, moventur quæ in ambientibus rela­<lb/>tive quie&longs;cunt. </s> <s>Et propterea motus verus & ab&longs;olutus definiri ne­<lb/>quit per tran&longs;lationem e vicinia corporum, quæ tanquam quie&longs;cen­<lb/>tia &longs;pectantur. </s> <s>Debent enim corpora externa non &longs;olum tanquam qui­<lb/>e&longs;centia &longs;pectari, &longs;ed etiam vere quie&longs;cere. </s> <s>Alioquin inclu&longs;a om­<lb/>nia, præter tran&longs;lationem e vicinia ambientium, participabunt <lb/>etiam ambientium motus veros; & &longs;ublata illa tran&longs;latione non <lb/>vere quie&longs;cent, &longs;ed tanquam quie&longs;centia &longs;olummodo &longs;pectabun­<lb/>tur. </s> <s>Sunt enim ambientia ad inclu&longs;a, ut totius pars exterior ad <lb/>partem interiorem, vel ut cortex ad nucleum. </s> <s>Moto autem cor­<lb/>tice, nucleus etiam, <expan abbr="ab&longs;q;">ab&longs;que</expan> tran&longs;latione de vicinia corticis, ceu pars <lb/>totius movetur. </s> </p> <p type="main"> <s>Præcedenti proprietati affinis e&longs;t, quod moto Loco movetur una <lb/>Locatum: adeoque corpus, quod de loco moto movetur, participat <lb/>etiam loci &longs;ui motum. </s> <s>Motus igitur omnes, qui de locis motis <lb/>fiunt, &longs;unt partes &longs;olummodo motuum integrorum & ab&longs;olutorum: <lb/>& motus omnis integer componitur ex motu corporis de loco &longs;uo <lb/>primo, & motu loci hujus de loco &longs;uo, & &longs;ic deinceps; u&longs;que dum <lb/>perveniatur ad locum immotum, ut in exemplo Nautæ &longs;upra me­<lb/>morato. </s> <s>Unde motus integri & ab&longs;oluti non ni&longs;i per loca immota <lb/>definiri po&longs;&longs;unt: & propterea hos ad loca immota, relativos ad mo­<lb/>bilia &longs;upra retuli. </s> <s>Loca autem immota non &longs;unt, ni&longs;i quæ omnia <lb/>ab infinito in infinitum datas &longs;ervant po&longs;itiones ad invicem; atque <lb/>adeo &longs;emper manent immota, &longs;patiumque con&longs;tituunt quod Immo­<lb/>bile appello. </s> </p> <p type="main"> <s>Cau&longs;æ, quibus motus veri & relativi di&longs;tinguuntur ab invicem, <lb/>&longs;unt Vires in corpora impre&longs;&longs;æ ad motum generandum. </s> <s>Motus <lb/>verus nec generatur nec mutatur, ni&longs;i per vires in ip&longs;um corpus mo­<lb/>tum impre&longs;&longs;as: at motus relativus generari & mutari pote&longs;t <expan abbr="ab&longs;q;">ab&longs;que</expan> <lb/>viribus impre&longs;&longs;is in hoc corpus. </s> <s>Sufficit enim ut imprimantur in <lb/>alia &longs;olum corpora ad quæ fit relatio, ut iis cedentibus mutetur <lb/>relatio illa in qua hujus quies vel motus relativus con&longs;i&longs;tit. </s> <s>Rur­<lb/>&longs;um motus verus a viribus in corpus motum impre&longs;&longs;is &longs;emper muta­<lb/>tur; at motus relativus ab his viribus non mutatur nece&longs;&longs;ario. </s> <s>Nam <lb/>&longs;i eædem vires in alia etiam corpora, ad quæ &longs;it relatio, &longs;ic impri- <pb pagenum="9"/>mantur ut &longs;itus relativus con&longs;ervetur, con&longs;ervabitur relatio in qua <lb/>motus relativus con&longs;i&longs;tit. </s> <s>Mutari igitur pote&longs;t motus omnis relati­<lb/>vus ubi verus con&longs;ervatur, & con&longs;ervari ubi verus mutatur; & prop­<lb/>terea motus verus in eju&longs;modi relationibus minime con&longs;i&longs;tit. </s> </p> <p type="main"> <s>Effectus quibus motus ab&longs;oluti & relativi di&longs;tinguuntur ab invi­<lb/>cem, &longs;unt vires recedendi ab axe motus circularis. </s> <s>Nam in motu <lb/>circulari nude relativo hæ vires nullæ &longs;unt, in vero autem & ab&longs;o­<lb/>luto majores vel minores pro quantitate motus. </s> <s>Si pendeat &longs;itula <lb/>a filo prælongo, agaturque perpetuo in orbem, donec filum a con­<lb/>tor&longs;ione admodum rige&longs;cat, dein impleatur aqua, & una cum aqua <lb/>quie&longs;cat; tum vi aliqua &longs;ubitanea agatur motu contrario in orbem, <lb/>& filo &longs;e relaxante, diutius per&longs;everet in hoc motu; &longs;uperficies a­<lb/>quæ &longs;ub initio plana erit, quemadmodum ante motum va&longs;is: at <lb/>po&longs;tquam, vi in aquam paulatim impre&longs;&longs;a, effecit vas, ut hæc quoque <lb/>&longs;en&longs;ibiliter revolvi incipiat; recedet ip&longs;a paulatim a medio, a&longs;cen­<lb/>detque ad latera va&longs;is, figuram concavam induens, (ut ip&longs;e exper­<lb/>tus &longs;um) & incitatiore &longs;emper motu a&longs;cendet magis & magis, do­<lb/>nec revolutiones in æqualibus cum va&longs;e temporibus peragendo, <lb/>quie&longs;cat in eodem relative. </s> <s>Indicat hic a&longs;cen&longs;us conatum rece­<lb/>dendi ab axe motus, & per talem conatum innote&longs;cit & men&longs;ura­<lb/>tur motus aquæ circularis verus & ab&longs;olutus, motuique relativo <lb/>hic omnino contrarius. </s> <s>Initio, ubi maximus erat aquæ motus rela­<lb/>tivus in va&longs;e, motus ille nullum excitabat conatum recedendi ab <lb/>axe: aqua non petebat circumferentiam a&longs;cendendo ad latera va­<lb/>&longs;is, &longs;ed plana manebat, & propterea motus illius circularis verus <lb/>nondum inceperat. </s> <s>Po&longs;tea vero, ubi aquæ motus relativus decre­<lb/>vit, a&longs;cen&longs;us ejus ad latera va&longs;is indicabat conatum recedendi ab <lb/>axe; atque hic conatus mon&longs;trabat motum illius circularem verum <lb/>perpetuo cre&longs;centem, ac tandem maximum factum ubi aqua quie­<lb/>&longs;cebat in va&longs;e relative. </s> <s>Igitur conatus i&longs;te non pendet a tran&longs;la­<lb/>tione aquæ re&longs;pectu corporum ambientium, & propterea motus cir­<lb/>cularis verus per tales tran&longs;lationes definiri nequit. </s> <s>Unicus e&longs;t cor­<lb/>poris cuju&longs;que revolventis motus vere circularis, conatui unico tan­<lb/>quam proprio & adæquato effectui re&longs;pondens: motus autem rela­<lb/>tivi pro variis relationibus ad externa innumeri &longs;unt; & relationum <lb/>in&longs;tar, effectibus veris omnino de&longs;tituuntur, ni&longs;i quatenus verum <lb/>illum & unicum motum participant. </s> <s>Unde & in Sy&longs;temate eorum <lb/>qui Cælos no&longs;tros infra Cælos Fixarum in orbem revolvi volunt, <lb/>& Planetas &longs;ecum deferre; &longs;ingulæ Cælorum partes, & Planetæ <lb/>qui relative quidem in Cælis &longs;uis proximis quie&longs;cunt, moven- <pb pagenum="10"/> <arrow.to.target n="note4"></arrow.to.target><lb/>tur vere. </s> <s>Mutant enim po&longs;itiones &longs;uas ad invicem (&longs;ecus quam fit <lb/>in vere quie&longs;centibus) unaque cum cælis delati participant eorum <lb/>motus, & ut partes revolventium totorum, ab eorum axibus rece­<lb/>dere conantur. </s> </p> <p type="margin"> <s><margin.target id="note4"></margin.target>NI­<lb/>ES.</s> </p> <p type="main"> <s>Igitur quantitates relativæ non &longs;unt eæ ip&longs;æ quantitates, quarum <lb/>nomina præ &longs;e ferunt, &longs;ed earum men&longs;uræ illæ &longs;en&longs;ibiles (veræ an <lb/>errantes) quibus vulgus loco quantitatum men&longs;uratarum utitur. </s> <s>At <lb/>&longs;i ex u&longs;u definiendæ &longs;unt verborum &longs;ignificationes; per nomina il­<lb/>la Temporis, Spatii, Loci & Motus proprie intelligendæ erunt hæ <lb/>men&longs;uræ; & &longs;ermo erit in&longs;olens & pure Mathematicus, &longs;i quantita­<lb/>tes men&longs;uratæ hic intelligantur. </s> <s>Proinde vim inferunt Sacris <lb/>Literis, qui voces ha&longs;ce de quantitatibus men&longs;uratis ibi interpre­<lb/>tantur. </s> <s>Neque minus contaminant Mathe&longs;in & Philo&longs;ophiam, <lb/>qui quantitates veras cum ip&longs;arum relationibus & vulgaribus men­<lb/>furis confundunt. </s> </p> <p type="main"> <s>Motus quidem veros corporum &longs;ingulorum cogno&longs;cere, & ab <lb/>apparentibus actu di&longs;criminare, difficillimum. </s> <s>e&longs;t propterea quod <lb/>partes &longs;patii illius immobilis, in quo corpora vere moventur, non <lb/>incurrunt in &longs;en&longs;us. </s> <s>Cau&longs;a tamen non e&longs;t pror&longs;us de&longs;perata. </s> <s>Nam <lb/>&longs;uppetunt argumenta, partim ex motibus apparentibus qui &longs;unt <lb/>motuum verorum differentiæ, partim ex viribus quæ &longs;unt mo­<lb/>tuum verorum cau&longs;æ & effectus. </s> <s>Ut &longs;i globi duo, ad datam ab in­<lb/>vicem di&longs;tantiam filo intercedente connexi, revolverentur circa <lb/>commune gravitatis centrum; innote&longs;ceret ex ten&longs;ione fili cona­<lb/>tus globorum recedendi ab axe motus, & inde quantitas motus <lb/>circularis computari po&longs;&longs;et. </s> <s>Deinde &longs;i vires quælibet æquales in <lb/>alternas globorum facies ad motum circularem augendum vel mi­<lb/>nuendum &longs;imul imprimerentur, innote&longs;ceret ex aucta vel diminuta <lb/>fili ten&longs;ione augmentum vel decrementum motus; & inde tandem <lb/>inveniri po&longs;&longs;ent facies globorum in quas vires imprimi deberent, <lb/>ut motus maxime augeretur; id e&longs;t, facies po&longs;ticæ, &longs;ive quæ in mo­<lb/>tu circulari &longs;equuntur. </s> <s>Cognitis autem faciebus quæ &longs;equuntur, <lb/>& faciebus oppo&longs;itis quæ præcedunt, cogno&longs;ceretur determinatio <lb/>motus. </s> <s>In hunc modum inveniri po&longs;&longs;et & quantitas & determi­<lb/>natio motus hujus circularis in vacuo quovis immen&longs;o, ubi nihil <lb/>extaret externum & &longs;en&longs;ibile quocum globi conferri po&longs;&longs;ent. </s> <s>Si <lb/>jam con&longs;tituerentur in &longs;patio illo corpora aliqua longinqua datam <lb/>inter &longs;e po&longs;itionem &longs;ervantia, qualia &longs;unt Stellæ Fixæ in regionibus <lb/>no&longs;tris: &longs;ciri quidem non po&longs;&longs;et ex relativa globorum tran&longs;latione <lb/>inter corpora, utrum his an illis tribuendus e&longs;&longs;et motus. </s> <s>At &longs;i <pb pagenum="11"/>attenderetur ad filum, & deprenderetur ten&longs;ionem ejus illam ip&longs;am <lb/>e&longs;&longs;e quam motus globorum requireret; concludere liceret mo­<lb/>tum e&longs;&longs;e globorum, & corpora quie&longs;cere; & tum demum ex <lb/>tran&longs;latione globorum inter corpora, determinationem hujus <lb/>motus colligere. </s> <s>Motus autem veros ex eorum cau&longs;is, effecti­<lb/>bus, & apparentibus differentiis colligere; & contra ex motibus <lb/>&longs;eu veris &longs;eu apparentibus eorum cau&longs;as & effectus, docebitur <lb/>fu&longs;ius in &longs;equentibus. </s> <s>Hunc enim in finem Tractatum &longs;equentem <lb/>compo&longs;ui. <pb pagenum="12"/> <arrow.to.target n="note5"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note5"></margin.target>TA,</s> </p> <p type="main"> <s><emph type="center"/>AXIOMATA, <lb/>SIVE <lb/>LEGES MOTUS.<emph.end type="center"/><lb/><gap desc="hr tag"/></s> </p> <p type="main"> <s><emph type="center"/>LEX I.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Corpus omne per&longs;everare in &longs;tatu &longs;uo quie&longs;cendi vel movendi uni­<lb/>formiter in directum, ni&longs;i quatenus a viribus impre&longs;&longs;is cogitur <lb/>&longs;tatum illum mutare.<emph.end type="italics"/></s> </p> <p type="main"> <s>PRojectilia per&longs;everant in motibus &longs;uis, ni&longs;i quatenus a re&longs;i­<lb/>&longs;tentia aeris retardantur, & vi gravitatis impelluntur deor&longs;um. </s> <s><lb/>Trochus, cujus partes cohærendo perpetuo retrahunt &longs;e&longs;e a mo­<lb/>tibus rectilineis, non ce&longs;&longs;at rotari, ni&longs;i quatenus ab aere retardatur. </s> <s><lb/>Majora autem Planetarum & Cometarum corpora motus &longs;uos & <lb/>progre&longs;&longs;ivos & circulares in &longs;patiis minus re&longs;i&longs;tentibus factos con­<lb/>&longs;ervant diutius. </s> </p> <p type="main"> <s><emph type="center"/>LEX II.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Mutationem motus proportionalem e&longs;&longs;e vi motrici impre&longs;&longs;æ, & fieri <lb/>&longs;ecundum lineam rectam qua vis illa imprimitur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Si vis aliqua motum quemvis generet; dupla duplum, tripla tri­<lb/>plum generabit, &longs;ive &longs;imul & &longs;emel, &longs;ive gradatim & &longs;ucce&longs;&longs;ive im­<lb/>pre&longs;&longs;a fuerit. </s> <s>Et hic motus (quoniam in eandem &longs;emper plagam <lb/>cum vi generat<gap/>ice determinatur) &longs;i corpus antea movebatur, mo­<lb/>tui ejus vel con&longs;piranti additur, vel contrario &longs;ubducitur, vel obli­<lb/>quo oblique adjicitur, & cum eo &longs;ecundum utriu&longs;que determina­<lb/>tionem componitur. </s> </p> <pb pagenum="13"/> <p type="main"> <s><emph type="center"/>LEX III.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Actioni contrariam &longs;emper & æqualem e&longs;&longs;e reactionem: &longs;ive cor­<lb/>porum duorum actiones in &longs;e mutuo &longs;emper e&longs;&longs;e æquales & in par­<lb/>tes contrarias dirigi.<emph.end type="italics"/></s> </p> <p type="main"> <s>Quicquid premit vel trahit alterum, tantundem ab eo premitur <lb/>vel trahitur. </s> <s>Si quis lapidem digito premit, premitur & hujus <lb/>digitus a lapide. </s> <s>Si equus lapidem funi alligatum trahit, retrahe­<lb/>tur etiam & equus (ut ita dicam) æqualiter in lapidem: nam funis <lb/>utrinque di&longs;tentus eodem relaxandi &longs;e conatu urgebit equum ver­<lb/>&longs;us lapidem, ac lapidem ver&longs;us equum; tantumque impediet pro­<lb/>gre&longs;&longs;um unius quantum promovet progre&longs;&longs;um alterius. </s> <s>Si corpus <lb/>aliquod in corpus aliud impingens, motum ejus vi &longs;ua quomodo­<lb/>cunque mutaverit, idem quoque vici&longs;&longs;im in motu proprio eandem <lb/>mutationem in partem contrariam vi alterius ob æqualitatem pre&longs;­<lb/>&longs;ionis mutuæ) &longs;ubibit. </s> <s>His actionibus æquales fiunt mutationes, <lb/>non velocitatum, &longs;ed motuum; &longs;cilicet in corporibus non aliunde <lb/>impeditis. </s> <s>Mutationes enim velocitatum, in contrarias itidem <lb/>partes factæ, quia motus æqualiter mutantur, &longs;unt corporibus re­<lb/>ciproce proportionales. </s> <s>Obtinet etiam hæc Lex in Attractionibus, <lb/>ut in Scholio proximo probabitur. </s> </p> <p type="main"> <s><emph type="center"/>COROLLARIUM I.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corpus viribus conjunctis diagonalem parallelogrammi eodem tem­<lb/>pore de&longs;cribere, quo latera &longs;eparatis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Si corpus dato tempore, vi &longs;ola <lb/> <arrow.to.target n="fig1"></arrow.to.target><lb/><emph type="italics"/>M<emph.end type="italics"/> in loco <emph type="italics"/>A<emph.end type="italics"/> impre&longs;&longs;a, ferretur uni­<lb/>formi cum motu ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>B<emph.end type="italics"/>; & vi <lb/>&longs;ola <emph type="italics"/>N<emph.end type="italics"/> in eodem loco impre&longs;&longs;a, fer­<lb/>retur ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>C:<emph.end type="italics"/> compleatur pa­<lb/>rallelogrammum <emph type="italics"/>ABDC,<emph.end type="italics"/> & vi utra­<lb/>que feretur id eodem tempore in diagonali ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>D.<emph.end type="italics"/> Nam quo­<lb/>niam vis <emph type="italics"/>N<emph.end type="italics"/> agit &longs;ecundum lineam <emph type="italics"/>AC<emph.end type="italics"/> ip&longs;i <emph type="italics"/>BD<emph.end type="italics"/> parallelam, hæc vis per <lb/>Legem 11 nihil mutabit velocitatem accedendi ad lineam illam <emph type="italics"/>BD<emph.end type="italics"/><lb/>a vi altera genitam. </s> <s>Accedet igitur corpus eodem tempore ad lineam <lb/><emph type="italics"/>BD,<emph.end type="italics"/> &longs;ive vis <emph type="italics"/>N<emph.end type="italics"/> imprimatur, &longs;ive non; atque adeo in fine illius tempo­<lb/>ris reperietur alicubi in linea illa <emph type="italics"/>BD.<emph.end type="italics"/> Eodem argumento in fine tem­<lb/>poris eju&longs;dem reperietur alicubi in linea <emph type="italics"/>CD,<emph.end type="italics"/> & idcirco in utriu&longs;que <lb/>lineæ concur&longs;u <emph type="italics"/>D<emph.end type="italics"/> reperiri nece&longs;&longs;e e&longs;t. </s> <s>Perget autem motu rectili­<lb/>neo ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>D<emph.end type="italics"/> per Legem 1. <pb pagenum="14"/> <arrow.to.target n="note6"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note6"></margin.target>TA, <lb/>E</s> </p> <figure id="fig1"></figure> <p type="main"> <s><emph type="center"/>COROLLARIUM II.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Et hinc patet compo&longs;itio vis directæ<emph.end type="italics"/> AD <emph type="italics"/>ex viribus quibu&longs;vis <lb/>obliquis<emph.end type="italics"/> AB <emph type="italics"/>&<emph.end type="italics"/> BD, <emph type="italics"/>& vici&longs;&longs;im re&longs;olutio vis cuju&longs;vis directæ<emph.end type="italics"/><lb/>AD <emph type="italics"/>in obliquas qua&longs;cunque<emph.end type="italics"/> AB <emph type="italics"/>&<emph.end type="italics"/> BD. <emph type="italics"/>Quæ quidem compo&longs;itio <lb/>& re&longs;olutio abunde confirmatur ex Mechanica.<emph.end type="italics"/></s> </p> <p type="main"> <s>Ut &longs;i de rotæ alicujus centro <emph type="italics"/>O<emph.end type="italics"/> exeuntes radii inæquales <emph type="italics"/>OM, <lb/>ON<emph.end type="italics"/> filis <emph type="italics"/>MA, NP<emph.end type="italics"/> &longs;u&longs;tineant pondera <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P,<emph.end type="italics"/> & quærantur vi­<lb/>res ponderum ad movendam rotam: Per centrum <emph type="italics"/>O<emph.end type="italics"/> agatur recta <lb/><emph type="italics"/>KOL<emph.end type="italics"/> filis perpendiculariter occurrens in <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>L,<emph.end type="italics"/> centroque <emph type="italics"/>O<emph.end type="italics"/> & <lb/>intervallorum <emph type="italics"/>OK, OL<emph.end type="italics"/> majore <emph type="italics"/>OL<emph.end type="italics"/><lb/> <arrow.to.target n="fig2"></arrow.to.target><lb/>de&longs;cribatur circulus occurrens filo <lb/><emph type="italics"/>MA<emph.end type="italics"/> in <emph type="italics"/>D:<emph.end type="italics"/> & actæ rectæ <emph type="italics"/>OD<emph.end type="italics"/> pa­<lb/>rallela &longs;it <emph type="italics"/>AC,<emph.end type="italics"/> & perpendicularis <lb/><emph type="italics"/>DC.<emph.end type="italics"/> Quoniam nihil refert, utrum <lb/>filorum puncta <emph type="italics"/>K, L, D<emph.end type="italics"/> affixa &longs;int <lb/>an non affixa ad planum rotæ; pon­<lb/>dera idem valebunt, ac &longs;i &longs;u&longs;pende­<lb/>rentur a punctis <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>L<emph.end type="italics"/> vel <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>L.<emph.end type="italics"/><lb/>Ponderis autem <emph type="italics"/>A<emph.end type="italics"/> exponatur vis to­<lb/>ta per lineam <emph type="italics"/>AD,<emph.end type="italics"/> & hæc re&longs;olvetur <lb/>in vires <emph type="italics"/>AC, CD,<emph.end type="italics"/> quarum <emph type="italics"/>AC<emph.end type="italics"/> trahendo radium <emph type="italics"/>OD<emph.end type="italics"/> directe a cen­<lb/>tro nihil valet ad movendam rotam; vis autem altera <emph type="italics"/>DC,<emph.end type="italics"/> trahen­<lb/>do radium <emph type="italics"/>DO<emph.end type="italics"/> perpendiculariter, idem valet ac &longs;i perpendiculari­<lb/>ter traheret radium <emph type="italics"/>OL<emph.end type="italics"/> ip&longs;i <emph type="italics"/>OD<emph.end type="italics"/> æqualem; hoc e&longs;t, idem atque <lb/>pondus <emph type="italics"/>P,<emph.end type="italics"/> &longs;i modo pondus illud &longs;it ad pondus <emph type="italics"/>A<emph.end type="italics"/> ut vis <emph type="italics"/>DC<emph.end type="italics"/> ad <lb/>vim <emph type="italics"/>DA,<emph.end type="italics"/> id e&longs;t (ob &longs;imilia triangula <emph type="italics"/>ADC, DOK,<emph.end type="italics"/>) ut <emph type="italics"/>OK<emph.end type="italics"/><lb/>ad <emph type="italics"/>OD<emph.end type="italics"/> &longs;eu <emph type="italics"/>OL.<emph.end type="italics"/> Pondera igitur <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P,<emph.end type="italics"/> quæ &longs;unt reciproce ut <lb/>radii in directum po&longs;iti <emph type="italics"/>OK<emph.end type="italics"/> & <emph type="italics"/>OL,<emph.end type="italics"/> idem pollebunt, & &longs;ic con&longs;i­<lb/>&longs;tent in æquilibrio: quæ e&longs;t proprietas noti&longs;&longs;ima Libræ, Vectis, & <lb/>Axis in Peritrochio. </s> <s>Sin pondus alterutrum &longs;it majus quam in hac <lb/>ratione, erit vis ejus ad movendam rotam tanto major. </s> </p> <figure id="fig2"></figure> <p type="main"> <s>Quod &longs;i pondus <emph type="italics"/>p<emph.end type="italics"/> ponderi <emph type="italics"/>P<emph.end type="italics"/> æquale partim &longs;u&longs;pendatur filo <emph type="italics"/>Np,<emph.end type="italics"/><lb/>partim incumbat plano obliquo <emph type="italics"/>pG:<emph.end type="italics"/> agantur <emph type="italics"/>pH, NH,<emph.end type="italics"/> prior ho­<lb/>rizonti, po&longs;terior plano <emph type="italics"/>pG<emph.end type="italics"/> perpendicularis; & &longs;i vis ponderis <emph type="italics"/>p<emph.end type="italics"/><lb/>deor&longs;um tendens, exponatur per lineam <emph type="italics"/>pH,<emph.end type="italics"/> re&longs;olvi pote&longs;t hæc in <lb/>vires <emph type="italics"/>pN, HN.<emph.end type="italics"/> Si filo <emph type="italics"/>pN<emph.end type="italics"/> perpendiculare e&longs;&longs;et planum aliquod <lb/><emph type="italics"/>pQ,<emph.end type="italics"/> &longs;ecans planum alterum <emph type="italics"/>pG<emph.end type="italics"/> in linea ad horizontem paral­<lb/>lela; & pondas <emph type="italics"/>p<emph.end type="italics"/> his planis <emph type="italics"/>pQ, pG<emph.end type="italics"/> &longs;olummodo incumberet; ur- <pb pagenum="15"/>geret illud hæc plana viribus <emph type="italics"/>pN, HN<emph.end type="italics"/> perpendiculariter, nimirun <lb/>planum <emph type="italics"/>pQ<emph.end type="italics"/> vi <emph type="italics"/>pN,<emph.end type="italics"/> & planum <emph type="italics"/>pG<emph.end type="italics"/> vi <emph type="italics"/>HN.<emph.end type="italics"/> Ideoque &longs;i tollatur pla­<lb/>num <emph type="italics"/>pQ,<emph.end type="italics"/> ut pondus tendat filum; quoniam filum &longs;u&longs;tinendo pon <lb/>dus jam vicem præ&longs;tat plani &longs;ublati, tendetur illud eadem vi <emph type="italics"/>pN,<emph.end type="italics"/><lb/>qua planum antea urgebatur. </s> <s>Unde ten&longs;io fili hujus obliqui erit <lb/>ad ten&longs;ionem &longs;ili alterius perpendicularis <emph type="italics"/>PN,<emph.end type="italics"/> ut <emph type="italics"/>pN<emph.end type="italics"/> ad <emph type="italics"/>pH.<emph.end type="italics"/> Id. </s> <s><lb/>eoque &longs;i pondus <emph type="italics"/>p<emph.end type="italics"/> &longs;it ad pondus <emph type="italics"/>A<emph.end type="italics"/> in ratione quæ componitur <gap/><lb/>ratione reciproca minimarum di&longs;tantiarum &longs;uorum &longs;uorum <emph type="italics"/>pN, <lb/>AM<emph.end type="italics"/> a centro rotæ, & ratione directa <emph type="italics"/>pH<emph.end type="italics"/> ad <emph type="italics"/>pN<emph.end type="italics"/>; pondera idem <lb/>valebunt ad rotam movendam, atque adeo &longs;e mutuo &longs;u&longs;tinebunt, <lb/>ut quilibet experiri pote&longs;t. </s> </p> <p type="main"> <s>Pondus autem <emph type="italics"/>p,<emph.end type="italics"/> planis illis duobus obliquis incumbens, rationem <lb/>habet cunei inter corporis fi&longs;&longs;i facies internas: & inde vires cunei <lb/>& mallei innote&longs;cunt: utpote cum vis qua pondus <emph type="italics"/>p<emph.end type="italics"/> urget planum <lb/><emph type="italics"/>pQ<emph.end type="italics"/> &longs;it ad vim, qua idem vel gravitate &longs;ua vel ictu mallei impellitur <lb/>&longs;ecundum lineam <emph type="italics"/>pH<emph.end type="italics"/> in plano, &c. </s> <s>ut <emph type="italics"/>pN<emph.end type="italics"/> and <emph type="italics"/>pH<emph.end type="italics"/>; atque ad vim, qua <lb/>urget planum alterum <emph type="italics"/>pG,<emph.end type="italics"/> ut <emph type="italics"/>pN<emph.end type="italics"/> ad <emph type="italics"/>NH.<emph.end type="italics"/> Sed & vis Cochleæ per <lb/>&longs;imilem virium divi&longs;ionem colligitur; quippe quæ cuneus e&longs;t a ve­<lb/>cte impul&longs;us. </s> <s>U&longs;us igitur Corollarii hujus lati&longs;&longs;ime patet, & late <lb/>patendo veritatem &longs;uam evincit; cum pendeat ex jam dictis Mecha­<lb/>nica tota ab Auctoribus diver&longs;imode demon&longs;trata. </s> <s>Ex hi&longs;ce enim <lb/>facile derivantur vires Machinarum, quæ ex Rotis, Tympanis, <lb/>Trochleis, Vectibus, nervis ten&longs;is & ponderibus directe vel obli­<lb/>que a&longs;cendentibus, cæteri&longs;que potentiis Mechanicis componi &longs;o­<lb/>lent, ut & vires Tendinum ad animalium o&longs;&longs;a movenda. </s> </p> <p type="main"> <s><emph type="center"/>COROLLARIUM III.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Quantitas motus quæ colligitur capiendo &longs;ummam motuum factorum <lb/>ad eandem partem, & differentiam factorum ad contrarias, non <lb/>mutatur ab actione corporum inter &longs;e.<emph.end type="italics"/></s> </p> <p type="main"> <s>Etenim actio eique contraria reactio æquales &longs;unt per Legem 111, <lb/>adeoque per Legem 11 æquales in motibus efficiunt mutationes ver­<lb/>&longs;us contrarias partes. </s> <s>Ergo &longs;i motus fiunt ad eandem partem; quic­<lb/>quid additur motui corporis fugientis, &longs;ubducetur motui corporis <lb/>in&longs;equentis &longs;ic, ut &longs;umma maneat eadem quæ prius. </s> <s>Sin corpora ob­<lb/>viam eant; æqualis erit &longs;ubductio de motu utriu&longs;que, adeoque diffe­<lb/>rentia motuum factorum in contrarias partes manebit eadem. </s> </p> <p type="main"> <s>Ut &longs;i corpus &longs;phæricum <emph type="italics"/>A<emph.end type="italics"/> &longs;it triplo majus corpore &longs;phærico <emph type="italics"/>B,<emph.end type="italics"/> ha­<lb/>beatque duas velocitatis partes; & <emph type="italics"/>B<emph.end type="italics"/> &longs;equatur in eadem recta cum ve- <pb pagenum="16"/> <arrow.to.target n="note7"></arrow.to.target><lb/>locitatis partibus decem, adeoque motus ip&longs;ius <emph type="italics"/>A<emph.end type="italics"/> &longs;it ad motum ip&longs;ius <lb/><emph type="italics"/>B,<emph.end type="italics"/> ut &longs;ex ad decem: ponantur motus illis e&longs;&longs;e partium &longs;ex & par­<lb/>tium decem, & &longs;umma erit partium &longs;exdecim. </s> <s>In corporum igitur <lb/>concur&longs;u, &longs;i corpus <emph type="italics"/>A<emph.end type="italics"/> lucretur motus partes tres vel quatuor vel <lb/>quinque, corpus <emph type="italics"/>B<emph.end type="italics"/> amittet partes totidem, adeoque perget corpus <lb/><emph type="italics"/>A<emph.end type="italics"/> po&longs;t reflexionem cum partibus novem vel decem vel undecim, <lb/>& <emph type="italics"/>B<emph.end type="italics"/> cum partibus &longs;eptem vel &longs;ex vel quinque, exi&longs;tente &longs;emper &longs;um­<lb/>ma partium &longs;exdecim ut prius. </s> <s>Si corpus <emph type="italics"/>A<emph.end type="italics"/> lucretur partes novem <lb/>vel decem vel undecim vel duodecim, adeoque progrediatur po&longs;t <lb/>concur&longs;um cum partibus quindecim vel &longs;exdecim vel &longs;eptendecim <lb/>vel octodecim; corpus <emph type="italics"/>B,<emph.end type="italics"/> amittendo tot partes quot <emph type="italics"/>A<emph.end type="italics"/> lucratur, <lb/>vel cum una parte progredietur ami&longs;&longs;is partibus novem, vel qui­<lb/>e&longs;cet ami&longs;&longs;o motu &longs;uo progre&longs;&longs;ivo partium decem, vel cum una par­<lb/>te regredietur ami&longs;&longs;o motu &longs;uo & (ut ita dicam) una parte amplius, <lb/>vel regredietur cum partibus duabus ob detractum motum progre&longs;­<lb/>&longs;ivum partium duodecim. </s> <s>Atque ita &longs;ummæ motuum con&longs;pirantium <lb/>15+1 vel 16+c, & differentiæ contrariorum 17-1 & 18-2 &longs;emper <lb/>erunt partium &longs;exdecim, ut ante concur&longs;um & reflexionem. </s> <s>Cogni­<lb/>tis autem motibus quibu&longs;cum corpora po&longs;t reflexionem pergent, in­<lb/>venietur cuju&longs;que velocitas, ponendo eam e&longs;&longs;e ad velocitatem ante <lb/>reflexionem, ut motus po&longs;t e&longs;t ad motum ante. </s> <s>Ut in ca&longs;u ultimo, ubi <lb/>corporis <emph type="italics"/>A<emph.end type="italics"/> motus erat partium &longs;ex ante reflexionem & partium octo­<lb/>decim po&longs;tea, & velocitas partium duarum ante reflexionem; in­<lb/>venietur ejus velocitas partium &longs;ex po&longs;t reflexionem, dicendo, ut <lb/>motus partes &longs;ex ante reflexionem ad motus partes octodecim po&longs;t­<lb/>ea, ita velocitatis partes duæ ante reflexionem ad velocitatis partes <lb/>&longs;ex po&longs;tea. </s> </p> <p type="margin"> <s><margin.target id="note7"></margin.target>TA,</s> </p> <p type="main"> <s>Quod &longs;i corpora vel non Sphærica vel diver&longs;is in rectis moventia <lb/>incidant in &longs;e mutuo oblique, & requirantur eorum motus po&longs;t refle­<lb/>xionem; cogno&longs;cendus e&longs;t &longs;itus plani a quo corpora concurrentia tan­<lb/>guntur in puncto concur&longs;us: dein corporis utriu&longs;que motus (per <lb/>Corol.11.) di&longs;tinguendus e&longs;t in duos, unum huic plano perpendicu­<lb/>larem, alterum eidem parallelum: motus autem paralleli, propter­<lb/>ea quod corpora agant in &longs;e invicem &longs;ecundum lineam huic plano <lb/>perpendicularem, retinendi &longs;unt iidem po&longs;t reflexionem atque an­<lb/>tea; & motibus perpendicularibus mutationes æquales in partes con­<lb/>trarias tribuendæ &longs;unt &longs;ic, ut &longs;umma con&longs;pirantium & differentia <lb/>contrariorum maneat eadem quæ prius. </s> <s>Ex huju&longs;modi reflexio­<lb/>nibus oriri etiam &longs;olent motus circulares corporum circa centra pro­<lb/>pria. </s> <s>Sed hos ca&longs;us in &longs;equentibus non con&longs;idero, & nimis longum <lb/>e&longs;&longs;et omnia huc &longs;pectantia demon&longs;trare. <pb pagenum="18"/> <arrow.to.target n="note8"></arrow.to.target><lb/>mutat &longs;tatum &longs;uum; & reliquorum, quibu&longs;cum actio illa non in­<lb/>tercedit, commune gravitatis centrum nihil inde patitur; di&longs;tantia <lb/>autem horum duorum centrorum dividitur a communi corporum <lb/>omnium centro in partes &longs;ummis totalibus corporum quorum <lb/>&longs;unt centra reciproce proportionales; adeoque centris illis duobus <lb/>&longs;tatum &longs;uum movendi vel quie&longs;cendi &longs;ervantibus, commune omni­<lb/>um centrum &longs;ervat etiam &longs;tatum &longs;uum: manife&longs;tum e&longs;t quod com­<lb/>mune illud omnium centrum ob actiones binorum corporum inter <lb/>&longs;e nunquam mutat &longs;tatum &longs;uum quoad motum & quietem. </s> <s>In tali <lb/>autem &longs;y&longs;temate actiones omnes corporum inter &longs;e, vel inter bina <lb/>&longs;unt corpora, vel ab actionibus inter bina compo&longs;itæ; & propterea <lb/>communi omnium centro mutationem in &longs;tatu motus ejus vel quie­<lb/>tis nunquam inducunt. </s> <s>Quare cum centrum illud ubi corpora non <lb/>agunt in &longs;e invicem, vel quie&longs;cit, vel in recta aliqua progreditur uni­<lb/>formiter; perget idem, non ob&longs;tantibus corporum actionibus inter <lb/>&longs;e, vel &longs;emper quie&longs;cere, vel &longs;emper progredi uniformiter in dire­<lb/>ctum; ni&longs;i a viribus in &longs;y&longs;tema extrin&longs;ecus impre&longs;&longs;is deturbetur de hoc <lb/>&longs;tatu. </s> <s>E&longs;t igitur &longs;y&longs;tematis corporum plurium Lex eadem quæ cor­<lb/>poris &longs;olitarii, quoad per&longs;everantiam in &longs;tatu motus vel quietis. </s> <s>Mo­<lb/>tus enim progre&longs;&longs;ivus &longs;eu corporis &longs;olitarii &longs;eu &longs;y&longs;tematis corporum <lb/>ex motu centri gravitatis æ&longs;timari &longs;emper debet. </s> </p> <p type="margin"> <s><margin.target id="note8"></margin.target>IATA, <lb/>VF.</s> </p> <p type="main"> <s><emph type="center"/>COROLLARIUM V.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Corporum dato &longs;patio inclu&longs;orum iidem &longs;unt motus inter &longs;e, &longs;ive &longs;pa­<lb/>tium illud quie&longs;cat, &longs;ive moveatur idem uniformiter in directum <lb/>ab&longs;que motu circulari.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam differentiæ motuum tendentium ad eandem partem, & &longs;um­<lb/>mæ tendentium ad contrarias, eædem &longs;unt &longs;ub initio in <expan abbr="utroq;">utroque</expan> ca&longs;u (ex <lb/>hypothe&longs;i) & ex his &longs;ummis vel differentiis oriuntur congre&longs;&longs;us & im­<lb/>petus quibus corpora &longs;e mutuo feriunt. </s> <s>Ergo per Legem 11 æquales e­<lb/>runt congre&longs;&longs;uum effectus in <expan abbr="utroq;">utroque</expan> ca&longs;u; & propterea manebunt mo­<lb/>tus inter &longs;e in uno ca&longs;u æquales motibus inter &longs;e in altero. </s> <s>Idem com­<lb/>probatur experimento luculento. </s> <s>Motus omnes eodem modo &longs;e ha­<lb/>bent in Navi, &longs;ive ea quie&longs;cat, &longs;ive moveatur uniformiter in directum. </s> </p> <p type="main"> <s><emph type="center"/>COROLLARIUM VI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si corpora <expan abbr="moveãtur">moveantur</expan> <expan abbr="quomodocunq;">quomodocunque</expan> inter&longs;e, & a viribus acceler atrici­<lb/>bus æqualibus &longs;ecundum lineas parallelas urgeantur; pergent omnia <lb/>eodem modo moveri inter &longs;e, ac &longs;i viribus illis non e&longs;&longs;ent incitata.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam vires illæ æqualiter (pro quantitatibus movendorum corpo- <pb pagenum="19"/>rum) & &longs;ecundum lineas parallelas agendo, corpora omnia æquali­<lb/>ter (quoad velocitatem) movebunt per Legem 11. adeoque nunquam <lb/>mutabunt po&longs;itiones & motus eorum inter &longs;e. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Hactenus principia tradidi a Mathematicis recepta & experien­<lb/>tia multiplici confirmata. </s> <s>Per Leges duas primas & Corollaria duo <lb/>prima <emph type="italics"/>Galilæus<emph.end type="italics"/> invenit de&longs;cen&longs;um Gravium e&longs;&longs;e in duplicata ratione <lb/>temporis, & motum Projectilium fieri in Parabola; con&longs;pirante ex­<lb/>perientia, ni&longs;i quatenus motus illi per aeris re&longs;i&longs;tentiam aliquantu­<lb/>lum retardantur. </s> <s>Ab ii&longs;dem Legibus & Corollariis pendent de­<lb/>mon&longs;trata de temporibus o&longs;cillantium Pendulorum, &longs;uffragante Ho­<lb/>rologiorum experientia quotidiana. </s> <s>Ex his ii&longs;dem & Lege tertia <lb/><emph type="italics"/>Chri&longs;tophorus Wrennus<emph.end type="italics"/> Eques Auratus, <emph type="italics"/>Jobannes Walli&longs;ius S.T.D.<emph.end type="italics"/><lb/>& <emph type="italics"/>Chri&longs;tianus Hugenius,<emph.end type="italics"/> hujus ætatis Geometrarum facile prin­<lb/>cipes, regulas congre&longs;&longs;uum & reflexionum duorum corporum &longs;e­<lb/>or&longs;im invenerunt, & eodem fere tempore cum <emph type="italics"/>Societate Regia<emph.end type="italics"/><lb/>communicarunt, inter &longs;e (quoad has leges) omnino con&longs;pirantes: <lb/>& primus quidem <emph type="italics"/>Walli&longs;ius,<emph.end type="italics"/> deinde <emph type="italics"/>Wrennus<emph.end type="italics"/> & <emph type="italics"/>Hugenius<emph.end type="italics"/> inven­<lb/>tum prodiderunt. </s> <s>Sed & veritas comprobata e&longs;t a <emph type="italics"/>Wrenno<emph.end type="italics"/> co­<lb/>ram <emph type="italics"/>Regia Societate<emph.end type="italics"/> per experimentum Pendulorum: quod etiam <lb/><emph type="italics"/>Clari&longs;&longs;imus Mariottus<emph.end type="italics"/> libro integro exponere mox dignatus e&longs;t. </s> <s>Ve­<lb/>rum, ut hoc experimentum cum Theoriis ad amu&longs;&longs;im congruat, ha­<lb/>benda e&longs;t ratio cum re&longs;i&longs;tentiæ aeris, tum etiam vis Ela&longs;ticæ con­<lb/>currentium corporum. </s> <s>Pendeant corpora <emph type="italics"/>A, B<emph.end type="italics"/> filis parallelis & <lb/>æqualibus <emph type="italics"/>AC, BD,<emph.end type="italics"/> a centris <emph type="italics"/>C, D.<emph.end type="italics"/> His centris & intervallis de­<lb/>&longs;cribantur &longs;emicirculi <emph type="italics"/>EAF, GBH<emph.end type="italics"/> radiis <emph type="italics"/>CA, DB<emph.end type="italics"/> bi&longs;ecti. </s> <s>Tra­<lb/>hatur corpus <emph type="italics"/>A<emph.end type="italics"/> ad arcus <emph type="italics"/>EAF<emph.end type="italics"/> punctum quodvis <emph type="italics"/>R,<emph.end type="italics"/> & (&longs;ubducto <lb/>corpore <emph type="italics"/>B<emph.end type="italics"/>) demittatur inde, redeatque po&longs;t unam o&longs;cillationem <lb/>ad punctum <emph type="italics"/>V.<emph.end type="italics"/> E&longs;t <emph type="italics"/>RV<emph.end type="italics"/> re­<lb/> <arrow.to.target n="fig3"></arrow.to.target><lb/>tardatio ex re&longs;i&longs;tentia aeris. </s> <s><lb/>Hujus <emph type="italics"/>RV<emph.end type="italics"/> fiat <emph type="italics"/>ST<emph.end type="italics"/> pars quar­<lb/>ta &longs;ita in medio, ita &longs;cilicet <lb/>ut <emph type="italics"/>RS<emph.end type="italics"/> & <emph type="italics"/>TV<emph.end type="italics"/> æquentur, &longs;it­<lb/>que <emph type="italics"/>RS<emph.end type="italics"/> ad <emph type="italics"/>ST<emph.end type="italics"/> ut 3 ad 2. <lb/>Et i&longs;ta <emph type="italics"/>ST<emph.end type="italics"/> exhibebit retarda­<lb/>tionem in de&longs;cen&longs;u ab <emph type="italics"/>S<emph.end type="italics"/> ad <emph type="italics"/>A<emph.end type="italics"/><lb/>quam proxime. </s> <s>Re&longs;tituatur <lb/>corpus <emph type="italics"/>B<emph.end type="italics"/> in locum &longs;uum. </s> <s>Cadat corpus <emph type="italics"/>A<emph.end type="italics"/> de puncto <emph type="italics"/>S,<emph.end type="italics"/> & velo­<lb/>citas ejus in loco reflexionis <emph type="italics"/>A,<emph.end type="italics"/> ab&longs;que errore &longs;en&longs;ibili, tanta erit ae <pb pagenum="20"/>&longs;i in vacuo cecidi&longs;&longs;et de loco <emph type="italics"/>T.<emph.end type="italics"/> Exponatur igitur hæc velocitas <lb/> <arrow.to.target n="note9"></arrow.to.target><lb/>per chordam arcus <emph type="italics"/>TA.<emph.end type="italics"/> Nam velocitatem Penduli in puncto in­<lb/>fimo e&longs;&longs;e ut chordam arcus quem cadendo de&longs;crip&longs;it, Propo&longs;itio e&longs;t <lb/>e&longs;t Geometris noti&longs;&longs;ima. </s> <s>Po&longs;t reflexionem perveniat corpus <emph type="italics"/>A<emph.end type="italics"/> ad <lb/>locum <emph type="italics"/>s,<emph.end type="italics"/> & corpus <emph type="italics"/>B<emph.end type="italics"/> ad locum <emph type="italics"/>k.<emph.end type="italics"/> Tollatur corpus <emph type="italics"/>B<emph.end type="italics"/> & invenia­<lb/>tur locus <emph type="italics"/>v<emph.end type="italics"/>; a quo &longs;i corpus <emph type="italics"/>A<emph.end type="italics"/> demittatur & po&longs;t unam o&longs;cillatio­<lb/>nem redeat ad locum <emph type="italics"/>r,<emph.end type="italics"/> &longs;it <emph type="italics"/>st<emph.end type="italics"/> pars quarta ip&longs;ius <emph type="italics"/>rv<emph.end type="italics"/> &longs;ita in medio, <lb/>ita videlicet ut <emph type="italics"/>rs<emph.end type="italics"/> & <emph type="italics"/>tu<emph.end type="italics"/> æquentur; & per chordam arcus <emph type="italics"/>tA<emph.end type="italics"/> ex­<lb/>ponatur velocitas quam corpus <emph type="italics"/>A<emph.end type="italics"/> proxime po&longs;t reflexionem habuit <lb/>in loco <emph type="italics"/>A.<emph.end type="italics"/> Nam <emph type="italics"/>t<emph.end type="italics"/> erit locus ille vcrus & correctus, ad quem cor­<lb/>pus <emph type="italics"/>A,<emph.end type="italics"/> &longs;ublata aeris re&longs;i&longs;tentia, a&longs;cendere debui&longs;&longs;et: Simili me­<lb/>thodo corrigendus erit locus <emph type="italics"/>k,<emph.end type="italics"/> ad quem corpus <emph type="italics"/>B<emph.end type="italics"/> a&longs;cendit, & in­<lb/>veniendus locus <emph type="italics"/>l,<emph.end type="italics"/> ad quem corpus illud a&longs;cendere debui&longs;&longs;et in va­<lb/>cuo. </s> <s>Hoc pacto experiri licet omnia perinde ac &longs;i in vacuo con­<lb/>&longs;tituti e&longs;&longs;emus. </s> <s>Tandem ducendum erit corpus <emph type="italics"/>A<emph.end type="italics"/> in chordam ar­<lb/>cus <emph type="italics"/>TA<emph.end type="italics"/> (quæ velocitatem ejus exhibet) ut habeatur motus ejus in <lb/>loco <emph type="italics"/>A<emph.end type="italics"/> proxime ante reflexionem; deinde in chordam arcus <emph type="italics"/>tA,<emph.end type="italics"/> ut <lb/>habeatur motus ejus in loco <emph type="italics"/>A<emph.end type="italics"/> proxime po&longs;t reflexionem. </s> <s>Et &longs;ic <lb/>corpus <emph type="italics"/>B<emph.end type="italics"/> ducendum erit in chordam arcus <emph type="italics"/>Bb,<emph.end type="italics"/> ut habeatur motus <lb/>ejus proxime po&longs;t reflexionem. </s> <s>Et &longs;imili methodo, ubi corpora duo <lb/>fimul demittuntur de locis diver&longs;is, inveniendi &longs;unt motus <expan abbr="utriu&longs;q;">utriu&longs;que</expan> <lb/>tam ante, quam po&longs;t reflexionem; & tum demum conferendi &longs;unt <lb/>motus inter &longs;e & colligendi effectus reflexionis. </s> <s>Hoc modo in <lb/>Pendulis pedum decem rem tentando, idque in corporibus tam <lb/>inæqualibus quam æqualibus, & faciendo ut corpora de intervallis <lb/>ampli&longs;&longs;imis, puta pedum octo vel duodecim vel &longs;exdecim, concurre­<lb/>rent; reperi &longs;emper &longs;ine errore trium digitorum in men&longs;uris, ubi <lb/>corpora &longs;ibi mutuo directe occurrebant, quod æquales erant muta­<lb/>tiones motuum corporibus in partes contrarias illatæ, atque adeo <lb/>quod actio & reactio &longs;emper <lb/> <arrow.to.target n="fig4"></arrow.to.target><lb/>erant æquales. </s> <s>Ut &longs;i corpus <lb/><emph type="italics"/>A<emph.end type="italics"/> incidebat in corpus <emph type="italics"/>B<emph.end type="italics"/> cum <lb/>novem partibus motus, & a­<lb/>mi&longs;&longs;is &longs;eptem partibus perge­<lb/>bat po&longs;t reflexionem cum du­<lb/>abus; corpus <emph type="italics"/>B<emph.end type="italics"/> re&longs;iliebat cum <lb/>partibus i&longs;tis &longs;eptem. </s> <s>Si cor­<lb/>pora obviam ibant <emph type="italics"/>A<emph.end type="italics"/> cum <lb/>duodecim partibus & <emph type="italics"/>B<emph.end type="italics"/> cum &longs;ex, & redibat <emph type="italics"/>A<emph.end type="italics"/> cum duabus; redi­<lb/>bat <emph type="italics"/>B<emph.end type="italics"/> cum octo, facta detractione partium quatuordecim utrin­<lb/>que. </s> <s>De motu ip&longs;ius <emph type="italics"/>A<emph.end type="italics"/> &longs;ubducantur partes duodecim, & re&longs;tabit <pb pagenum="21"/>nihil: &longs;ubducantur aliæ partes duæ, & fiet motus duarum partium <lb/>in plagam contrariam: & &longs;ic de motu corporis <emph type="italics"/>B<emph.end type="italics"/> partium &longs;ex &longs;ub­<lb/>ducendo partes quatuordecim, fient partes octo in plagam contra­<lb/>riam. </s> <s>Quod &longs;i corpora ibant ad eandam plagam, <emph type="italics"/>A<emph.end type="italics"/> velocius cum <lb/>partibus quatuordecim, & <emph type="italics"/>B<emph.end type="italics"/> tardius cum partibus quinque, & po&longs;t <lb/>reflexionem pergebat <emph type="italics"/>A<emph.end type="italics"/> cum quinque partibus; pergebat <emph type="italics"/>B<emph.end type="italics"/> cum qua­<lb/>tuordecim, facta tran&longs;latione partium novem de <emph type="italics"/>A<emph.end type="italics"/> in <emph type="italics"/>B.<emph.end type="italics"/> Et &longs;ic <lb/>in reliquis. </s> <s>A congre&longs;&longs;u & colli&longs;ione corporum nunquam muta­<lb/>batur quantitas motus, quæ ex &longs;umma motuum con&longs;pirantium & <lb/>differentia contrariorum colligebatur. </s> <s>Nam errorem digiti unius <lb/>& alterius in men&longs;uris tribuerim difficultati peragendi &longs;ingula <lb/>&longs;atis accurate. </s> <s>Difficile erat, tum pendula &longs;imul demittere fic, ut <lb/>corpora in &longs;e mutuo impingerent in loco infimo <emph type="italics"/>AB<emph.end type="italics"/>; tum loca <emph type="italics"/>s, <lb/>k<emph.end type="italics"/> notare, ad quæ corpora a&longs;cendebant po&longs;t concur&longs;um. </s> <s>Sed & in <lb/>ip&longs;is pilis inæqualis partium den&longs;itas, & textura aliis de cau&longs;is irre­<lb/>gularis, errores inducebant. </s> </p> <p type="margin"> <s><margin.target id="note9"></margin.target><gap/>ATA, <lb/>VE</s> </p> <figure id="fig3"></figure> <figure id="fig4"></figure> <p type="main"> <s>Porro nequis objiciat Regulam, ad quam probandam inventum <lb/>e&longs;t hoc experimentum, præ&longs;upponere corpora vel ab&longs;olute dura <lb/>e&longs;&longs;e, vel &longs;altem perfecte ela&longs;tica, cuju&longs;modi nulla reperiuntur in <lb/>compo&longs;itionibus naturalibus; addo quod Experimenta jam de&longs;crip­<lb/>ta &longs;uccedunt in corporibus mollibus æque ac in duris, nimirum a <lb/>conditione duritiei neutiquam pendentia. </s> <s>Nam &longs;i Regula illa in <lb/>corporibus non perfecte duris tentanda e&longs;t, debebit &longs;olummodo <lb/>reflexio minui in certa proportione pro quantitate vis Ela&longs;ticæ. </s> <s>In <lb/>Theoria <emph type="italics"/>Wrenni<emph.end type="italics"/> & <emph type="italics"/>Hugenii<emph.end type="italics"/> corpora ab&longs;olute dura redeunt ab invi­<lb/>cem cum velocitate congre&longs;&longs;us. </s> <s>Certius id affirmabitur de perfecte <lb/>Ela&longs;ticis. </s> <s>In imperfecte Ela&longs;ticis velocitas reditus minuenda e&longs;t &longs;i­<lb/>mul cum vi Ela&longs;tica; propterea quod vis illa; (ni&longs;i ubi partes cor­<lb/>porum ex congre&longs;&longs;u læduntur, vel exten&longs;ionem aliqualem qua&longs;i &longs;ub <lb/>malleo patiuntur,) certa ac determinata &longs;it (quantum &longs;entio) faci­<lb/>atque corpora redire ab invicem cum velocitate relativa, quæ &longs;it ad <lb/>relativam velocitatem concur&longs;us in data ratione. </s> <s>Id in pilis ex lana <lb/>arcte conglomerata & fortiter con&longs;tricta &longs;ic tentavi. </s> <s>Primum demit­<lb/>tendo Pendula & men&longs;urando reflexionem, inveni quantitatem vis <lb/>Ela&longs;ticæ; deinde per hanc vim determinavi reflexiones in aliis ca­<lb/>&longs;ibus concur&longs;uum, & re&longs;pondebant Experimenta. </s> <s>Redibant &longs;emper <lb/>pilæ ab invicem cum velocitate relativa, quæ e&longs;&longs;et ad velocitatem <lb/>relativam concur&longs;us ut 5 ad 9 circiter. </s> <s>Eadem fere cum velocitate <lb/>redibant pilæ ex chalybe: aliæ ex &longs;ubere cum paulo minore: in vi­<lb/>treis autem proportio erat 15 ad 16 circiter. </s> <s>Atque hoc pacto Lex <lb/>tertia quoad ictus & reflexiones per Theoriam comprobata e&longs;t, quæ <lb/>cum experientia plane congruit. <pb pagenum="22"/> <arrow.to.target n="note10"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note10"></margin.target><gap/>ATA <lb/><gap/>E</s> </p> <p type="main"> <s>In Attractionibus rem &longs;ic breviter o&longs;tendo. </s> <s>Corporibus duobus <lb/>quibu&longs;vis <emph type="italics"/>A, B<emph.end type="italics"/> &longs;e mutuo trahentibus, concipe ob&longs;taculum quodvis <lb/>interponi quo congre&longs;&longs;us eorum impediatur. </s> <s>Si corpus alterutrum <lb/><emph type="italics"/>A<emph.end type="italics"/> magis trahitur ver&longs;us corpus alterum <emph type="italics"/>B,<emph.end type="italics"/> quam illud alterum <emph type="italics"/>B<emph.end type="italics"/><lb/>in prius <emph type="italics"/>A,<emph.end type="italics"/> ob&longs;taculum magis urgebitur pre&longs;&longs;ione corporis <emph type="italics"/>A<emph.end type="italics"/> quam <lb/>pre&longs;&longs;ione corporis <emph type="italics"/>B<emph.end type="italics"/>; proindeque non manebit in æquilibrio. </s> <s>Præ­<lb/>valebit pre&longs;&longs;io fortior, facietque ut &longs;y&longs;tema corporum duorum & <lb/>ob&longs;taculi moveatur in directum in partes ver&longs;us <emph type="italics"/>B,<emph.end type="italics"/> motuque in &longs;patiis <lb/>liberis &longs;emper accelerato abeat in infinitum. </s> <s>Quod e&longs;t ab&longs;urdum & <lb/>Legi primæ contrarium. </s> <s>Nam per Legem primam debebit &longs;y&longs;tema <lb/>per&longs;everare in &longs;tatu &longs;uo quie&longs;cendi vel movendi uniformiter in di­<lb/>rectum, proindeque corpora æqualiter urgebunt ob&longs;taculum, & id­<lb/>circo æqualiter trahentur in invicem. </s> <s>Tentavi hoc in Magnete & <lb/>Ferro. </s> <s>Si hæc in va&longs;culis propriis &longs;e&longs;e contingentibus &longs;eor&longs;im po­<lb/>&longs;ita, in aqua &longs;tagnante juxta fluitent; neutrum propellet alterum, <lb/>&longs;ed æqualitate attractionis utrinque &longs;u&longs;tinebunt conatus in &longs;e mu­<lb/>tuos, ac tandem in æquilibrio con&longs;tituta quie&longs;cent. </s> </p> <p type="main"> <s>Sic etiam gravitas inter Terram & ejus partes, mutua e&longs;t. </s> <s>Se­<lb/>cetur Terra <emph type="italics"/>FI<emph.end type="italics"/> plano quovis <emph type="italics"/>EG<emph.end type="italics"/> in partes duas <emph type="italics"/>EGF<emph.end type="italics"/> & <emph type="italics"/>EGI:<emph.end type="italics"/><lb/>& æqualia erunt harum pondera in &longs;e mu­<lb/> <arrow.to.target n="fig5"></arrow.to.target><lb/>tuo. </s> <s>Nam &longs;i plano alio <emph type="italics"/>HK<emph.end type="italics"/> quod priori <lb/><emph type="italics"/>EG<emph.end type="italics"/> parallelum &longs;it, pars major <emph type="italics"/>EGI<emph.end type="italics"/> &longs;e­<lb/>cetur in partes duas <emph type="italics"/>EGKH<emph.end type="italics"/> & <emph type="italics"/>HKI,<emph.end type="italics"/><lb/>quarum <emph type="italics"/>HKI<emph.end type="italics"/> æqualis &longs;it parti prius ab­<lb/>&longs;ci&longs;&longs;æ <emph type="italics"/>EFG:<emph.end type="italics"/> manife&longs;tum e&longs;t quod pars <lb/>media <emph type="italics"/>EGKH<emph.end type="italics"/> pondere proprio in neu­<lb/>tram partium extremarum propendebit, <lb/>&longs;ed inter utramque in æquilibrio, ut ita <lb/>dicam, &longs;u&longs;pendetur, & quie&longs;cet. </s> <s>Pars autem extrema <emph type="italics"/>HKI<emph.end type="italics"/> toto <lb/>&longs;uo pondere incumbet in partem mediam, & urgebit illam in <lb/>partom alteram extremam <emph type="italics"/>EGF<emph.end type="italics"/>; ideoque vis qua partium <lb/><emph type="italics"/>HKI<emph.end type="italics"/> & <emph type="italics"/>EGKH<emph.end type="italics"/> &longs;umma <emph type="italics"/>EGI<emph.end type="italics"/> tendit ver&longs;us partem tertiam <lb/><emph type="italics"/>EGF,<emph.end type="italics"/> æqualis e&longs;t ponderi partis <emph type="italics"/>HKI,<emph.end type="italics"/> id e&longs;t ponderi partis ter­<lb/>tiæ <emph type="italics"/>EGF.<emph.end type="italics"/> Et propterea pondera partium duarum <emph type="italics"/>EGI, EGF<emph.end type="italics"/><lb/>in &longs;e mutuo &longs;unt æqualia, uti volui o&longs;tendere. </s> <s>Et ni&longs;i pondera illa <lb/>æqualia e&longs;&longs;ent, Terra tota in libero æthere fluitans ponderi majori <lb/>cederet, & ab eo fugiendo abiret in infinitum. </s> </p> <figure id="fig5"></figure> <p type="main"> <s>Ut corpora in concur&longs;u & reflexione idem pollent, quorum ve­<lb/>locitates &longs;unt reciproce ut vires in&longs;itæ: &longs;ic in movendis In&longs;tru­<lb/>mentis Mechanicis agentia idem pollent & conatibus contrariis &longs;e <lb/>mutuo &longs;u&longs;tinent, quorum velocitates &longs;ecundum determinationem <pb pagenum="23"/>virium æ&longs;timatæ, &longs;unt reciproce ut vires. </s> <s>Sie pondera æquipollent <lb/>ad movenda brachia Libræ, quæ o&longs;cillante Libra &longs;unt reciproce ut <lb/>eorum velocitates &longs;ur&longs;um & deor&longs;um: hoc e&longs;t, pondera, &longs;i recta <lb/>a&longs;cendunt & de&longs;cendunt, æquipollent, quæ &longs;unt reciproce ut pun­<lb/>ctorum a quibus &longs;u&longs;penduntur di&longs;tantiæ ab axe Libræ; &longs;in planis <lb/>obliquis alii&longs;ve admotis ob&longs;taculis impedi<gap/>a a&longs;cendunt vel de&longs;cen­<lb/>dunt oblique, æquipollent quæ &longs;unt reciproce ut a&longs;cen&longs;us & de&longs;cen­<lb/>&longs;us, quatenus facti &longs;ecundum perpendiculum: id adeo ob determi­<lb/>nationem gravitatis deor&longs;um. </s> <s>Similiter in Trochlea &longs;eu Poly&longs;pa&longs;to <lb/>vis manus funem directe trahentis, quæ &longs;it ad pondus vel directe <lb/>vel oblique a&longs;cendens ut velocitas a&longs;cen&longs;us perpendicularis ad ve­<lb/>locitatem manus funem trahentis, &longs;u&longs;tinebit pondus. </s> <s>In Horolo­<lb/>giis & &longs;imilibus in&longs;trumentis, quæ ex rotulis commi&longs;&longs;is con&longs;tructa <lb/>&longs;unt, vires contrariæ ad motum rotularum promovendum & impe­<lb/>diendum, &longs;i &longs;unt reciproce ut velocitates partium rotularum in quas <lb/>imprimuntur, &longs;u&longs;tinebunt &longs;e mutuo. </s> <s>Vis Cochleæ ad premendum <lb/>corpus e&longs;t ad vim manus manubrium circumagentis, ut circularis <lb/>velocitas manubrii ea in parte ubi a manu urgetur, ad velocitatem <lb/>progre&longs;&longs;ivam cochleæ ver&longs;us corpus pre&longs;&longs;um. </s> <s>Vires quibus Cu­<lb/>neus urget partes duas ligni fi&longs;&longs;i &longs;unt ad vim mallei in cuueum, ut <lb/>progre&longs;&longs;us cunei &longs;ecundum determinationem vis a malleo in ip&longs;um <lb/>impre&longs;&longs;æ, ad velocitatem qua partes <gap/>gni cedunt cuneo, &longs;ecundum <lb/>lineas faciebus cunei perpendiculares. </s> <s>Et par e&longs;t ratio Machina­<lb/>rum omnium. </s> </p> <p type="main"> <s>Harum efficacia & u&longs;us in eo &longs;olo con&longs;i&longs;tit, ut diminuendo velo­<lb/>citatem augeamus vim, & contra: Unde &longs;olvitur in omni aptorum <lb/>in&longs;trumentorum genere Problema, <emph type="italics"/>Datum pondus data vi moven­<lb/>di,<emph.end type="italics"/> aliamve datam re&longs;i&longs;tentiam vi data &longs;uperandi. </s> <s>Nam &longs;i Ma­<lb/>chinæ ita formentur, ut velocitates Agentis & Re&longs;i&longs;tentis &longs;ine reci­<lb/>proce ut vires; Agens re&longs;i&longs;tentiam &longs;u&longs;tinebit: & majori cum veloci­<lb/>tatum di&longs;paritate eandem vincet. </s> <s>Certe &longs;i tanta &longs;ic velocitatum <lb/>di&longs;paritas, ut vincatur etiam re&longs;i&longs;tentia omnis, quæ tam ex conti­<lb/>guorum & inter &longs;e labentium corporum attritione, quam ex con­<lb/>tinuorum & ab invicem &longs;eparandorum cohæ&longs;ione & elevandorum <lb/>ponderibus orirj &longs;olet; &longs;uperata omni ea re&longs;i&longs;tentia, vis redun­<lb/>dans accelerationem motus &longs;ibi proportionalem, partim in parti­<lb/>bus machinæ, partim in corpore re&longs;i&longs;tente producet. </s> <s>Ceterum <lb/>Mechanicam tractare non e&longs;t hujus in&longs;tituti. </s> <s>Hi&longs;ce volui tan­<lb/>tum o&longs;tendere, quam late pateat quamque certa &longs;it Lex tertia <lb/>Motus. </s> <s>Nam &longs;i æ&longs;timetur Agentis actio ex ejus vi & veloci-</s> </p> <pb pagenum="24"/> <p type="main"> <s> <arrow.to.target n="note11"></arrow.to.target><lb/>tate conjunctim; & &longs;imiliter Re&longs;i&longs;tentis reactio æ&longs;timetur conjun­<lb/>ctim ex ejus partium &longs;ingularum velocitatibus & viribus re&longs;i&longs;tendi <lb/>ab earum attritione, cohæ&longs;ione, pondere, & acceleratione ori­<lb/>undis; erunt actio & reactio, in omni in&longs;trumentorum u&longs;u, <lb/>&longs;ibi invicem &longs;emper æquales. </s> <s>Et quatenus actio propagatur per <lb/>in&longs;trumentum & ultimo imprimitur in corpus omne re&longs;i&longs;tens, <lb/>ejus ultima determinatio determinationi reactionis &longs;emper erit <lb/>contraria. <lb/><gap desc="hr tag"/></s> </p> <p type="margin"> <s><margin.target id="note11"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>DE <lb/>MOTU CORPORUM <lb/>LIBER PRIMUS.<emph.end type="center"/><lb/><gap desc="hr tag"/></s> </p> <p type="main"> <s><emph type="center"/>SECTIO I.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Methodo Rationum primarum & ultimarum, cujus ope &longs;equentia <lb/>demon&longs;trantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>LEMMA I.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>QUantitates, ut & quantitatum rationes, quæ ad æqualitatem <lb/>tempore quovis finito con&longs;tanter tendunt, & ante finem tempo­<lb/>ris illius propius ad invicem accedunt quam pro data quavis diffe­<lb/>tia, fiunt ultimo æquales.<emph.end type="italics"/></s> </p> <p type="main"> <s>Si negas; fiant ultimò inequales, & &longs;it earum ultima differentia <lb/><emph type="italics"/>D.<emph.end type="italics"/> Ergo nequeunt propius ad æqualitatem accedere quam pro <lb/>data differentia <emph type="italics"/>D:<emph.end type="italics"/> contra hypothe&longs;in. </s> </p> <pb pagenum="2"/> <p type="main"> <s><emph type="center"/>LEMMA II.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si in Figura quavis<emph.end type="italics"/> AacE, <emph type="italics"/>rectis<emph.end type="italics"/> Aa, AE <emph type="italics"/>& curva<emph.end type="italics"/> acE <emph type="italics"/>com <lb/>preben&longs;a, in&longs;cribantur parallelogramma quotcunque<emph.end type="italics"/> Ab, Bc, Cd <lb/>&c. <emph type="italics"/>&longs;ub ba&longs;ibus<emph.end type="italics"/> AB, BC, CD, &c. <emph type="italics"/>æqualibus, & lateribu<emph.end type="italics"/><lb/>Bb, Cc, Dd, &c. <emph type="italics"/>Figuræ lateri<emph.end type="italics"/> Aa <emph type="italics"/>pa­<lb/>rallelis contenta; & compleantur paral-<emph.end type="italics"/><lb/> <arrow.to.target n="fig6"></arrow.to.target><lb/><emph type="italics"/>lelogramma<emph.end type="italics"/> aKbl, bLcm, cMdn, &c. <lb/><emph type="italics"/>Dein boru<gap/> parallelogr ammorum lati­<lb/>tudo minuatur, & numerus augeatur <lb/>in infinitum: dico quod ultimæ rationes, <lb/>quas babent ad &longs;e invicem Figura in­<lb/>&longs;cripta<emph.end type="italics"/> AKbLcMdD, <emph type="italics"/>circum&longs;cripta<emph.end type="italics"/><lb/>AalbmcndoE, <emph type="italics"/>& curvilinea<emph.end type="italics"/> AbcdE, <lb/><emph type="italics"/>&longs;unt rationes æqualitatis.<emph.end type="italics"/></s> </p> <figure id="fig6"></figure> <p type="main"> <s>Nam Figuræ in&longs;criptæ & circum&longs;criptæ differentia e&longs;t &longs;umma pa­<lb/>rallelogrammorum <emph type="italics"/>Kl, Lm, Mn, Do,<emph.end type="italics"/> hoc e&longs;t (ob æquales om­<lb/>nium ba&longs;es) rectangulum &longs;ub unius ba&longs;i <emph type="italics"/>Kb<emph.end type="italics"/> & altitudinum &longs;umma <lb/><emph type="italics"/>Aa,<emph.end type="italics"/> id e&longs;t, rectangulum <emph type="italics"/>ABla.<emph.end type="italics"/> Sed hoc rectangulum, eo quod <lb/>latitudo ejus <emph type="italics"/>AB<emph.end type="italics"/> in infinitum minuitur, fit minus quovis dato. </s> <s>Er­<lb/>go (per Lemma 1) Figura in&longs;cripta & circum&longs;cripta & multo magis <lb/>Figura curvilinea intermedia fiunt ultimo æquales. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>LEMMA III.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Eædem rationes ultimæ &longs;unt etiam rationes æqualitatis, ubi par al­<lb/>lelogr ammorum latitudines<emph.end type="italics"/> AB, BC, CD, &c. <emph type="italics"/>&longs;unt inæquales, <lb/>& omnes minuuntur in infinitum.<emph.end type="italics"/></s> </p> <p type="main"> <s>Sit enim <emph type="italics"/>AF<emph.end type="italics"/> æqualis latitudini maximæ, & compleatur paralle­<lb/>logrammum <emph type="italics"/>FAaf.<emph.end type="italics"/> Hoc erit majus quam differentia Figuræ in­<lb/>&longs;criptæ & Figuræ circum&longs;criptæ; at latitudine &longs;ua <emph type="italics"/>AF<emph.end type="italics"/> in infinitum <lb/>diminuta, minus fiet quam datum quodvis rectangulum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;umma ultima parallelogrammorum evane&longs;centium <lb/>coincidit omni ex parte cum Figura curvilinea. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et multo magis Figura rectilinea, quæ chordis evane&longs;- <pb pagenum="26"/> <arrow.to.target n="note12"></arrow.to.target><lb/>centium arcuum <emph type="italics"/>ab, bc, cd, &c.<emph.end type="italics"/> comprehenditur, coincidit ultimo <lb/>cum Figura curvilinea. </s> </p> <p type="margin"> <s><margin.target id="note12"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Ut & Figura rectilinea circum&longs;cripta quæ tangentibus <lb/>eorundem arcuum comprehenditur. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Et propterea hæ Figuræ ultimæ (quoad perimetros <emph type="italics"/>acE,<emph.end type="italics"/>) <lb/>non &longs;unt rectilineæ, &longs;ed rectilinearum limites curvilinei. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA IV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si in duabus Figuris<emph.end type="italics"/> AacE, PprT, <emph type="italics"/>in&longs;cribantur (ut &longs;upra) duæ <lb/>parallelogrammorum &longs;eries, &longs;itque idem amborum numerus, & ubi <lb/>latitudines in infinitum diminuuntur, rationes ultimæ parallelo­<lb/>grammorum in una Figura ad parallelogramma in altera, &longs;ingulorum <lb/>ad fingula, &longs;int eædem; dico quod Figuræ duæ<emph.end type="italics"/> AacE, PprT, <lb/><emph type="italics"/>&longs;unt ad invicem in eadem illa ratione.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s>Etenim ut &longs;unt parallelogramma &longs;ingula ad &longs;ingula, ita (compo­<lb/>nendo) fit &longs;umma omnium ad &longs;ummam omnium, & ita Figura ad <lb/>Figuram; exi&longs;tente nimirum Figura priore (per Lemma 111) ad &longs;um­<lb/>mam priorem, & Figura po&longs;teriore ad &longs;ummam po&longs;teriorem in ra­<lb/>tione æqualitatis. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Hinc &longs;i duæ cuju&longs;cunque generis quantitates in eundem <lb/>partium numerum utcunque dividantur; & partes illæ, ubi numerus <lb/>earum augetur & magnitudo diminuitur in infinitum, datam obti­<lb/>neant rationem ad invicem, prima ad primam, &longs;ecunda ad &longs;ecundam, <lb/>cæteræque &longs;uo ordine ad cæteras: erunt tota ad invicem in eadem <lb/>illa data ratione. </s> <s>Nam &longs;i in Lemmatis hujus Figuris &longs;umantur pa- <pb pagenum="27"/>rallelogramma inter &longs;e ut partes, &longs;ummæ partium &longs;emper erunt ut <lb/>&longs;ummæ parallelogrammorum; atque adeo, ubi partium & paralle­<lb/>logrammorum numerus augetur & magnitudo diminuitur in infini­<lb/>tum, in ultima ratione parallelogrammi ad parallelogrammum, id <lb/>e&longs;t (per hypothe&longs;in) in ultima ratione partis ad partem. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA V.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Similium Figurarum latera omnia, quæ &longs;ibi mutuo re&longs;pondent, &longs;unt <lb/>proportionalia, tam curvilinea quam rectilinea; & areæ &longs;unt in <lb/>duplicata ratione laterum.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>LEMMA VI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si arcus quilibet po&longs;itione datus<emph.end type="italics"/> AB <emph type="italics"/>&longs;ub-<emph.end type="italics"/><lb/> <arrow.to.target n="fig7"></arrow.to.target><lb/><emph type="italics"/>tendatur chorda<emph.end type="italics"/> AB, <emph type="italics"/>& in puncto <lb/>aliquo<emph.end type="italics"/> A, <emph type="italics"/>in medio curvaturæ continuæ, <lb/>tangatur a recta utrinque producta<emph.end type="italics"/><lb/>AD; <emph type="italics"/>dein puncta<emph.end type="italics"/> A, B <emph type="italics"/>ad invicem <lb/>accedant & coëant; dico quod angulus<emph.end type="italics"/><lb/>BAD, <emph type="italics"/>&longs;ub chorda & tangente conten­<lb/>tus, minuetur in infinitum & ultimo e­<lb/>vane&longs;cet.<emph.end type="italics"/></s> </p> <figure id="fig7"></figure> <p type="main"> <s>Nam &longs;i angulus ille non evane&longs;cit, continebit arcus <emph type="italics"/>AB<emph.end type="italics"/> cum tan­<lb/>gente <emph type="italics"/>AD<emph.end type="italics"/> angulum rectilineo æqualem, & propterea curvatura ad <lb/>ad punctum <emph type="italics"/>A<emph.end type="italics"/> non erit continua, contra hypothe&longs;in. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA VII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis; dico quod ultima ratio arcus, chordæ, & tangentis <lb/>ad invicem est ratio æqualitatis.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam dum punctum <emph type="italics"/>B<emph.end type="italics"/> ad punctum <emph type="italics"/>A<emph.end type="italics"/> accedit, intelligantur &longs;emper <lb/><emph type="italics"/>AB<emph.end type="italics"/> & <emph type="italics"/>AD<emph.end type="italics"/> ad puncta longinqua <emph type="italics"/>b<emph.end type="italics"/> ac <emph type="italics"/>d<emph.end type="italics"/> product, & &longs;ecanti <emph type="italics"/>BD<emph.end type="italics"/><lb/>parallela agatur <emph type="italics"/>bd.<emph.end type="italics"/> Sitque arcus <emph type="italics"/>Ab<emph.end type="italics"/> &longs;emper &longs;imilis arcui <emph type="italics"/>AB.<emph.end type="italics"/><lb/>Et punctis <emph type="italics"/>A, B<emph.end type="italics"/> coeuntibus, angulus <emph type="italics"/>dAb,<emph.end type="italics"/> per Lemma &longs;uperius, <lb/>evane&longs;cet; adeoque rectæ &longs;emper &longs;initæ <emph type="italics"/>Ab, Ad<emph.end type="italics"/> & arcus interme­<lb/>dius <emph type="italics"/>Ab<emph.end type="italics"/> coincident, & propterea æquales erunt. </s> <s>Unde & hi&longs;ce <lb/>&longs;emper proportionales rectæ <emph type="italics"/>AB, AD,<emph.end type="italics"/> & arcus intermedius <emph type="italics"/>AB<emph.end type="italics"/> <pb pagenum="28"/> <arrow.to.target n="note13"></arrow.to.target><lb/>evane&longs;cent, & rationem ultimam habebunt æqualitatis. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note13"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Unde &longs;i per <emph type="italics"/>B<emph.end type="italics"/> ducatur tangenti parallela <emph type="italics"/>BF,<emph.end type="italics"/> rectam <lb/>quamvis <emph type="italics"/>AF<emph.end type="italics"/> per <emph type="italics"/>A<emph.end type="italics"/> tran&longs;e­<lb/> <arrow.to.target n="fig8"></arrow.to.target><lb/>untem perpetuo &longs;ecans in <emph type="italics"/>F,<emph.end type="italics"/><lb/>hæc <emph type="italics"/>BF<emph.end type="italics"/> ultimo ad arcum e­<lb/>vane&longs;centem <emph type="italics"/>AB<emph.end type="italics"/> rationem <lb/>habebit æqualitatis, eo quod <lb/>completo parallelogrammo <emph type="italics"/>AFBD<emph.end type="italics"/> rationem &longs;emper habet æqua­<lb/>litatis ad <emph type="italics"/>AD.<emph.end type="italics"/></s> </p> <figure id="fig8"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et &longs;i per <emph type="italics"/>B<emph.end type="italics"/> & <emph type="italics"/>A<emph.end type="italics"/> ducantur plures rectæ <emph type="italics"/>BE, BD, AF, <lb/>AG,<emph.end type="italics"/> &longs;ecantes tangentem <emph type="italics"/>AD<emph.end type="italics"/> & ip&longs;ius parallelam <emph type="italics"/>BF<emph.end type="italics"/>; ratio ulti­<lb/>ma ab&longs;ci&longs;&longs;arum omnium <emph type="italics"/>AD, AE, BF, BG,<emph.end type="italics"/> chordæque & ar­<lb/>cus <emph type="italics"/>AB<emph.end type="italics"/> ad invicem erit ratio æqualitatis. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Et propterea hæ omnes lineæ, in omni de rationibus ul­<lb/>timis argumentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA VIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si rectæ datæ<emph.end type="italics"/> AR, BR <emph type="italics"/>cum arcu<emph.end type="italics"/> AB, <emph type="italics"/>chorda<emph.end type="italics"/> AB <emph type="italics"/>& tangente<emph.end type="italics"/><lb/>AD, <emph type="italics"/>triangula tria<emph.end type="italics"/> ARB, ARB, ARD <emph type="italics"/>con&longs;tituunt, dein <lb/>puncta<emph.end type="italics"/> A, B <emph type="italics"/>accedunt ad invicem: dico quod ultima forma <lb/>triangulorum evane&longs;centium est &longs;imilitudinis, & ultima ratio <lb/>æqualitatis.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam dum punctum <emph type="italics"/>B<emph.end type="italics"/> ad punctum <emph type="italics"/>A<emph.end type="italics"/><lb/> <arrow.to.target n="fig9"></arrow.to.target><lb/>accedit, <expan abbr="intelligãtur">intelligantur</expan> &longs;emper <emph type="italics"/>AB, AD, AR<emph.end type="italics"/><lb/>ad puncta longinqua <emph type="italics"/>b, d<emph.end type="italics"/> & <emph type="italics"/>r<emph.end type="italics"/> produci, <lb/>ip&longs;ique <emph type="italics"/>RD<emph.end type="italics"/> parallela agi <emph type="italics"/>rbd,<emph.end type="italics"/> & arcui <lb/><emph type="italics"/>AB<emph.end type="italics"/> &longs;imilis &longs;emper &longs;it arcus <emph type="italics"/>Ab.<emph.end type="italics"/> Et coe­<lb/>untibus punctis <emph type="italics"/>A, B,<emph.end type="italics"/> angulus <emph type="italics"/>bAd<emph.end type="italics"/> eva­<lb/>ne&longs;cet, & propterea triangula tria &longs;emper <lb/>finita <emph type="italics"/>rAb, rAb, rAd<emph.end type="italics"/> coincident, &longs;unt­<lb/>que eo nomine &longs;imilia & æqualia. </s> <s>Unde <lb/>& hi&longs;ce &longs;emper &longs;imilia & proportionalia <lb/><emph type="italics"/>RAB, RAB, RAD<emph.end type="italics"/> &longs;ient ultimo &longs;ibi <lb/>invicem &longs;imilia & æqualia. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <figure id="fig9"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Et hinc triangula illa, in omni de rationibus ultimis argu­<lb/>mentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt. </s> </p> <pb pagenum="29"/> <p type="main"> <s><emph type="center"/>LEMMA IX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si recta<emph.end type="italics"/> AE <emph type="italics"/>& curva<emph.end type="italics"/> ABC <emph type="italics"/>po&longs;itione datæ &longs;e mutuo &longs;ecent in <lb/>angulo dato<emph.end type="italics"/> A, <emph type="italics"/>& ad rectam illam in alio dato angulo ordina­<lb/>tim applicentur<emph.end type="italics"/> BD, CE, <emph type="italics"/>curvæ occurrentes in<emph.end type="italics"/> B, C; <emph type="italics"/>dein <lb/>puncta<emph.end type="italics"/> B, C <emph type="italics"/>&longs;imul accedant ad punctum<emph.end type="italics"/> A: <emph type="italics"/>dico quod areæ tri­<lb/>angulorum<emph.end type="italics"/> ABD, ACE <emph type="italics"/>erunt ultimo ad invicem in duplicata <lb/>ratione laterum.<emph.end type="italics"/></s> </p> <p type="main"> <s>Etenim dum puncta <emph type="italics"/>B, C<emph.end type="italics"/> acce­<lb/> <arrow.to.target n="fig10"></arrow.to.target><lb/>dunt ad punctum <emph type="italics"/>A,<emph.end type="italics"/> intelligatur <lb/>&longs;emper <emph type="italics"/>AD<emph.end type="italics"/> produci ad puncta lon­<lb/>ginqua <emph type="italics"/>d<emph.end type="italics"/> & <emph type="italics"/>e,<emph.end type="italics"/> ut &longs;int <emph type="italics"/>Ad, Ae<emph.end type="italics"/> ip­<lb/>&longs;is <emph type="italics"/>AD, AE<emph.end type="italics"/> proportionales, & e­<lb/>rigantur ordinatæ <emph type="italics"/>db, ec<emph.end type="italics"/> ordina­<lb/>tis <emph type="italics"/>DB, EC<emph.end type="italics"/> parallelæ quæ occur­<lb/>rant ip&longs;is <emph type="italics"/>AB, AC<emph.end type="italics"/> productis in <lb/><emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>c.<emph.end type="italics"/> Duci intelligatur, tum curva <lb/><emph type="italics"/>Abc<emph.end type="italics"/> ip&longs;i <emph type="italics"/>ABC<emph.end type="italics"/> &longs;imilis, tum recta <lb/><emph type="italics"/>Ag,<emph.end type="italics"/> quæ tangat curvam utramque <lb/>in <emph type="italics"/>A,<emph.end type="italics"/> & &longs;ecet ordinatim applica­<lb/>tas <emph type="italics"/>DB, EC, db, ec<emph.end type="italics"/> in <emph type="italics"/>F, G, f, g.<emph.end type="italics"/><lb/>Tum manente longitudine <emph type="italics"/>Ae<emph.end type="italics"/> coeant puncta <emph type="italics"/>B, C<emph.end type="italics"/> cum puncto <emph type="italics"/>A<emph.end type="italics"/>; <lb/>& angulo <emph type="italics"/>cAg<emph.end type="italics"/> evane&longs;cente, coincident areæ curvilineæ <emph type="italics"/>Abd, Ace<emph.end type="italics"/><lb/>cum rectilineis <emph type="italics"/>Afd, Age:<emph.end type="italics"/> adeoque (per Lemma v) erunt in dupli­<lb/>cata ratione laterum <emph type="italics"/>Ad, A<gap/>:<emph.end type="italics"/> Sed his areis proportionales &longs;emper <lb/>&longs;unt areæ <emph type="italics"/>ABD, ACE,<emph.end type="italics"/> & his lateribus latera <emph type="italics"/>AD, AE.<emph.end type="italics"/> Ergo & <lb/>areæ <emph type="italics"/>ABD, ACE<emph.end type="italics"/> &longs;unt ultimo in duplicata ratione laterum <emph type="italics"/>AD, <lb/>AE. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <figure id="fig10"></figure> <p type="main"> <s><emph type="center"/>LEMMA X.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Spatia quæ corpus urgente quacunque Vi finita de&longs;cribit, five Vis <lb/>illa determinata & immutabilis &longs;it, five eadem continuo auge­<lb/>atur vel continuo diminuatur, &longs;unt ip&longs;o motus initio in duplica­<lb/>ta ratione Temporum.<emph.end type="italics"/></s> </p> <p type="main"> <s>Exponantur tempora per lineas <emph type="italics"/>AD, AE,<emph.end type="italics"/> & velocitates genitæ <lb/>per ordinatas <emph type="italics"/>DB, EC<emph.end type="italics"/>; & &longs;patia his velocitatibus de&longs;cripta, erunt <lb/>ut areæ <emph type="italics"/>ABD, ACE<emph.end type="italics"/> his ordinatis de&longs;criptæ, hoc e&longs;t, ip&longs;o motus <lb/>initio (per Lemma IX) in duplicata ratione remporum <emph type="italics"/>AD, AE. <lb/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/> <pb pagenum="30"/> <arrow.to.target n="note14"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note14"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Et hinc facile colligitur, quod corporum &longs;imiles &longs;imi­<lb/>lium Figurarum partes temporibus proportionalibus de&longs;cribentium <lb/>Errores, qui viribus quibu&longs;vis æqualibus ad corpora &longs;imiliter ap­<lb/>plicatis generantur, & men&longs;urantur per di&longs;tantias corporum a Fi­<lb/>gurarum &longs;imilium locis illis ad quæ corpora eadem temporibus ii&longs;­<lb/>dem proportionalibus ab&longs;que viribus i&longs;tis pervenirent, &longs;unt ut qua­<lb/>drata temporum in quibus generantur quam proxime. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Errores autem qui viribus proportionalibus ad &longs;imiles <lb/>Figurarum &longs;imilium partes &longs;imiliter applicatis generantur, &longs;unt ut <lb/>vires & quadrata temporum conjunctim. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Idem intelligendum e&longs;t de &longs;patiis quibu&longs;vis quæ corpo­<lb/>ra urgentibus diver&longs;is viribus de&longs;cribunt. </s> <s>Hæc &longs;unt, ip&longs;o motus ini­<lb/>tio, ut vires & quadrata temporum conjunctim. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Ideoque vires &longs;unt ut &longs;patia, ip&longs;o motus initio, de&longs;cripta <lb/>directe & quadrata temporum inver&longs;e. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Et quadrata temporum &longs;unt ut de&longs;cripta &longs;patia directe <lb/>& vires inver&longs;e. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Si quantitates indeterminatæ diver&longs;orum generum conferantur <lb/>inter &longs;e, & earum aliqua dicatur e&longs;&longs;e ut e&longs;t alia quævis directe vel <lb/>inver&longs;e: &longs;en&longs;us e&longs;t, quod prior augetur vel diminuitur in eadem <lb/>ratione cum po&longs;teriore, vel cum ejus reciproca. </s> <s>Et &longs;i earum aliqua <lb/>dicatur e&longs;&longs;e ut &longs;unt aliæ duæ vel plures directe vel inver&longs;e: &longs;en&longs;us <lb/>e&longs;t, quod prima augetur vel diminuitur in ratione quæ componitur <lb/>ex rationibus in quibus aliæ vel aliarum reciprocæ augentur vel di­<lb/>minuuntur. </s> <s>Ut &longs;i A dicatur e&longs;&longs;e ut B directe & C directe & D in­<lb/>ver&longs;e: &longs;en&longs;us e&longs;t, quod A augetur vel diminuitur in eadem ratione <lb/>cum BXCX1/D, hoc e&longs;t, quod A & (BC/D) &longs;unt ad invicem in ratio­<lb/>ne data. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA XI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Subten&longs;a evane&longs;cens anguli contactus, in curvis omnibus curvatu­<lb/>ram finitam ad punctum contactus habentibus, est ultimo in ra­<lb/>tione duplicata &longs;ubten&longs;æ arcus contermini.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/> 1. Sit arcus ille <emph type="italics"/>AB,<emph.end type="italics"/> tangens ejus <emph type="italics"/>AD,<emph.end type="italics"/> &longs;ubten&longs;a anguli con­<lb/>tactus ad tangentem perpendicularis <emph type="italics"/>BD,<emph.end type="italics"/> &longs;ubten&longs;a arcus <emph type="italics"/>AB.<emph.end type="italics"/> Huic <lb/>&longs;ubten&longs;æ <emph type="italics"/>AB<emph.end type="italics"/> & tangenti <emph type="italics"/>AD<emph.end type="italics"/> perpendiculares erigantur <emph type="italics"/>AG, BG,<emph.end type="italics"/> <pb pagenum="31"/>concurrentes in <emph type="italics"/>G<emph.end type="italics"/>; dein accedant puncta <emph type="italics"/>D, B, G,<emph.end type="italics"/> ad puncta <emph type="italics"/>d, b, g,<emph.end type="italics"/><lb/>&longs;itque <emph type="italics"/>J<emph.end type="italics"/> inter&longs;ectio linearum <emph type="italics"/>BG, AG<emph.end type="italics"/> ultimo facta ubi puncta <emph type="italics"/>D, B<emph.end type="italics"/><lb/>accedunt u&longs;que ad <emph type="italics"/>A.<emph.end type="italics"/> Manife&longs;tum e&longs;t quod di&longs;tantia <emph type="italics"/>GJ<emph.end type="italics"/> minor <lb/>e&longs;&longs;e pote&longs;t quam a&longs;&longs;ignata quævis. </s> <s>E&longs;t autem (ex natura circulorum <lb/>per puncta <emph type="italics"/>ABG, Abg<emph.end type="italics"/> tran&longs;euntium) <emph type="italics"/>ABquad.<emph.end type="italics"/><lb/> <arrow.to.target n="fig11"></arrow.to.target><lb/>æquale <emph type="italics"/>AGXBD,<emph.end type="italics"/> & <emph type="italics"/>Ab quad.<emph.end type="italics"/> æquale <emph type="italics"/>AgXbd,<emph.end type="italics"/><lb/>adeoque ratio <emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad.<emph.end type="italics"/> compo­<lb/>nitur ex rationibus <emph type="italics"/>AG<emph.end type="italics"/> ad <emph type="italics"/>Ag<emph.end type="italics"/> & <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd.<emph.end type="italics"/><lb/>Sed quoniam <emph type="italics"/>GJ<emph.end type="italics"/> a&longs;&longs;umi pote&longs;t minor longitu­<lb/>dine quavis a&longs;&longs;ignata, fieri pote&longs;t ut ratio <emph type="italics"/>AG<emph.end type="italics"/><lb/>ad <emph type="italics"/>Ag<emph.end type="italics"/> minus differat a ratione æqualitatis quam <lb/>pro differentia quavis a&longs;&longs;ignata, adeoque ut ra­<lb/>tio <emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad.<emph.end type="italics"/> minus differat a ra­<lb/>tione <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd<emph.end type="italics"/> quam pro differentia quavis <lb/>a&longs;&longs;ignata. </s> <s>E&longs;t ergo, per Lemma 1, ratio ultima <lb/><emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad.<emph.end type="italics"/> æqualis rationi ultimæ <lb/><emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <figure id="fig11"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Inclinetur jam <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>AD<emph.end type="italics"/> in angulo <lb/>quovis dato, & eadem &longs;emper erit ratio ultima <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd<emph.end type="italics"/> quæ <lb/>prius, adeoque eadem ae <emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 3. Et quamvis angulus <emph type="italics"/>D<emph.end type="italics"/> non detur, &longs;ed recta <emph type="italics"/>BD<emph.end type="italics"/> ad da­<lb/>tum punctum convergente, vel alia quacunque lege con&longs;tituatur; <lb/>tamen anguli <emph type="italics"/>D, d<emph.end type="italics"/> communi lege con&longs;tituti ad æqualitatem &longs;emper <lb/>vergent & propius accedent ad invicem quam pro differentia qua­<lb/>vis a&longs;&longs;ignata, adeoque ultimo æquales erunt, per Lem. <gap/> & prop­<lb/>terea lineæ <emph type="italics"/>BD, bd<emph.end type="italics"/> &longs;unt in eadem ratione ad invicem ac prius. <lb/><emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Unde eum tangentes <emph type="italics"/>AD, Ad,<emph.end type="italics"/> arcus <emph type="italics"/>AB, Ab,<emph.end type="italics"/> & eo­<lb/>rum &longs;inus <emph type="italics"/>BC, bc<emph.end type="italics"/> fiant ultimo chordis <emph type="italics"/>AB, Ab<emph.end type="italics"/> æquales; erunt <lb/>etiam illorum quadrata ultimo ut &longs;ubten&longs;æ <emph type="italics"/>BD, bd.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Eorundem quadrata &longs;unt etiam ultimo ut &longs;unt arcuum <lb/>&longs;agittæ quæ chordas bi&longs;ecant & ad datum punctum conver gunt. </s> <s><lb/>Nam &longs;agittæ illæ &longs;unt ut &longs;ubten&longs;æ <emph type="italics"/>BD, bd.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Ideoque &longs;agitta e&longs;t in duplicata ratione temporis quo <lb/>corpus data velocitate de&longs;cribit arcum. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Triangula rectilinea <emph type="italics"/>ADB, Adb<emph.end type="italics"/> &longs;unt ultimo in tripli­<lb/>cata ratione laterum <emph type="italics"/>AD, Ad,<emph.end type="italics"/> inque &longs;e&longs;quiplicata laterum <emph type="italics"/>DB, <lb/>db<emph.end type="italics"/>; utpote in compo&longs;ita ratione laterum <emph type="italics"/>AD,<emph.end type="italics"/> & <emph type="italics"/>DB, Ad<emph.end type="italics"/> & <emph type="italics"/>db<emph.end type="italics"/><lb/>exi&longs;tentia. </s> <s>Sic & triangula <emph type="italics"/>ABC, Abc<emph.end type="italics"/> &longs;unt ultimo in triplicata <lb/>ratione laterum <emph type="italics"/>BC, bc.<emph.end type="italics"/> Rationem vero Se&longs;quiplicatam voco tri­<lb/>plicatæ &longs;ubduplicatam, quæ nempe ex &longs;implici & &longs;ubduplicata com­<lb/>ponitur, quamque alias Se&longs;quialteram dicunt. </s> </p> <pb pagenum="32"/> <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/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Et quoniam <emph type="italics"/>DB, db<emph.end type="italics"/> &longs;unt ultimo parallelæ & in dupli­<lb/>cata ratione ip&longs;arum <emph type="italics"/>AD, Ad:<emph.end type="italics"/> erunt areæ ultimæ curvilineæ <emph type="italics"/>ADB, <lb/>Adb<emph.end type="italics"/> (ex natura Parabolæ) duæ tertiæ partes triangulorum rectili­<lb/>neorum <emph type="italics"/>ADB, Adb<emph.end type="italics"/>; & &longs;egmenta <emph type="italics"/>AB, Ab<emph.end type="italics"/> partes tertiæ eo­<lb/>rundem triangulorum. </s> <s>Et inde hæ areæ & hæc &longs;egmenta erunt in <lb/>triplicata ratione tum tangentium <emph type="italics"/>AD, Ad<emph.end type="italics"/>; tum chordarum & <lb/>arcuum <emph type="italics"/>AB, Ab.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Cæterum in his omnibus &longs;upponimus angulum contactus nec in­<lb/>finite majorem e&longs;&longs;e angulis contactuum, quos Circuli continent cum <lb/>tangentibus &longs;uis, nec ii&longs;dem infinite minorem; hoc e&longs;t, curvaturam <lb/>ad punctum <emph type="italics"/>A,<emph.end type="italics"/> nec infinite parvam e&longs;&longs;e nec infinite magnam, &longs;eu <lb/>intervallum <emph type="italics"/>AJ<emph.end type="italics"/> finitæ e&longs;&longs;e magnitudinis. </s> <s>Capi enim pote&longs;t <emph type="italics"/>DB<emph.end type="italics"/><lb/>ut <emph type="italics"/>AD<emph type="sup"/>3<emph.end type="sup"/>:<emph.end type="italics"/> quo in ca&longs;u Circulus nullus per punctum <emph type="italics"/>A<emph.end type="italics"/> inter tangen­<lb/>tem <emph type="italics"/>AD<emph.end type="italics"/> & curvam <emph type="italics"/>AB<emph.end type="italics"/> duci pote&longs;t, proindeque angulus contactus <lb/>erit infinite minor Circularibus. </s> <s>Et &longs;imili argumento &longs;i fiat <emph type="italics"/>DB<emph.end type="italics"/><lb/>&longs;ucce&longs;&longs;ive ut <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>4<emph.end type="sup"/>, <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>5<emph.end type="sup"/>, <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>6<emph.end type="sup"/>, <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>7<emph.end type="sup"/>, &c. </s> <s>habebitur &longs;eries an­<lb/>gulorum contactus pergens in infinitum, quorum quilibet po&longs;te­<lb/>rior e&longs;t infinite minor priore. </s> <s>Et &longs;i fiat <emph type="italics"/>DB<emph.end type="italics"/> &longs;ucce&longs;&longs;ive ut <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>2<emph.end type="sup"/>, <lb/><emph type="italics"/>AD<emph.end type="italics"/>3/2, <emph type="italics"/>AD<emph.end type="italics"/>4/3, <emph type="italics"/>AD<emph.end type="italics"/>5/4, <emph type="italics"/>AD<emph.end type="italics"/>6/5, <emph type="italics"/>AD<emph.end type="italics"/>7/6, &c. </s> <s>habebitur alia &longs;eries infinita <lb/>angulorum contactus, quorum primus e&longs;t eju&longs;dem generis cum Cir­<lb/>cularibus, &longs;ecundus infinite major, & quilibet po&longs;terior infinite ma­<lb/>jor priore. </s> <s>Sed & inter duos quo&longs;vis ex his angulis pote&longs;t &longs;eries <lb/>utrinque in infinitum pergens angulorum intermediorum in&longs;eri, <lb/>quorum quilibet po&longs;terior erit infinite major minorve priore. </s> <s>Ut <lb/>&longs;i inter terminos <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>2<emph.end type="sup"/> & <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/> in&longs;eratur &longs;eries <emph type="italics"/>AD<emph.end type="italics"/>(13/6), <emph type="italics"/>AD<emph.end type="italics"/>(1<gap/>/5), <lb/><emph type="italics"/>AD<emph.end type="italics"/>9/4, <emph type="italics"/>AD<emph.end type="italics"/>7/3, <emph type="italics"/>AD<emph.end type="italics"/>5/2, <emph type="italics"/>AD<emph.end type="italics"/><gap/>/3, <emph type="italics"/>AD<emph.end type="italics"/>(11/4), <emph type="italics"/>AD<emph.end type="italics"/>(14/5), <emph type="italics"/>AD<emph.end type="italics"/>(17/6), &c. </s> <s>Et rur­<lb/>&longs;us inter binos quo&longs;vis angulos hujus &longs;eriei in&longs;eri pote&longs;t &longs;eries no­<lb/>va angulorum intermediorum ab invicem infinitis intervallis diffe­<lb/>rentium. </s> <s>Neque novit natura limitem. </s> </p> <p type="main"> <s>Quæ de curvis lineis deque &longs;uperficiebus comprehen&longs;is demon­<lb/>&longs;trata &longs;unt, facile applicantur ad &longs;olidorum &longs;uperficies curvas & <lb/>contenta. </s> <s>Præmi&longs;i vero hæc Lemmata, ut effugerem tædium dedu­<lb/>cendi perplexas demon&longs;trationes, more veterum Geometrarum, ad <lb/>ab&longs;urdum. </s> <s>Contractiores enim redduntur demon&longs;trationes per me­<lb/>thodum Indivi&longs;ibilium. </s> <s>Sed quoniam durior e&longs;t Indivi&longs;ibilium hy­<lb/>pothe&longs;is, & propterea methodus illa minus Geometrica cen&longs;etur; <lb/>malui demon&longs;trationes rerum &longs;equentium ad ultimas quantitatum <pb pagenum="33"/>evane&longs;centium &longs;ummas & rationes, prima&longs;que na&longs;centium, id e&longs;t, <lb/>ad limites &longs;ummarum & rationum deducere; & propterea limitum <lb/>illorum demon&longs;trationes qua potui brevitate præmittere. </s> <s>His enim <lb/>idem præ&longs;tatur quod per methodum Indivi&longs;ibilium; & principiis de­<lb/>mon&longs;tratis jam tutius utemur. </s> <s>Proinde in &longs;equentibus, &longs;iquando <lb/>quantitates tanquam ex particulis con&longs;tantes con&longs;ideravero, vel &longs;i <lb/>pro rectis u&longs;urpavero lineolas curvas; nolim indivi&longs;ibilia, &longs;ed eva­<lb/>ne&longs;centia divi&longs;ibilia, non &longs;ummas & rationes partium determinata­<lb/>rum, &longs;ed &longs;ummarum & rationum limites &longs;emper intelligi; vimque <lb/>talium demon&longs;trationum ad methodum præcedentium Lemmatum <lb/>&longs;emper revocari. </s> </p> <p type="main"> <s>Objectio e&longs;t, quod quantitatum evane&longs;centium nulla &longs;it ultima <lb/>proportio; quippe quæ, antequam evanuerunt, non e&longs;t ultima, ubi <lb/>evanuerunt, nulla e&longs;t. </s> <s>Sed & eodem argumento æque contendi po&longs;&longs;et <lb/>nullam e&longs;&longs;e corporis ad certum locum pervenientis velocitatem ul­<lb/>timam: hanc enim, antequam corpus attingit locum, non e&longs;&longs;e ulti­<lb/>mam, ubiattingit, nullam e&longs;&longs;e. </s> <s>Et re&longs;pon&longs;io facilis e&longs;t: Per velocita­<lb/>tem ultimam intelligi eam, qua corpus movetur neque antequam <lb/>attingit locum ultimum & motus ce&longs;&longs;at, neque po&longs;tea, &longs;ed tunc <lb/>cum attingit; id e&longs;t, illam ip&longs;am velocitatem quacum corpus attin­<lb/>git locum ultimum & quacum motus ce&longs;&longs;at. </s> <s>Et &longs;imiliter per ulti­<lb/>mam rationem quantitatum evane&longs;centium, intelligendam e&longs;&longs;e ratio­<lb/>nem quantitatum non antequam evane&longs;cunt, non po&longs;tea, &longs;ed qua­<lb/>cum evane&longs;cunt. </s> <s>Pariter & ratio prima na&longs;centium e&longs;t ratio qua­<lb/>cum na&longs;cuntur. </s> <s>Et &longs;umma prima & ultima e&longs;t quacum e&longs;&longs;e (vel <lb/>augeri & minui) incipiunt & ce&longs;&longs;ant. </s> <s>Extat limes quem velocitas <lb/>in fine motus attingere pote&longs;t, non autem tran&longs;gredi. </s> <s>Hæc e&longs;t <lb/>velocitas ultima. </s> <s>Et par e&longs;t ratio limitis quantitatum & propor­<lb/>tionum omnium incipientium & ce&longs;&longs;antium. </s> <s>Cumque hic limes <lb/>&longs;it certus & definitus, Problema e&longs;t vere Geometricum eundem de­<lb/>terminare. </s> <s>Geometrica vero omnia in aliis Geometricis determi­<lb/>nandis ac demon&longs;trandis legitime u&longs;urpantur. </s> </p> <p type="main"> <s>Contendi etiam pote&longs;t, quod &longs;i dentur ultimæ quantitatum eva­<lb/>ne&longs;centium rationes, dabuntur & ultimæ magnitudines: & &longs;ic quan­<lb/>titas omnis con&longs;tabit ex Indivi&longs;ibilibus, contra quam <emph type="italics"/>Euclides<emph.end type="italics"/> de <lb/>Incommen&longs;urabilibus, in libro decimo Elementorum, demon&longs;travit. </s> <s><lb/>Verum hæc Objectio fal&longs;æ innititur hypothe&longs;i. </s> <s>Ultimæ rationes <lb/>illæ quibu&longs;cum quantitates evane&longs;cunt, revera non &longs;unt rationes <lb/>quantitatum ultimarum, &longs;ed limites ad quos quantitatum &longs;ine limi­<lb/>te decre&longs;centium rationes &longs;emper appropinquant; & quas propius <lb/>a&longs;&longs;equi po&longs;&longs;unt quam pro data quavis differentia, nunquam vero </s> </p> <pb pagenum="34"/> <p type="main"> <s> <arrow.to.target n="note16"></arrow.to.target><lb/>tran&longs;gredi, neque prius attingere quam quantitates diminuuntur in <lb/>infinitum. </s> <s>Res clarius intelligetur in infinite magnis. </s> <s>Si quantitates <lb/>duæ quarum data e&longs;t differentia auges ntur in infinitum, dabitur <lb/>harum ultima ratio, nimirum ratio æqualitatis, nec tamen ideo da­<lb/>buntur quantitates ultimæ &longs;eu maximæ quarum i&longs;ta e&longs;t ratio. </s> <s>Igitur <lb/>in &longs;equentibus, &longs;iquando facili rerum conceptui con&longs;ulens dixero <lb/>quantitates quam minimas, vel evane&longs;centes, vel ultimas; cave in­<lb/>telligas quantitates magnitudine determinatas, &longs;ed cogita &longs;emper <lb/>diminuendas &longs;ine limite. </s> </p> <p type="margin"> <s><margin.target id="note16"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>SECTIO II.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Inventione Virium Centripetarum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO I. THE OREMA I.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Areas, quas corpora in gyros acta radiis ad immobile centrum virium <lb/>ductis de&longs;cribunt, & in planis immobilibus con&longs;i&longs;tere, & e&longs;&longs;e tem­<lb/>poribus proportionales.<emph.end type="italics"/></s> </p> <p type="main"> <s>Dividatur tempus in partes æquales, & prima temporis parte de­<lb/>&longs;eribat corpus vi in&longs;ita rectam <emph type="italics"/>AB.<emph.end type="italics"/> Idem &longs;ecunda temporis parte, &longs;i <lb/>nil impediret, recta <lb/> <arrow.to.target n="fig12"></arrow.to.target><lb/>pergeret ad <emph type="italics"/>c,<emph.end type="italics"/> (per <lb/>Leg. </s> <s>1.) de&longs;cribens <lb/>lineam <emph type="italics"/>Bc<emph.end type="italics"/> æqualem <lb/>ip&longs;i <emph type="italics"/>AB<emph.end type="italics"/>; adeo ut ra­<lb/>diis <emph type="italics"/>AS, BS, cS<emph.end type="italics"/> ad <lb/>centrum actis, con­<lb/>fectæ forent æqua­<lb/>les areæ <emph type="italics"/>ASB, BSc.<emph.end type="italics"/><lb/>Verum ubi corpus <lb/>venitad <emph type="italics"/>B,<emph.end type="italics"/> agat vis <lb/>centripeta impul­<lb/>&longs;u unico &longs;ed mag­<lb/>no, efficiatque ut <lb/>corpus de recta <emph type="italics"/>Bc<emph.end type="italics"/><lb/>declinet & pergat <lb/>in recta <emph type="italics"/>BC.<emph.end type="italics"/> Ip&longs;i <lb/><emph type="italics"/>BS<emph.end type="italics"/> parallela agatur <emph type="italics"/>cC,<emph.end type="italics"/> occurens <emph type="italics"/>BC<emph.end type="italics"/> in <emph type="italics"/>C<emph.end type="italics"/>; & completa &longs;ecunda <lb/>temporis parte, corpus (per Legum Corol. </s> <s>1.) reperietur in <emph type="italics"/>C,<emph.end type="italics"/> in <pb pagenum="35"/>eodem plano cum triangulo <emph type="italics"/>ASB.<emph.end type="italics"/> Junge <emph type="italics"/>SC<emph.end type="italics"/>; & triangulum <emph type="italics"/>SBC,<emph.end type="italics"/><lb/>ob parallelas <emph type="italics"/>SB, Cc,<emph.end type="italics"/> æquale erit triangulo <emph type="italics"/>SBc,<emph.end type="italics"/> atque adeo etiam <lb/>triangulo <emph type="italics"/>SAB.<emph.end type="italics"/> Simili argumento &longs;i vis centripeta &longs;ucce&longs;&longs;ive agat <lb/>in <emph type="italics"/>C, D, E,<emph.end type="italics"/> &c. </s> <s>faciens ut corpus &longs;ingulis temporis particulis &longs;in­<lb/>gulas de&longs;eribat rectas <emph type="italics"/>CD, DE, EF,<emph.end type="italics"/> &c. </s> <s>jacebunt hæ omnes in <lb/>eodem plano; & triangulum <emph type="italics"/>SCD<emph.end type="italics"/> triangulo <emph type="italics"/>SBC,<emph.end type="italics"/> & <emph type="italics"/>SDE<emph.end type="italics"/> ip&longs;i <lb/><emph type="italics"/>SCD,<emph.end type="italics"/> & <emph type="italics"/>SEF<emph.end type="italics"/> ip&longs;i <emph type="italics"/>SDE<emph.end type="italics"/> æquale erit. </s> <s>Æqualibus igitur tempori­<lb/>bus æquales areæ in plano immoto de&longs;cribuntur: & componendo, <lb/>&longs;unt arearum &longs;ummæ quævis <emph type="italics"/>SADS, SAFS<emph.end type="italics"/> inter &longs;e, ut &longs;unt tem­<lb/>pora de&longs;criptionum. </s> <s>Augeatur jam numerus & minuatur latitudo <lb/>triangulorum in infinitum; & eorum ultima perimeter <emph type="italics"/>ADF,<emph.end type="italics"/> (per <lb/>Corollarium quartum Lemmatis tertii) erit linea curva: adeoque vis <lb/>centripeta, qua corpus a tangente hujus curvæ perpetuo retrahitur, <lb/>aget inde&longs;inenter; areæ vero quævis de&longs;criptæ <emph type="italics"/>SADS, SAFS<emph.end type="italics"/><lb/>temporibus de&longs;criptionum &longs;emper proportionales, erunt ii&longs;dem tem­<lb/>poribus in hoc ca&longs;u proportionales. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <figure id="fig12"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Velocitas corporis in centrum immobile attracti e&longs;t in <lb/>&longs;patiis non re&longs;i&longs;tentibus reciproce ut perpendiculum a centro illo in <lb/>Orbis tangentem rectilineam demi&longs;&longs;um. </s> <s>E&longs;t enim velocitas in locis <lb/>illis <emph type="italics"/>A, B, C, D, E,<emph.end type="italics"/> ut &longs;unt ba&longs;es æqualium triangulorum <emph type="italics"/>AB, BC, <lb/>CD, DE, EF<emph.end type="italics"/>; & hæ ba&longs;es &longs;unt reciproce ut perpendicula in ip&longs;as <lb/>demi&longs;&longs;a. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Si arcuum duorum æqualibus temporibus in &longs;patiis non <lb/>re&longs;i&longs;tentibus ab eodem corpore &longs;ucce&longs;&longs;ive de&longs;criptorum chordæ <emph type="italics"/>AB, <lb/>BC<emph.end type="italics"/> compleantur in parallelogrammum <emph type="italics"/>ABCU,<emph.end type="italics"/> & hujus diagona­<lb/>lis <emph type="italics"/>BU<emph.end type="italics"/> in ea po&longs;itione quam ultimo habet ubi arcus illi in infini­<lb/>tum diminuuntur, producatur utrinque; tran&longs;ibit eadem per cen­<lb/>trum virium. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Si arcuum æqualibus temporibus in &longs;patiis non re&longs;i&longs;ten­<lb/>tibus de&longs;criptorum chordæ <emph type="italics"/>AB, BC<emph.end type="italics"/> ac <emph type="italics"/>DE, EF<emph.end type="italics"/> compleantur in <lb/>parallelogramma <emph type="italics"/>ABCU, DEFZ<emph.end type="italics"/>; vires in <emph type="italics"/>B<emph.end type="italics"/> & <emph type="italics"/>E<emph.end type="italics"/> &longs;unt ad invi­<lb/>cem in ultima ratione diagonalium <emph type="italics"/>BU, EZ,<emph.end type="italics"/> ubi arcus i&longs;ti in infi­<lb/>nitum diminuuntur. </s> <s>Nam corporis motus <emph type="italics"/>BC<emph.end type="italics"/> & <emph type="italics"/>EF<emph.end type="italics"/> componun­<lb/>tur (per Legum Corol. </s> <s>1.) ex motibus <emph type="italics"/>Bc, BU<emph.end type="italics"/> & <emph type="italics"/>Ef, EZ:<emph.end type="italics"/> at­<lb/>qui <emph type="italics"/>BU<emph.end type="italics"/> & <emph type="italics"/>EZ,<emph.end type="italics"/> ip&longs;is <emph type="italics"/>Cc<emph.end type="italics"/> & <emph type="italics"/>Ff<emph.end type="italics"/> æquales, in Demon&longs;tratione Pro­<lb/>po&longs;itionis hujus generabantur ab impul&longs;ibus vis centripetæ in B & <lb/><emph type="italics"/>E,<emph.end type="italics"/> ideoque &longs;unt his impul&longs;ibus proportionales. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Vires quibus corpora quælibet in &longs;patiis non re&longs;i&longs;tenti­<lb/>bus a motibus rectilineis retrahuntur ac detorquentur in orbes cur­<lb/>vos &longs;unt inter &longs;e ut arcuum æqualibus temporibus de&longs;criptorum &longs;a­<lb/>gittæ illæ quæ convergunt ad centrum virium, & chordas bi&longs;ecant <pb pagenum="36"/> <arrow.to.target n="note17"></arrow.to.target><lb/>ubi arcus illi in infinitum diminuuntur. </s> <s>Nam hæ &longs;agittæ &longs;unt &longs;e­<lb/>mi&longs;&longs;es diagonalium de quibus egimus in Corollario tertio. </s> </p> <p type="margin"> <s><margin.target id="note17"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Ideoque vires eædem &longs;unt ad vim gravitatis, ut hæ &longs;a­<lb/>gittæ ad &longs;agittas horizonti perpendiculares arcuum Parabolicorum <lb/>quos projectilia eodem tempore de&longs;cribunt. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 6. Eadem omnia obtinent per Legum Corol. </s> <s>IV, ubi plana <lb/>in quibus corpora moventur, una cum centris virium quæ in ip&longs;is <lb/>fita &longs;unt, non quie&longs;cunt, &longs;ed moventur uniformiter in directum. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO II. THEOREMA II.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Corpus omne, quod movetur in linea aliqua curva in plano de­<lb/>&longs;cripta, & radio ducto ad punctum vel immobile, vel motu rectili­<lb/>neo uniformiter progrediens, de&longs;cribit areas circa punctum illud <lb/>temporibus proportionales, urgetur a vi centripeta tendente ad idem <lb/>punctum.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Nam corpus omne quod movetur in linea curva, detor­<lb/>quetur de cur&longs;u rectilineo per vim aliquam in ip&longs;um agentem (per <lb/>Leg. </s> <s>1.) Et vis illa qua corpus de cur&longs;u rectilineo detorquetur, & <lb/>cogitur triangula quam minima <emph type="italics"/>SAB, SBC, SCD,<emph.end type="italics"/> &c. </s> <s>circa <lb/>punctum immobile <emph type="italics"/>S<emph.end type="italics"/> temporibus æqualibus æqualia de&longs;cribere, a­<lb/>git in loco <emph type="italics"/>B<emph.end type="italics"/> &longs;ecundum lineam parallelam ip&longs;i <emph type="italics"/>cC<emph.end type="italics"/> (per Prop. </s> <s>XL, <lb/>Lib. </s> <s>1 Elem. </s> <s>& Leg. </s> <s>11.) hoc e&longs;t, &longs;ecundum lineam <emph type="italics"/>BS<emph.end type="italics"/>; & in loco <lb/><emph type="italics"/>C<emph.end type="italics"/> &longs;ecundum lineam ip&longs;i <emph type="italics"/>dD<emph.end type="italics"/> parallelam, hoc e&longs;t, &longs;ecundum lineam <lb/><emph type="italics"/>SC,<emph.end type="italics"/> &c. </s> <s>Agit ergo &longs;emper &longs;ecundum lineas tendentes ad punctum <lb/>illud immobile <emph type="italics"/>S. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Et, per Legum Corollarium quintum, perinde e&longs;t &longs;ive <lb/>quie&longs;cat &longs;uperficies in qua corpus de&longs;cribit figuram curvilineam, <lb/>&longs;ive moveatur eadem una cum corpore, figura de&longs;cripta, & puncto <lb/>&longs;uo <emph type="italics"/>S<emph.end type="italics"/> uniformiter in directum. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. In Spatiis vel Mediis non re&longs;i&longs;tentibus, &longs;i areæ non &longs;unt <lb/>temporibus proportionales, vires non tendunt ad concur&longs;um radio­<lb/>rum; &longs;ed inde declinant in con&longs;equentia &longs;eu ver&longs;us plagam in quam <lb/>fit motus, &longs;i modo arearum de&longs;criptio acceleratur: &longs;in retardatur, de­<lb/>clinant in antecedentia. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. In Mediis etiam re&longs;i&longs;tentibus, &longs;i arearum de&longs;criptio accele­<lb/>ratur, virium directiones declinant a concur&longs;u radiorum ver&longs;us plagam <lb/>in quam &longs;it motus. </s> </p> <pb pagenum="37"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Urgeri pote&longs;t corpus a vi centripeta compo&longs;ita ex pluribus viri­<lb/>bus. </s> <s>In hoc ca&longs;u &longs;en&longs;us Propo&longs;itionis e&longs;t, quod vis illa quæ ex om­<lb/>nibus componitur, tendit ad punctum <emph type="italics"/>S.<emph.end type="italics"/> Porro &longs;i vis aliqua agat <lb/>perpetuo &longs;ecundum lineam &longs;uperficiei de&longs;criptæ perpendicularem; <lb/>hæc faciet ut corpus deflectatur a plano &longs;ui motus: &longs;ed quantita­<lb/>tem &longs;uperficiei de&longs;criptæ nec augebit nec minuet, & propterea in <lb/>compo&longs;itione virium negligenda e&longs;t. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO III. THEOREMA III.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Corpus omne, quod radio ad centrum corporis alterius utcunque moti <lb/>ducto de&longs;cribit areas circa centrum illud temporibus proportiona­<lb/>les, urgetur vi compo&longs;ita ex vi centripeta tendente ad corpus il­<lb/>lud alterum, & ex vi omni acceleratrice qua corpus illud alterum <lb/>urgetur.<emph.end type="italics"/></s> </p> <p type="main"> <s>Sit corpus primum <emph type="italics"/>L<emph.end type="italics"/> & corpus alterum <emph type="italics"/>T:<emph.end type="italics"/> & (per Legum Corol. </s> <s><lb/>VI.) &longs;i vi nova, quæ æqualis & contraria &longs;it illi qua corpus alterum <lb/><emph type="italics"/>T<emph.end type="italics"/> urgetur, urgeatur corpus utrumque &longs;ecundum lineas parallelas; <lb/>perget corpus primum <emph type="italics"/>L<emph.end type="italics"/> de&longs;cribere circa corpus alterum <emph type="italics"/>T<emph.end type="italics"/> areas <lb/>ea&longs;dem ac prius: vis autem, qua corpus alterum <emph type="italics"/>T<emph.end type="italics"/> urgebatur, jam <lb/>de&longs;truetur per vim &longs;ibi æqualem & contrariam; & propterea (per <lb/>Leg. </s> <s>1.) corpus illud alterum <emph type="italics"/>T<emph.end type="italics"/> &longs;ibimet ip&longs;i jam relictum vel qui­<lb/>e&longs;cet vel movebitur uniformiter in directum: & corpus primum <emph type="italics"/>L<emph.end type="italics"/><lb/>urgente differentia virium, id e&longs;t, urgente vi reliqua perget areas <lb/>temporibus proportionales circa corpus alterum <emph type="italics"/>T<emph.end type="italics"/> de&longs;cribere. </s> <s>Ten­<lb/>dit igitur (per Theor. </s> <s>11.) differentia virium ad corpus illud alte­<lb/>rum <emph type="italics"/>T<emph.end type="italics"/> ut centrum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i corpus unum <emph type="italics"/>L<emph.end type="italics"/> radio ad alterum <emph type="italics"/>T<emph.end type="italics"/> ducto de­<lb/>&longs;cribit areas temporibus proportionales; atque de vi tota (&longs;ive &longs;im­<lb/>plici, &longs;ive ex viribus pluribus, juxta Legum Corollarium &longs;ecundum, <lb/>compo&longs;ita,) qua corpus prius <emph type="italics"/>L<emph.end type="italics"/> urgetur, &longs;ubducatur (per idem Le­<lb/>gum Corollarium) vis tota acceleratrix qua corpus alterum urgetur: <lb/>vis omnis reliqua qua corpus prius urgetur tendet ad corpus alte­<lb/>rum <emph type="italics"/>T<emph.end type="italics"/> ut centrum. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et, &longs;i areæ illæ &longs;unt temporibus quamproxime propor­<lb/>tionales, vis reliqua tendet ad corpus alterum <emph type="italics"/>T<emph.end type="italics"/> quamproxime. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Et vice ver&longs;a, &longs;i vis reliqua tendit quamproxime ad <pb pagenum="38"/> <arrow.to.target n="note18"></arrow.to.target><lb/>corpus alterum <emph type="italics"/>T,<emph.end type="italics"/> erunt areæ illæ temporibus quamproxime pro­<lb/>portionales. </s> </p> <p type="margin"> <s><margin.target id="note18"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Si corpus <emph type="italics"/>L<emph.end type="italics"/> radio ad alterum corpus <emph type="italics"/>T<emph.end type="italics"/> ducto de&longs;cri­<lb/>bit areas quæ, cum temporibus collatæ, &longs;unt valde inæquales; & <lb/>corpus illud alterum <emph type="italics"/>T<emph.end type="italics"/> vel quie&longs;cit vel movetur uniformiter in di­<lb/>rectum: actio vis centripetæ ad corpus illud alterum <emph type="italics"/>T<emph.end type="italics"/> tendentis, <lb/>vel nulla e&longs;t, vel mi&longs;cetur & componitur cum actionibus admodum <lb/>potentibus aliarum virium: Vi&longs;que tota ex omnibus, &longs;i plures &longs;unt <lb/>vires, compo&longs;ita, ad aliud (&longs;ive immobile &longs;ive mobile) centrum <lb/>dirigitur. </s> <s>Idem obtinet, ubi corpus alterum motu quocunque mo­<lb/>vetur; &longs;i modo vis centripeta &longs;umatur, quæ re&longs;tat po&longs;t &longs;ubductio­<lb/>nem vis totius in corpus illud alterum <emph type="italics"/>T<emph.end type="italics"/> agentis. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Quoniam æquabilis arearum de&longs;criptio Index e&longs;t Centri, quod <lb/>vis illa re&longs;picit qua corpus maxime afficitur, quaque retrahitur a mo­<lb/>tu rectilineo & in orbita &longs;ua retinetur: quidni u&longs;urpemus in &longs;equen­<lb/>tibus æquabilem arearum de&longs;criptionem, ut Indicem Centri circum <lb/>quod motus omnis circularis in &longs;patiis liberis peragitur? </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO IV. THEOREMA IV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Corporum, quæ diver&longs;os circulos æquabili motu de&longs;cribunt, vires cen­<lb/>tripetas ad centra eorundem circulorum tendere; & e&longs;&longs;e inter &longs;e, <lb/>ut &longs;unt arcuum &longs;imul de&longs;criptorum quadrata applicata ad circulo­<lb/>rum radios.<emph.end type="italics"/></s> </p> <p type="main"> <s>Tendunt hæ vires ad centra circulorum per Prop.II. & Corol. </s> <s>II. <lb/>Prop. </s> <s>1; & &longs;unt inter &longs;e ut arcuum æqualibus temporibus quam mini­<lb/>mis de&longs;criptorum &longs;inus ver&longs;i per Corol. </s> <s>IV. Prop. </s> <s>I; hoc e&longs;t, ut qua­<lb/>drata arcuum eorundem ad diametros circulorum applicata per <lb/>Lem. </s> <s>VII: & propterea, cum hi arcus &longs;int ut arcus temporibus <lb/>quibu&longs;vis æqualibus de&longs;cripti, & diametri &longs;int ut eorum radii; vi­<lb/>res erunt ut arcuum quorumvis &longs;imul de&longs;criptorum quadrata ap­<lb/>plicata ad radios circulorum. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Igitur, cum arcus illi &longs;int ut velocitates corporum, vi­<lb/>res centripetæ &longs;unt ut velocitatum quadrata applicata ad radios <lb/>circulorum: hoc e&longs;t, ut cum Geometris loquar, vires &longs;unt in ra­<lb/>tione compo&longs;ita ex duplicata ratione velocitatum directe & ratione <lb/>&longs;implici radiorum inver&longs;e. </s> </p> <pb pagenum="39"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et, cum tempora periodica &longs;int in ratione compo&longs;ita ex <lb/>ratione radiorum directe & ratione velocitatum inver&longs;e, vires cen­<lb/>tripetæ &longs;unt reciproce ut quadrata temporum periodicorum appli­<lb/>cata ad circulorum radios; hoc e&longs;t, in ratione compo&longs;ita ex ratione <lb/>radiorum directe & ratione duplicata temporum periodicorum in­<lb/>ver&longs;e. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Unde, &longs;i tempora periodica æquentur & propterea ve­<lb/>locitates &longs;int ut radii; erunt etiam vires centripetæ ut radii: & <lb/>contra. </s> </p> <p type="main"> <s><emph type="italics"/>Cor.<emph.end type="italics"/> 4. Si & tempora periodica & velocitates &longs;int in ratione &longs;ub­<lb/>duplicata radiorum; æquales erunt vires centripetæ inter &longs;e: & <lb/>contra. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Si tempora periodica &longs;int ut radii & propterea veloci­<lb/>tates æquales; vires centriperæ erunt reciproce ut radii: & contra. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 6. Si tempora periodica &longs;int in ratione &longs;e&longs;quiplicata radio­<lb/>rum & propterea velocitates reciproce in radiorum ratione &longs;ubdu­<lb/>plicata; vires centripetæ erunt reciproce ut quadrata radiorum: <lb/>& contra. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 7. Et univer&longs;aliter, &longs;i tempus periodicum &longs;it ut Radii <emph type="italics"/>R<emph.end type="italics"/><lb/>pote&longs;tas quælibet <emph type="italics"/>R<emph type="sup"/>n<emph.end type="sup"/>,<emph.end type="italics"/> & propterea velocitas reciproce ut Radii <lb/>pote&longs;tas <emph type="italics"/>R<emph type="sup"/>n-1<emph.end type="sup"/><emph.end type="italics"/>; erit vis centripeta reciproce ut Radii pote&longs;tas <emph type="italics"/>R<emph type="sup"/>2n-1<emph.end type="sup"/>:<emph.end type="italics"/><lb/>& contra. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 8. Eadem omnia de temporibus, velocitatibus, & viribus, qui­<lb/>bus corpora &longs;imiles figurarum quarumcunque &longs;imilium, centraque <lb/>in figuris illis &longs;imiliter po&longs;ita habentium, partes de&longs;cribunt, con&longs;e­<lb/>quuntur ex Demon&longs;tratione præcedentium ad ho&longs;ce ca&longs;us applicata. </s> <s><lb/>Applicatur autem &longs;ub&longs;tituendo æquabilem arearum de&longs;criptionem <lb/>pro æquabili motu, & di&longs;tantias corporum a centris pro radiis u&longs;ur­<lb/>pando. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 9. Ex eadem demon&longs;tratione con&longs;equitur etiam; quod ar­<lb/>cus, quem corpus in circulo data vi centripeta uniformiter revolven­<lb/>do tempore quovis de&longs;cribit, medius e&longs;t proportionalis inter dia­<lb/>metrum circuli, & de&longs;cen&longs;um corporis eadem data vi eodem que tem­<lb/>pore cadendo confectum. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Ca&longs;us Corollarii &longs;exti obtinet in corporibus cæle&longs;tibus, (ut &longs;eor­<lb/>&longs;um collegerunt etiam no&longs;trates <emph type="italics"/>Wrennus, Hookius<emph.end type="italics"/> & <emph type="italics"/>Hallæus<emph.end type="italics"/>) & <lb/>propterea quæ &longs;pectant ad vim centripetam decre&longs;centem in dupli­<lb/>cata ratione di&longs;tantiarum a centris, decrevi fu&longs;ius in &longs;equentibus <lb/>exponere. <pb pagenum="40"/> <arrow.to.target n="note19"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note19"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s>Porro præcedentis propo&longs;itionis & corollariorum ejus beneficio, <lb/>colligitur etiam proportio vis centripetæ ad vim quamlibet notam, <lb/>qualis e&longs;t ea Gravitatis. </s> <s>Nam &longs;i corpus in circulo Terræ concen­<lb/>trico vi gravitatis &longs;uæ revolvatur, hæc gravitas e&longs;t ip&longs;ius vis centri­<lb/>peta. </s> <s>Datur autem, ex de&longs;cen&longs;u gravium, & tempus revolutionis <lb/>unius, & arcus dato quovis tempore de&longs;criptus, per hujus Corol. </s> <s><lb/>IX. </s> <s>Et huju&longs;modi propo&longs;itionibus <emph type="italics"/>Hugenius,<emph.end type="italics"/> in eximio &longs;uo Tracta­<lb/>tu <emph type="italics"/>de Horologio O&longs;cillatorio,<emph.end type="italics"/> vim gravitatis cum revolventium vi­<lb/>ribus centrifugis contulit. </s> </p> <p type="main"> <s>Demon&longs;trari etiam po&longs;&longs;unt præcedentia in hunc modum. </s> <s>In cir­<lb/>culo quovis de&longs;cribi intelligatur Polygonum laterum quotcunque. </s> <s><lb/>Et &longs;i corpus, in polygoni lateribus data cum velocitate movendo, <lb/>ad ejus angulos &longs;ingulos a circulo reflectatur; vis qua &longs;ingulis re­<lb/>flexionibus impingit in circulum erit ut ejus velocitas: adeoque <lb/>&longs;umma virium in dato tempore erit ut velocitas illa & numerus re­<lb/>flexionum conjunctim: hoc e&longs;t (&longs;i polygonum detur &longs;pecie) ut longi­<lb/>tudo dato illo tempore de&longs;cripta & longitudo eadem applicata ad <lb/>Radium circuli; id e&longs;t, ut quadratum longitudinis illius applicatum <lb/>ad Radium: adeoque, &longs;i polygonum lateribus infinite diminutis co­<lb/>incidat cum circulo, ut quadratum arcus dato tempore de&longs;cripti ap­<lb/>plicatum ad radium. </s> <s>Hæc e&longs;t vis centrifuga, qua corpus urget cir­<lb/>culum: & huic æqualis e&longs;t vis contraria, qua circulus continuo re­<lb/>pellit corpus centrum ver&longs;us. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO. V. PROBLEMA I.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Data quibu&longs;cunque in locis velocitate, qua corpus figuram datam vi­<lb/>ribus ad commune aliquod centrum tendentibus de&longs;cribit, centrum <lb/>illud invenire.<emph.end type="italics"/></s> </p> <p type="main"> <s>Figuram de&longs;criptam tangant rectæ tres <emph type="italics"/>PT, TQV, VR<emph.end type="italics"/> in <lb/>punctis totidem <emph type="italics"/>P, Q, R,<emph.end type="italics"/> concurrentes in <emph type="italics"/>T<emph.end type="italics"/> & <emph type="italics"/>V.<emph.end type="italics"/> Ad tangentes <lb/>erigantur perpendicula <emph type="italics"/>PA, QB, RC,<emph.end type="italics"/> velocitatibus corporis in <lb/>punctis illis <emph type="italics"/>P, Q, R<emph.end type="italics"/> a quibus eriguntur reciproce proportionalia; <lb/>id e&longs;t, ita ut &longs;it <emph type="italics"/>PA<emph.end type="italics"/> ad <emph type="italics"/>QB<emph.end type="italics"/> ut velocitas in <emph type="italics"/>Q<emph.end type="italics"/> ad velocitatem in <lb/><emph type="italics"/>P,<emph.end type="italics"/> & <emph type="italics"/>QB<emph.end type="italics"/> ad <emph type="italics"/>RC<emph.end type="italics"/> ut velocitas in <emph type="italics"/>R<emph.end type="italics"/> ad velocitatem in <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> Per <lb/>perpendiculorum terminos <emph type="italics"/>A, B, C<emph.end type="italics"/> ad angulos rectos ducantur <emph type="italics"/>AD, <lb/>DBE, EC<emph.end type="italics"/> concurrentes in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>E:<emph.end type="italics"/> Et actæ <emph type="italics"/>TD, VE<emph.end type="italics"/> concur­<lb/>rent in centro q<gap/>&longs;ito <emph type="italics"/>S.<emph.end type="italics"/></s> </p> <pb pagenum="41"/> <figure></figure> <p type="main"> <s>Nam perpendicula a centro <emph type="italics"/>S<emph.end type="italics"/><lb/>in tangentes <emph type="italics"/>PT, QT<emph.end type="italics"/> demi&longs;&longs;a (per <lb/>Corol. </s> <s>1. Prop.I.) &longs;unt reciproce <lb/>ut velocitates corporis in punctis <lb/><emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>V<emph.end type="italics"/>; &c. </s> <s>adeoque per con&longs;tructio­<lb/>nem ut perpendicula <emph type="italics"/>AP, BQ<emph.end type="italics"/> di­<lb/>recte, id e&longs;t ut perpendicula a pun­<lb/>cto <emph type="italics"/>D<emph.end type="italics"/> in tangentes demi&longs;&longs;a. </s> <s>Un­<lb/>de facile colligitur quod puncta <lb/><emph type="italics"/>S, D, T,<emph.end type="italics"/> &longs;unt in una recta. </s> <s>Et &longs;imili <lb/>argumento puncta <emph type="italics"/>S, E, V<emph.end type="italics"/> &longs;unt eti­<lb/>am in una recta; & propterea centrum <emph type="italics"/>S<emph.end type="italics"/> in concur&longs;u rectarum <emph type="italics"/>TD, VE<emph.end type="italics"/><lb/>ver&longs;atur. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO VI. THEOREMA V.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si corpus in &longs;patio non re&longs;i&longs;tente circa centrum immobile in Orbe quocun­<lb/>que revolvatur, & arcum quemvis jamjam na&longs;centem tempore quàm <lb/>minimo de&longs;cribat, & &longs;agitta arcus duci intelligatur quæ chor dam bi­<lb/>&longs;ecet, & producta tran&longs;eat per centrum virium: erit vis centripeta <lb/>in medio arcus, ut &longs;agitta directe & tempus bis inver&longs;e.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam &longs;agitta dato tempore e&longs;t ut vis (per Corol.4 Prop.I,) & augen­<lb/>do tempus in ratione quavis, ob auctum arcum in eadem ratione &longs;a­<lb/>gitta augetur in ratione illa duplicata (per Corol. </s> <s>2 & 3, Lem. </s> <s>XI,) ad­<lb/>eoque e&longs;t ut vis &longs;emel & tempus bis. </s> <s>Subducatur duplicata ratio tempo­<lb/>ris utrinque, & fiet vis ut &longs;agitta directe & tempus bis inver&longs;e. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s>Idem facile demon&longs;tratur etiam per Corol. </s> <s>4 Lem. </s> <s>X. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Si corpus <emph type="italics"/>P<emph.end type="italics"/> revolvendo <lb/> <arrow.to.target n="fig13"></arrow.to.target><lb/>circa centrum <emph type="italics"/>S<emph.end type="italics"/> de&longs;cribat lineam <lb/>curvam <emph type="italics"/>APQ,<emph.end type="italics"/> tangat verò recta <lb/><emph type="italics"/>ZPR<emph.end type="italics"/> curvam illam in puncto <lb/>quovis <emph type="italics"/>P,<emph.end type="italics"/> & ad tangentem ab alio <lb/>quovis Curvæ puncto <emph type="italics"/>Q<emph.end type="italics"/> agatur <lb/><emph type="italics"/>QR<emph.end type="italics"/> di&longs;tantiæ <emph type="italics"/>SP<emph.end type="italics"/> parallela, ac <lb/>demittatur <emph type="italics"/>QT<emph.end type="italics"/> perpendicularis <lb/>ad di&longs;tantiam illam <emph type="italics"/>SP:<emph.end type="italics"/> vis cen­<lb/>tripeta erit reciproce ut &longs;olidum <lb/>(<emph type="italics"/>SP quad.XQT quad./QR<emph.end type="italics"/>) &longs;i modo &longs;olidi illius ea &longs;emper &longs;umatur quan­<lb/>titas, quæ ultimò fit ubi coeunt puncta <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> Nam <emph type="italics"/>QR<emph.end type="italics"/> æqualis </s> </p> <pb pagenum="42"/> <figure id="fig13"></figure> <p type="main"> <s> <arrow.to.target n="note20"></arrow.to.target><lb/>e&longs;t &longs;agittæ dupli arcus <emph type="italics"/>QP,<emph.end type="italics"/> in cujus medio e&longs;t <emph type="italics"/>P,<emph.end type="italics"/> & duplum trian­<lb/>guli <emph type="italics"/>SQP<emph.end type="italics"/> &longs;ive <emph type="italics"/>SPXQT,<emph.end type="italics"/> tempori quo arcus i&longs;te duplus de&longs;cribitur <lb/>proportionale e&longs;t, ideoque pro temporis exponente &longs;cribi pote&longs;t. </s> </p> <p type="margin"> <s><margin.target id="note20"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Eodem argumento vis centripeta e&longs;t reciprocè ut &longs;olidum <lb/>(<emph type="italics"/>SYqXQPq/QR<emph.end type="italics"/>), &longs;i modo <emph type="italics"/>SY<emph.end type="italics"/> perpendiculum &longs;it a centro virium in Or­<lb/>bis tangentem <emph type="italics"/>PR<emph.end type="italics"/> demi&longs;&longs;um. </s> <s>Nam rectangula <emph type="italics"/>SYXQP<emph.end type="italics"/> & <emph type="italics"/>SPXQT<emph.end type="italics"/><lb/>æquantur. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Si Orbis vel circulus e&longs;t, vel angulum contactus cum cir­<lb/>culo quam minimum continet, eandem habens curvaturam eundem­<lb/>que radium curvaturæ ad punctum contactus <emph type="italics"/>P<emph.end type="italics"/>; & &longs;i <emph type="italics"/>PV<emph.end type="italics"/> chorda <lb/>&longs;it circuli hujus a corpore per centrum virium acta: erit vis centri­<lb/>peta reciproce ut &longs;olidum <emph type="italics"/>SYqXPV.<emph.end type="italics"/> Nam <emph type="italics"/>PV<emph.end type="italics"/> e&longs;t (<emph type="italics"/>QPq/QR<emph.end type="italics"/>). </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Ii&longs;dem po&longs;itis, e&longs;t vis centripeta ut velocitas bis directe, <lb/>& chorda illa inver&longs;e. </s> <s>Nam velocitas e&longs;t reciproce ut perpendicu­<lb/>lum <emph type="italics"/>SY<emph.end type="italics"/> per Corol. </s> <s>I Prop. </s> <s>I. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Hinc &longs;i detur figura quævis curvilinea <emph type="italics"/>APQ,<emph.end type="italics"/> & in ea <lb/>detur etiam punctum <emph type="italics"/>S<emph.end type="italics"/> ad quod vis centripeta perpetuo dirigitur, <lb/>inveniri pote&longs;t lex vis centripetæ, qua corpus quodvis <emph type="italics"/>P<emph.end type="italics"/> a cur&longs;u <lb/>rectilineo perpetuò retractum in figuræ illius perimetro detinebitur <lb/>eamque revolvendo de&longs;cribet. </s> <s>Nimirum computandum e&longs;t vel &longs;o­<lb/>lidum (<emph type="italics"/>SPqXQTq/QR<emph.end type="italics"/>) vel &longs;olidum <emph type="italics"/>SYqXPV<emph.end type="italics"/> huic vi reciproce pro­<lb/>portionale. </s> <s>Ejus rei dabimus exempla in Problematis &longs;equentibus. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO VII. PROBLEMA II.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Gyretur corpus in circumferentia Circuli, requiritur Lex vis centri­<lb/>petæ tendentis ad punctum quodcunque datum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>E&longs;to Circuli circumferentia <lb/> <arrow.to.target n="fig14"></arrow.to.target><lb/><emph type="italics"/>VQPA,<emph.end type="italics"/> punctum datum ad <lb/>quod vis ceu ad <expan abbr="centrũ">centrum</expan> <expan abbr="&longs;uũ">&longs;uum</expan> ten­<lb/>dit <emph type="italics"/>S,<emph.end type="italics"/> corpus in circumferentia <lb/>latum <emph type="italics"/>P,<emph.end type="italics"/> locus proximus in quem <lb/>movebitur <emph type="italics"/>Q,<emph.end type="italics"/> & circuli tangens <lb/>ad locum priorem <emph type="italics"/>PRZ.<emph.end type="italics"/> Per <lb/>punctum <emph type="italics"/>S<emph.end type="italics"/> ducatur chorda <emph type="italics"/>PV,<emph.end type="italics"/><lb/>& acta circuli diametro <emph type="italics"/>VA<emph.end type="italics"/> jun­<lb/>gatur <emph type="italics"/>AP,<emph.end type="italics"/> & ad <emph type="italics"/>SP<emph.end type="italics"/> demittatur <lb/>perpendiculum <emph type="italics"/>QT,<emph.end type="italics"/> quod productum occurrat tangenti <emph type="italics"/>PR<emph.end type="italics"/> in <emph type="italics"/>Z,<emph.end type="italics"/> <pb pagenum="43"/>ac denique per punctum <emph type="italics"/>Q<emph.end type="italics"/> agatur <emph type="italics"/>LR<emph.end type="italics"/> quæ ip&longs;i <emph type="italics"/>SP<emph.end type="italics"/> parallela <lb/>&longs;it & occurrat tum circulo in <emph type="italics"/>L<emph.end type="italics"/> tum tangenti <emph type="italics"/>PZ<emph.end type="italics"/> in <emph type="italics"/>R.<emph.end type="italics"/> Et <lb/>ob &longs;imilia triangula <emph type="italics"/>ZQR, ZTP, VPA<emph.end type="italics"/>; erit <emph type="italics"/>RP quad.<emph.end type="italics"/> hoc <lb/>e&longs;t <emph type="italics"/>QRL<emph.end type="italics"/> ad <emph type="italics"/>QT quad.<emph.end type="italics"/> ut <emph type="italics"/>AV quad.<emph.end type="italics"/> ad <emph type="italics"/>PV quad.<emph.end type="italics"/> Ideoque <lb/>(<emph type="italics"/>QRLXPV quad./AV quad.<emph.end type="italics"/>) æquatur <emph type="italics"/>QT quad.<emph.end type="italics"/> Ducantur hæc æqualia in <lb/>(<emph type="italics"/>SP quad./QR<emph.end type="italics"/>) &, punctis <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>Q<emph.end type="italics"/> coeuntibus, &longs;cribatur <emph type="italics"/>PV<emph.end type="italics"/> pro <emph type="italics"/>RL.<emph.end type="italics"/><lb/>Sic fiet (<emph type="italics"/>SP quad.XPV cub./AV quad.<emph.end type="italics"/>) æquale (<emph type="italics"/>SP quad.XQT quad./QR<emph.end type="italics"/>) Ergo (per <lb/>Corol.1 & 5 Prop.VI.) vis centripeta e&longs;t reciproce ut (<emph type="italics"/>SPqXPV cub./AV quad<emph.end type="italics"/>) <lb/>id e&longs;t, (ob datum <emph type="italics"/>AV quad.<emph.end type="italics"/>) reciproce ut quadratum di&longs;tantiæ &longs;eu <lb/>altitudinis <emph type="italics"/>SP<emph.end type="italics"/> & cubus chordæ <emph type="italics"/>PV<emph.end type="italics"/> conjunctim. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <figure id="fig14"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Ad tangentem <emph type="italics"/>PR<emph.end type="italics"/> productam demittatur perpendiculum <emph type="italics"/>SY,<emph.end type="italics"/><lb/>& ob &longs;imilia triangula <emph type="italics"/>SYP, VPA<emph.end type="italics"/>; erit <emph type="italics"/>AV<emph.end type="italics"/> ad <emph type="italics"/>PV<emph.end type="italics"/> ut <emph type="italics"/>SP<emph.end type="italics"/> ad <lb/><emph type="italics"/>SY,<emph.end type="italics"/> ideoque (<emph type="italics"/>SPXPV/AV<emph.end type="italics"/>) æquale <emph type="italics"/>SY,<emph.end type="italics"/> & (<emph type="italics"/>SP quad.XPV cub./AV quad.<emph.end type="italics"/>) æquale <lb/><emph type="italics"/>SY quad.XPV.<emph.end type="italics"/> Et propterea (per Corol.3 & 5 Prop.VI.) vis centri­<lb/>peta e&longs;t reciproce ut (<emph type="italics"/>SPqXPV cub./AVq<emph.end type="italics"/>) hoc e&longs;t, ob datam <emph type="italics"/>AV,<emph.end type="italics"/> reci­<lb/>proce ut <emph type="italics"/>SPqXPV cub. </s> <s><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i punctum datum <emph type="italics"/>S<emph.end type="italics"/> ad quod vis centripeta &longs;em­<lb/>per tendit, locetur in circumferentia hujus circuli, puta ad <emph type="italics"/>V<emph.end type="italics"/>; erit <lb/>vis centripeta reciproce ut quadrato cubus altitudinis <emph type="italics"/>SP.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Vis qua corpus <emph type="italics"/>P<emph.end type="italics"/> in cir­<lb/> <arrow.to.target n="fig15"></arrow.to.target><lb/>culo <emph type="italics"/>APTV<emph.end type="italics"/> circum virium centrum <lb/><emph type="italics"/>S<emph.end type="italics"/> revolvitur, e&longs;t ad vim qua corpus <lb/>idem <emph type="italics"/>P<emph.end type="italics"/> in eodem circulo & eodem <lb/>tempore periodico circum aliud quod­<lb/>vis virium centrum <emph type="italics"/>R<emph.end type="italics"/> revolvi pote&longs;t, <lb/>ut <emph type="italics"/>RP quad.XSP<emph.end type="italics"/> ad cubum rectæ <emph type="italics"/>SG<emph.end type="italics"/><lb/>quæ a primo virium centro <emph type="italics"/>S<emph.end type="italics"/> ad or­<lb/>bis tangentem <emph type="italics"/>PG<emph.end type="italics"/> ducitur, & di&longs;tan­<lb/>tiæ corporis a &longs;ecundo virium centro <lb/>parallela e&longs;t. </s> <s>Nam, per con&longs;tructionem hujus Propo&longs;itionis, vis <lb/>prior e&longs;t ad vim po&longs;teriorem, ut <emph type="italics"/>RPqXPT cub.<emph.end type="italics"/> ad <emph type="italics"/>SPqXPV cub.<emph.end type="italics"/> <pb pagenum="44"/> <arrow.to.target n="note21"></arrow.to.target><lb/>id e&longs;t, ut <emph type="italics"/>SPXRPq<emph.end type="italics"/> ad (<emph type="italics"/>SP cub.XPV cub/PT cub.<emph.end type="italics"/>) &longs;ive (ob &longs;imilia <lb/>triangula <emph type="italics"/>PSG, TPV<emph.end type="italics"/>) ad <emph type="italics"/>SG cub.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note21"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig15"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Vis, qua corpus <emph type="italics"/>P<emph.end type="italics"/> in Orbe quocunque circum virium <lb/>centrum <emph type="italics"/>S<emph.end type="italics"/> revolvitur, e&longs;t ad vim qua corpus idem <emph type="italics"/>P<emph.end type="italics"/> in eodem <lb/>orbe eodemque tempore periodico circum aliud quodvis virium <lb/>centrum <emph type="italics"/>R<emph.end type="italics"/> revolvi pote&longs;t, ut <emph type="italics"/>SPXRPq<emph.end type="italics"/> contentum utique &longs;ub di­<lb/>&longs;tantia corporis a primo virium centro <emph type="italics"/>S<emph.end type="italics"/> & quadrato di&longs;tantiæ ejus <lb/>a &longs;ecundo virium centro <emph type="italics"/>R<emph.end type="italics"/> ad cubum rectæ <emph type="italics"/>SG<emph.end type="italics"/> quæ a primo vi­<lb/>rium centro <emph type="italics"/>S<emph.end type="italics"/> ad orbis tangentem <emph type="italics"/>PG<emph.end type="italics"/> ducitur, & corporis a &longs;e­<lb/>cundo virium centro di&longs;tantiæ <emph type="italics"/>RP<emph.end type="italics"/> parallela e&longs;t. </s> <s>Nam vires in <lb/>hoc Orbe, ad ejus punctum quodvis <emph type="italics"/>P,<emph.end type="italics"/> eædem &longs;unt ac in Circulo <lb/>eju&longs;dem curvaturæ. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO. VIII. PROBLEMA. III.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Moveatur corpus in Circulo<emph.end type="italics"/> PQA: <emph type="italics"/>ad hunc effectum requiritur Lex <lb/>vis centripetæ tendentis ad punctum adeo longinquum<emph.end type="italics"/> S, <emph type="italics"/>ut lineæ <lb/>omnes<emph.end type="italics"/> PS, RS <emph type="italics"/>ad id ductæ, pro parallelis haberi po&longs;&longs;int.<emph.end type="italics"/></s> </p> <p type="main"> <s>A Circuli centro <emph type="italics"/>C<emph.end type="italics"/> agatur &longs;emidiameter <emph type="italics"/>CA<emph.end type="italics"/> parallelas i&longs;tas <lb/>perpendiculariter &longs;ecans in <emph type="italics"/>M<emph.end type="italics"/> & <lb/> <arrow.to.target n="fig16"></arrow.to.target><lb/><emph type="italics"/>N,<emph.end type="italics"/> & jungatur <emph type="italics"/>CP.<emph.end type="italics"/> Ob &longs;imilia <lb/>triangula <emph type="italics"/>CPM, PZT<emph.end type="italics"/> & <emph type="italics"/>RZQ<emph.end type="italics"/><lb/>e&longs;t <emph type="italics"/>CPq<emph.end type="italics"/> ad <emph type="italics"/>PMq<emph.end type="italics"/> ut <emph type="italics"/>PRq<emph.end type="italics"/> ad <lb/><emph type="italics"/>QTq<emph.end type="italics"/> & ex natura Circuli <emph type="italics"/>PRq<emph.end type="italics"/><lb/>æquale e&longs;t rectangulo <emph type="italics"/>QRX√RN+QN<emph.end type="italics"/> &c. <lb/>&longs;ive coeuntibus punctis <emph type="italics"/>P, Q<emph.end type="italics"/> rect­<lb/>angulo <emph type="italics"/>QRX2PM.<emph.end type="italics"/> Ergo e&longs;t <lb/><emph type="italics"/>CPq<emph.end type="italics"/> ad <emph type="italics"/>PM quad.<emph.end type="italics"/> ut <emph type="italics"/>QRX2PM<emph.end type="italics"/><lb/>ad <emph type="italics"/>QT quad.<emph.end type="italics"/> adeoque (<emph type="italics"/>QT quad./QR<emph.end type="italics"/>) <lb/>æquale (2<emph type="italics"/>PM cub./CP quad.<emph.end type="italics"/>), & (<emph type="italics"/>QT quad.XSP quad./QR<emph.end type="italics"/>) æquale (2<emph type="italics"/>PM cub.XSP qu./CP quad.<emph.end type="italics"/>) <lb/>E&longs;t ergo (per Corol. </s> <s>1 & 5 Prop. </s> <s>VI.) vis centripeta reciproce ut <lb/>(2<emph type="italics"/>PMcub.XSP quad./CP quad.<emph.end type="italics"/>) hoc e&longs;t (neglecta ratione determinata (2<emph type="italics"/>SP quad./CP quad.<emph.end type="italics"/>)) <lb/>reciproce ut <emph type="italics"/>PM cub. </s> <s><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <figure id="fig16"></figure> <p type="main"> <s>Idem facile colligitur etiam ex Propo&longs;itione præcedente. </s> </p> <pb pagenum="45"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Et &longs;imili argumento corpus movebitur in Ellip&longs;i vel etiam in <lb/>Hyperbola vel Parabola, vi centripeta quæ &longs;it reciproce ut cu­<lb/>bus ordinatim applicatæ ad centrum virium maxime longinquum <lb/>tendentis. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO IX. PROBLEMA IV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Gyretur corpus in Spirali<emph.end type="italics"/> PQS <emph type="italics"/>&longs;ecante radios omnes<emph.end type="italics"/> SP, SQ, <emph type="italics"/>&c.<emph.end type="italics"/><lb/> <arrow.to.target n="fig17"></arrow.to.target><lb/><emph type="italics"/>in angulo dato: requiritur Lex <lb/>vis centripetæ tendentis ad <lb/>centrum Spiralis.<emph.end type="italics"/></s> </p> <figure id="fig17"></figure> <p type="main"> <s>Detur angulus indefinite par­<lb/>vus <emph type="italics"/>PSQ,<emph.end type="italics"/> & ob datos omnes <lb/>angulos dabitur &longs;pecie figura <emph type="italics"/>SPQRT.<emph.end type="italics"/> Ergo datur ratio (<emph type="italics"/>QT/QR<emph.end type="italics"/>), e&longs;tque <lb/>(<emph type="italics"/>QT quad./QR<emph.end type="italics"/>) ut <emph type="italics"/>QT,<emph.end type="italics"/> hoc e&longs;t ut <emph type="italics"/>SP.<emph.end type="italics"/> Mutetur jam uteunque angulus <emph type="italics"/>PSQ,<emph.end type="italics"/><lb/>& recta <emph type="italics"/>QR<emph.end type="italics"/> angulum contactus <emph type="italics"/>QPR<emph.end type="italics"/> &longs;ubtendens mutabitur (per <lb/>Lemma XI.) in duplicata ratione ip&longs;ius <emph type="italics"/>PR<emph.end type="italics"/> vel <emph type="italics"/>QT.<emph.end type="italics"/> Ergo manebit <lb/>(<emph type="italics"/>QT quad./QR<emph.end type="italics"/>) eadem quæ prius, hoc e&longs;t ut <emph type="italics"/>SP.<emph.end type="italics"/> Quare (<emph type="italics"/>QTq.XSPq/QR<emph.end type="italics"/>) <lb/>e&longs;t ut <emph type="italics"/>SP cub.<emph.end type="italics"/> adeoque (per Corol. </s> <s>1 & 5 Prop. </s> <s>VI.) vis centripeta e&longs;t <lb/>reciproce ut cubus di&longs;tantiæ <emph type="italics"/>SP. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Perpendiculum <emph type="italics"/>SY<emph.end type="italics"/> in tangentem demi&longs;&longs;um, & circuli Spiralem <lb/>tangentis chorda <emph type="italics"/>PV<emph.end type="italics"/> &longs;unt ad altitudinem <emph type="italics"/>SP<emph.end type="italics"/> in datis rationibus; <lb/>ideoque <emph type="italics"/>SP cub.<emph.end type="italics"/> e&longs;t ut <emph type="italics"/>SYqXPV,<emph.end type="italics"/> hoc e&longs;t (per Corol. </s> <s>3 & 5 Prop.VI.) <lb/>reciproce ut vis centripeta. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA XII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Parallelogramma omnia, circa datæ Ellip&longs;eos vel Hyperbolæ diametros <lb/>qua&longs;vis conjugatas de&longs;cripta, e&longs;&longs;e inter &longs;e æqualia.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Con&longs;tat ex Conicis. <pb pagenum="46"/> <arrow.to.target n="note22"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note22"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO X. PROBLEMA. V.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Gyretur corpus in Ellip&longs;i: requiritur lex vis centripetæ tendentis ad <lb/>centrum Ellip&longs;eos.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Sunto <emph type="italics"/>CA, CB<emph.end type="italics"/> &longs;emiaxes Ellip&longs;eos; <emph type="italics"/>GP, DK<emph.end type="italics"/> diametri conju­<lb/>gatæ; <emph type="italics"/>PF, Qt<emph.end type="italics"/> perpendicula ad diametros; <emph type="italics"/>Qv<emph.end type="italics"/> ordinatim appli­<lb/>cata ad diametrum <lb/> <arrow.to.target n="fig18"></arrow.to.target><lb/><emph type="italics"/>GP<emph.end type="italics"/>; & &longs;i compleatur <lb/>parallelogrammum <lb/><emph type="italics"/>QvPR,<emph.end type="italics"/> erit (ex Coni­<lb/>cis) <emph type="italics"/>PvG<emph.end type="italics"/> ad <emph type="italics"/>Qv quad.<emph.end type="italics"/><lb/>ut <emph type="italics"/>PC quad.<emph.end type="italics"/> ad <emph type="italics"/>CD <lb/>quad.<emph.end type="italics"/> & (ob &longs;imilia <lb/>triangula <emph type="italics"/>Qvt, PCF<emph.end type="italics"/>) <lb/><emph type="italics"/>Qv quad.<emph.end type="italics"/> e&longs;t ad <emph type="italics"/>Qt <lb/>quad.<emph.end type="italics"/> ut <emph type="italics"/>PC quad.<emph.end type="italics"/> ad <lb/><emph type="italics"/>PF quad.<emph.end type="italics"/> & conjun­<lb/>ctis rationibus, <emph type="italics"/>PvG<emph.end type="italics"/><lb/>ad <emph type="italics"/>Qt quad.<emph.end type="italics"/> ut <emph type="italics"/>PC <lb/>quad.<emph.end type="italics"/> ad <emph type="italics"/>CD quad.<emph.end type="italics"/><lb/>& <emph type="italics"/>PC quad.<emph.end type="italics"/> ad <emph type="italics"/>PF <lb/>quad.<emph.end type="italics"/> id e&longs;t, <emph type="italics"/>vG<emph.end type="italics"/> ad <lb/>(<emph type="italics"/>Qt quad./Pv<emph.end type="italics"/>) ut <emph type="italics"/>PC quad.<emph.end type="italics"/><lb/>ad (<emph type="italics"/>CDqXPFq/PCq<emph.end type="italics"/>). Scribe <emph type="italics"/>QR<emph.end type="italics"/> pro <emph type="italics"/>Pv,<emph.end type="italics"/> & (per Lemma XII.) <emph type="italics"/>BCXCA<emph.end type="italics"/><lb/>pro <emph type="italics"/>CDXPF,<emph.end type="italics"/> nec non, punctis <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>Q<emph.end type="italics"/> coeuntibus, 2<emph type="italics"/>PC<emph.end type="italics"/> pro <lb/><emph type="italics"/>vG,<emph.end type="italics"/> & ductis extremis & mediis in &longs;e mutuo, fiet (<emph type="italics"/>Qt quad.XPCq/QR<emph.end type="italics"/>) <lb/>æquale (2<emph type="italics"/>BCqXCAq/PC<emph.end type="italics"/>). E&longs;t ergo (per Corol. </s> <s>5 Prop. </s> <s>VI.) vis centri­<lb/>peta reciproce ut (2<emph type="italics"/>BCqXGAq;/PC<emph.end type="italics"/>) id e&longs;t (ob datum 2<emph type="italics"/>BCqXCAq<emph.end type="italics"/>) <lb/>reciproce ut (1/<emph type="italics"/>PC<emph.end type="italics"/>); hoc e&longs;t, directe ut di&longs;tantia <emph type="italics"/>PC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <figure id="fig18"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>In <emph type="italics"/>PG<emph.end type="italics"/> ab altera parte puncti <emph type="italics"/>t<emph.end type="italics"/> po&longs;ita intelligatur <emph type="italics"/>tu<emph.end type="italics"/> æqualis ip&longs;i <lb/><emph type="italics"/>tv<emph.end type="italics"/>; deinde cape <emph type="italics"/>uV<emph.end type="italics"/> quæ &longs;it ad <emph type="italics"/>vG<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>DC quad.<emph.end type="italics"/> ad <emph type="italics"/>PC quad.<emph.end type="italics"/><lb/>Et quoniam ex Conicis <gap/> <emph type="italics"/>Qv quad.<emph.end type="italics"/> ad <emph type="italics"/>PvG,<emph.end type="italics"/> ut <emph type="italics"/>DC quad.<emph.end type="italics"/> ad <lb/><emph type="italics"/>PC quad:<emph.end type="italics"/> erit <emph type="italics"/>Qv quad.<emph.end type="italics"/> æquale <emph type="italics"/>PvXuV.<emph.end type="italics"/> Unde quadratum chor- <pb pagenum="47"/>dæ arcus <emph type="italics"/>PQ<emph.end type="italics"/> erit æquale rectangulo <emph type="italics"/>VPv<emph.end type="italics"/>; adeoque Circulus qui <lb/> <arrow.to.target n="note23"></arrow.to.target><lb/>tangit Sectionem Conicam in <emph type="italics"/>P<emph.end type="italics"/> & tran&longs;it per punctum <emph type="italics"/>Q,<emph.end type="italics"/> tran&longs;ibit <lb/>etiam per punctum <emph type="italics"/>V.<emph.end type="italics"/> Coeant puncta <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>Q,<emph.end type="italics"/> & hic circulus <lb/>eju&longs;dem erit curvaturæ cum &longs;ectione conica in <emph type="italics"/>P,<emph.end type="italics"/> & <emph type="italics"/>PV<emph.end type="italics"/> æqualis erit <lb/>(2<emph type="italics"/>DCq/PC<emph.end type="italics"/>). Proinde vis qua corpus <emph type="italics"/>P<emph.end type="italics"/> in Ellip&longs;i revolvitur, erit reci­<lb/>proce ut (2<emph type="italics"/>DCq/PC<emph.end type="italics"/>) in <emph type="italics"/>PFq<emph.end type="italics"/> (per Corol. </s> <s>3 Prop. </s> <s>VI.) hoc e&longs;t (ob <lb/>datum 2<emph type="italics"/>DCq<emph.end type="italics"/> in <emph type="italics"/>PFq<emph.end type="italics"/>) directe ut <emph type="italics"/>PC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note23"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. E&longs;t igitur vis ut di&longs;tantia corporis a centro Ellip&longs;eos: & <lb/>vici&longs;&longs;im, &longs;i vis &longs;it ut di&longs;tantia, movebitur corpus in Ellip&longs;i centrum <lb/>habente in centro virium, aut forte in Circulo, in quem utique <lb/>Ellip&longs;is migrare pote&longs;t. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et æqualia erunt revolutionum in Ellip&longs;ibus univer&longs;is cir­<lb/>cum centrum idem factarum periodica tempora. </s> <s>Nam tempora <lb/>illa in Ellip&longs;ibus &longs;imilibus æqualia &longs;unt per Corol. </s> <s>3 & 8, Prop. </s> <s>IV: <lb/>in Ellip&longs;ibus autem communem habentibus axem majorem, &longs;unt ad <lb/>invicem ut Ellip&longs;eon areæ totæ directe & arearum particulæ &longs;imul <lb/>de&longs;criptæ inver&longs;e; id e&longs;t, ut axes minores directe & corporum ve­<lb/>locitates in verticibus principalibus inver&longs;e; hoc e&longs;t, ut axes illi mi­<lb/>nores directe & ordinatim applicatæ ad axes alteros inver&longs;e; & prop­<lb/>terea (ob æqualitatem rationum directarum & inver&longs;arum) in ra­<lb/>tione æqualitatis. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Si Ellip&longs;is, centro in infinitum abeunte vertatur in Parabolam, <lb/>corpus movebitur in hac Parabola; & vis ad centrum infinite di­<lb/>&longs;tans jam tendens evadet æquabilis. </s> <s>Hoc e&longs;t Theorema <emph type="italics"/>Galilæi.<emph.end type="italics"/><lb/>Et &longs;i coni &longs;ectio Parabolica, inclinatione plani ad conum &longs;ectum <lb/>mutata, vertatur in Hyperbolam, movebitur corpus in hujus pe­<lb/>rimetro, vi centripeta in centrifugam ver&longs;a. </s> <s>Et quemadmo­<lb/>dum in Circulo vel Ellip&longs;i, &longs;i vires tendunt ad centrum figuræ <lb/>in Ab&longs;ci&longs;&longs;a po&longs;itum, hæ vires augendo vel diminuendo Ordinatas in <lb/>ratione quacunque data, vel etiam mutando angulum inclinationis <lb/>Ordinatarum ad Ab&longs;ci&longs;&longs;am, &longs;emper augentur vel diminuuntur in <lb/>ratione di&longs;tantiarum a centro, &longs;i modo tempora periodica maneant <lb/>æqualia: &longs;ic etiam in figuris univer&longs;is, &longs;i Ordinatæ augeantur vel di­<lb/>minuantur in ratione quacunque data, vel angulus ordinationis ut­<lb/>cunque mutetur, manente tempore periodico; vires ad centrum <lb/>quodcunque in Ab&longs;ci&longs;&longs;a po&longs;itum tendentes a binis quibu&longs;vis figurarum locis, ad quæ termi­<lb/>nantur Ordinatæ corre&longs;pondentibus Ab&longs;ci&longs;&longs;arum punctis in&longs;i&longs;tentes, augentur vel &c. </s> <s>augentur vel diminuun­<lb/>tur in ratione di&longs;tantiarum a centro. <pb pagenum="48"/> <arrow.to.target n="note24"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note24"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>SECTIO III.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De motu Corporum in Conicis Sectionibus excentricis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XI. PROBLEMA VI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Revolvatur corpus in Ellip&longs;i: requiritur Lex vis centripetæ tenden­<lb/>tis ad umbilicum Ellip&longs;eos.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>E&longs;to Ellip&longs;eos umbilicus <emph type="italics"/>S.<emph.end type="italics"/> Agatur <emph type="italics"/>SP<emph.end type="italics"/> &longs;ecans Ellip&longs;eos <lb/>tum diametrum <emph type="italics"/>DK<emph.end type="italics"/> in <emph type="italics"/>E,<emph.end type="italics"/> tum ordinatim applicatam <emph type="italics"/>Qv<emph.end type="italics"/> in <lb/><emph type="italics"/>x,<emph.end type="italics"/> & compleatur parallelogrammum <emph type="italics"/>QxPR.<emph.end type="italics"/> Patet <emph type="italics"/>EP<emph.end type="italics"/> æqua­<lb/>lem e&longs;&longs;e &longs;emiaxi ma­<lb/> <arrow.to.target n="fig19"></arrow.to.target><lb/>jori <emph type="italics"/>AC,<emph.end type="italics"/> eo quod <lb/>acta ab altero Ellip­<lb/>&longs;eos umbilico <emph type="italics"/>H<emph.end type="italics"/> li­<lb/>nea <emph type="italics"/>HI<emph.end type="italics"/> ip&longs;i <emph type="italics"/>EC<emph.end type="italics"/> pa­<lb/>rallela, (ob æquales <lb/><emph type="italics"/>CS, CH<emph.end type="italics"/>) æquentur <lb/><emph type="italics"/>ES, EI,<emph.end type="italics"/> adeo ut <emph type="italics"/>EP<emph.end type="italics"/><lb/>&longs;emi&longs;umma &longs;it ip&longs;a­<lb/>rum <emph type="italics"/>PS, PI,<emph.end type="italics"/> id e&longs;t <lb/>(ob parallelas <emph type="italics"/>HI, <lb/>PR<emph.end type="italics"/> & angulos æqua­<lb/>les <emph type="italics"/>IPR, HPZ<emph.end type="italics"/>) <lb/>ip&longs;arum <emph type="italics"/>PS, PH,<emph.end type="italics"/><lb/>quæ <expan abbr="cõjunctim">conjunctim</expan> axem <lb/>totum 2<emph type="italics"/>AC<emph.end type="italics"/> adæ­<lb/>quant. </s> <s>Ad <emph type="italics"/>SP<emph.end type="italics"/> de­<lb/>mittatur perpendicularis <emph type="italics"/>QT,<emph.end type="italics"/> & Ellip&longs;eos latere recto principali <lb/>(&longs;eu (2<emph type="italics"/>BC quad./AC<emph.end type="italics"/>)) dicto <emph type="italics"/>L,<emph.end type="italics"/> erit <emph type="italics"/>LXQR<emph.end type="italics"/> ad <emph type="italics"/>LXPv<emph.end type="italics"/> ut <emph type="italics"/>QR<emph.end type="italics"/> ad <lb/><emph type="italics"/>Pv,<emph.end type="italics"/> id e&longs;t ut <emph type="italics"/>PE<emph.end type="italics"/> &longs;eu <emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>PC<emph.end type="italics"/>; & <emph type="italics"/>LXPv<emph.end type="italics"/> ad <emph type="italics"/>GvP<emph.end type="italics"/> ut <emph type="italics"/>L<emph.end type="italics"/> ad <lb/><emph type="italics"/>Gv<emph.end type="italics"/>; & <emph type="italics"/>GvP<emph.end type="italics"/> ad <emph type="italics"/>Qv quad.<emph.end type="italics"/> ut <emph type="italics"/>PC quad.<emph.end type="italics"/> ad <emph type="italics"/>CD quad<emph.end type="italics"/>; & (per Corol. </s> <s><lb/>2 Lem. </s> <s>VII.) <emph type="italics"/>Qv quad.<emph.end type="italics"/> ad <emph type="italics"/>Qx quad,<emph.end type="italics"/> punctis <emph type="italics"/>Q<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> coeuntibus, <lb/>e&longs;t ratio æqualitatis; & <emph type="italics"/>Qx quad.<emph.end type="italics"/> &longs;eu <emph type="italics"/>Qv quad.<emph.end type="italics"/> e&longs;t ad <emph type="italics"/>QT quad.<emph.end type="italics"/><lb/>ut <emph type="italics"/>EP quad.<emph.end type="italics"/> ad <emph type="italics"/>PF quad,<emph.end type="italics"/> id e&longs;t ut <emph type="italics"/>CA quad.<emph.end type="italics"/> ad <emph type="italics"/>PF quad.<emph.end type="italics"/> &longs;ive (per <lb/>Lem XII.) ut <emph type="italics"/>CD quad.<emph.end type="italics"/> ad <emph type="italics"/>CB quad.<emph.end type="italics"/> Et conjunctis his omnibus ratio­<lb/>nibus, <emph type="italics"/>LXQR<emph.end type="italics"/> fit ad <emph type="italics"/>QT quad.<emph.end type="italics"/> ut <emph type="italics"/><expan abbr="ACXLXPCq.XCDq.">ACXLXPCq.XCDque</expan><emph.end type="italics"/> &longs;eu 2<emph type="italics"/><expan abbr="CBq.">CBque</expan> <lb/><expan abbr="XPCq.XCDq.">XPCq.XCDque</expan><emph.end type="italics"/> ad <emph type="italics"/><expan abbr="PCXGvXCDq.XCBq.">PCXGvXCDq.XCBque</expan><emph.end type="italics"/> &longs;ive ut 2<emph type="italics"/>PC<emph.end type="italics"/> ad <emph type="italics"/>Gv.<emph.end type="italics"/> <pb pagenum="49"/>Sed, punctis <emph type="italics"/>Q<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> coeuntibus, <expan abbr="æquãtur">æquantur</expan> 2<emph type="italics"/>PC<emph.end type="italics"/> & <emph type="italics"/>Gv.<emph.end type="italics"/> Ergo & his pro­<lb/> <arrow.to.target n="note25"></arrow.to.target><lb/>portionalia <emph type="italics"/>LXQR<emph.end type="italics"/> & <emph type="italics"/>QT quad.<emph.end type="italics"/> æquantur. </s> <s>Ducantur hæc æqualia in <lb/>(<emph type="italics"/>SPq/QR<emph.end type="italics"/>) & fiet <emph type="italics"/><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"/> æquale (<emph type="italics"/>SPq.XQTq/QR<emph.end type="italics"/>). Ergo (per Corol. </s> <s>1 <lb/>& 5 Prop. </s> <s>VI.) vis centripeta reciproce e&longs;t ut <emph type="italics"/><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"/> id e&longs;t, reci­<lb/>proce in ratione duplicata di&longs;tantiæ <emph type="italics"/>SP. q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note25"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig19"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Cum vis ad centrum Ellip&longs;eos tendens, qua corpus <emph type="italics"/>P<emph.end type="italics"/> in Ellip&longs;i <lb/>illa revolvi pote&longs;t, &longs;it (per Corol. </s> <s>I Prop. </s> <s>X) ut <emph type="italics"/>CP<emph.end type="italics"/> di&longs;tantia cor­<lb/>poris ab Ellip&longs;eos centro <emph type="italics"/>C<emph.end type="italics"/>; ducatur <emph type="italics"/>CE<emph.end type="italics"/> parallela Ellip&longs;eos tan­<lb/>genti <emph type="italics"/>PR:<emph.end type="italics"/> & vis qua corpus idem <emph type="italics"/>P,<emph.end type="italics"/> circum aliud quodvis Ellip­<lb/>&longs;eos punctum <emph type="italics"/>S<emph.end type="italics"/> revolvi pote&longs;t, &longs;i <emph type="italics"/>CE<emph.end type="italics"/> & <emph type="italics"/>PS<emph.end type="italics"/> concurrant in <emph type="italics"/>E,<emph.end type="italics"/> erit ut <lb/>(<emph type="italics"/>PE cub./SPq<emph.end type="italics"/>) (per Corol. </s> <s>3 Prop. </s> <s>VII,) hoc e&longs;t, &longs;i punctum <emph type="italics"/>S<emph.end type="italics"/> &longs;it umbili­<lb/>cus Ellip&longs;eos, adeoque <emph type="italics"/>PE<emph.end type="italics"/> detur, ut <emph type="italics"/>SPq<emph.end type="italics"/> reciproce. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="main"> <s>Eadem brevitate qua traduximus Problema quintum ad Parabo­<lb/>lam, & Hyperbolam, liceret idem hic facere: verum ob dignita­<lb/>tem Problematis & u&longs;um ejus in &longs;equentibus, non pigebit ca&longs;us ce­<lb/>teros demon&longs;tratione confirmare. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XII. PROBLEMA. VII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Moveatur corpus in Hyperbola: requiritur Lex vis centripetæ ten­<lb/>dentis ad umbilicum figuræ.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Sunto <emph type="italics"/>CA, CB<emph.end type="italics"/> &longs;emi-axes Hyperbolæ; <emph type="italics"/>PG, KD<emph.end type="italics"/> diametri con­<lb/>jugatæ; <emph type="italics"/>PF, Qt<emph.end type="italics"/> perpendicula ad diametros; & <emph type="italics"/>Qv<emph.end type="italics"/> ordinatim <lb/>applicata ad diametrum <emph type="italics"/>GP.<emph.end type="italics"/> Agatur <emph type="italics"/>SP<emph.end type="italics"/> &longs;ecans cum diametrum <lb/><emph type="italics"/>DK<emph.end type="italics"/> in <emph type="italics"/>E,<emph.end type="italics"/> tum ordinatim applicatam <emph type="italics"/>Qv<emph.end type="italics"/> in <emph type="italics"/>x,<emph.end type="italics"/> & compleatur pa­<lb/>rallelogrammum <emph type="italics"/>QRPx.<emph.end type="italics"/> Patet <emph type="italics"/>EP<emph.end type="italics"/> æqualem e&longs;&longs;e &longs;emiaxi tran&longs;­<lb/>ver&longs;o <emph type="italics"/>AC,<emph.end type="italics"/> eo quod, acta ab altero Hyperbolæ umbilico <emph type="italics"/>H<emph.end type="italics"/> linea <lb/><emph type="italics"/>HI<emph.end type="italics"/> ip&longs;i <emph type="italics"/>EC<emph.end type="italics"/> parallela, ob æquales <emph type="italics"/>CS, CH,<emph.end type="italics"/> æquentur <emph type="italics"/>ES, EI<emph.end type="italics"/>; <lb/>adeo ut <emph type="italics"/>EP<emph.end type="italics"/> &longs;emidifferentia &longs;it ip&longs;arum <emph type="italics"/>PS, PI,<emph.end type="italics"/> id e&longs;t (ob pa­<lb/>rallelas <emph type="italics"/>IH, PR<emph.end type="italics"/> & angulos æquales <emph type="italics"/>IPR, HPZ<emph.end type="italics"/>) ip&longs;arum <emph type="italics"/>PS, <lb/>PH,<emph.end type="italics"/> quarum differentia axem totum 2<emph type="italics"/>AC<emph.end type="italics"/> adæquat. </s> <s>Ad <emph type="italics"/>SP<emph.end type="italics"/> de­<lb/>mittatur perpendicularis <emph type="italics"/>QT.<emph.end type="italics"/> Et Hyperbolæ latere recto princi­<lb/>pali (&longs;eu (2<emph type="italics"/>BCq/AC<emph.end type="italics"/>)) dicto <emph type="italics"/>L,<emph.end type="italics"/> erit <emph type="italics"/>LXQR<emph.end type="italics"/> ad <emph type="italics"/>LXPv<emph.end type="italics"/> ut <emph type="italics"/>QR<emph.end type="italics"/> ad <emph type="italics"/>Pv,<emph.end type="italics"/><lb/>id e&longs;t, ut <emph type="italics"/>PE<emph.end type="italics"/> &longs;eu <emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>PC<emph.end type="italics"/>; Et <emph type="italics"/>LXPv<emph.end type="italics"/> ad <emph type="italics"/>GvP<emph.end type="italics"/> ut <emph type="italics"/>L<emph.end type="italics"/> ad <pb pagenum="50"/> <arrow.to.target n="note26"></arrow.to.target><lb/><emph type="italics"/>Gv<emph.end type="italics"/>; & <emph type="italics"/>GvP<emph.end type="italics"/> ad <emph type="italics"/>Qv quad.<emph.end type="italics"/> ut <emph type="italics"/><expan abbr="PCq.">PCque</expan><emph.end type="italics"/> ad <emph type="italics"/>CDq<emph.end type="italics"/>; & (per Corol. </s> <s>2. <lb/>Lem. </s> <s>VII.) <emph type="italics"/>Qv quad.<emph.end type="italics"/> ad <emph type="italics"/>Qx quad.<emph.end type="italics"/> punctis <emph type="italics"/>Q<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> coeuntibus fit <lb/>ratio æqualitatis; & <emph type="italics"/>Qx quad.<emph.end type="italics"/> &longs;eu <emph type="italics"/>Qv quad.<emph.end type="italics"/> e&longs;t ad <emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/> ut <emph type="italics"/><expan abbr="EPq.">EPque</expan><emph.end type="italics"/><lb/>ad <emph type="italics"/>PFq,<emph.end type="italics"/> id e&longs;t ut <emph type="italics"/>CAq,<emph.end type="italics"/> ad <emph type="italics"/>PFq,<emph.end type="italics"/> &longs;ive (per Lem. </s> <s>XII.) ut <emph type="italics"/>CDq,<emph.end type="italics"/><lb/>ad <emph type="italics"/>CBq:<emph.end type="italics"/> & conjunctis his omnibus rationibus <emph type="italics"/>LXQR<emph.end type="italics"/> fit ad <lb/><emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/> ut <emph type="italics"/>ACXLXPCqXCDq<emph.end type="italics"/> &longs;eu 2<emph type="italics"/>CBqXPCqXCDq<emph.end type="italics"/> ad <lb/><emph type="italics"/>PCXGvXCDqXCB quad.<emph.end type="italics"/> &longs;ive ut 2<emph type="italics"/>PC<emph.end type="italics"/> ad <emph type="italics"/>Gv.<emph.end type="italics"/> Sed punctis <lb/><emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>Q<emph.end type="italics"/> cocuntibus æquantur 2<emph type="italics"/>PC<emph.end type="italics"/> & <emph type="italics"/>Gv.<emph.end type="italics"/> Ergo & his propor­<lb/>tionalia <emph type="italics"/>LXQR<emph.end type="italics"/> & <emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/> æquantur. </s> <s>Ducantur hæc æqualia in <lb/>(<emph type="italics"/>SPq/QR<emph.end type="italics"/>). & fiet <emph type="italics"/><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"/> æquale (<emph type="italics"/>SPqXQTq/QR<emph.end type="italics"/>). Ergo (per Corol. </s> <s>I <lb/> <arrow.to.target n="fig20"></arrow.to.target><lb/>& 5 Prop. </s> <s>VI.) vis centripeta reciproce e&longs;t ut <emph type="italics"/>LXSPq,<emph.end type="italics"/> id e&longs;t <lb/>reciproce in ratione duplicata di&longs;tantiæ <emph type="italics"/>SP. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/> <pb pagenum="51"/> <arrow.to.target n="note27"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note26"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="margin"> <s><margin.target id="note27"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig20"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Inveniatur vis quæ tendit ab Hyperbolæ centro <emph type="italics"/>C.<emph.end type="italics"/> Prodibit hæc <lb/>di&longs;tantiæ <emph type="italics"/>CP<emph.end type="italics"/> proportionalis. </s> <s>Inde vero (per Corol. </s> <s>3 Prop. </s> <s>VII.) <lb/>vis ad umbilicum <emph type="italics"/>S<emph.end type="italics"/> tendens erit ut (<emph type="italics"/>PEcub/SPq<emph.end type="italics"/>), hoc e&longs;t, ob datam <emph type="italics"/>PE,<emph.end type="italics"/><lb/>reciproce ut <emph type="italics"/><expan abbr="SPq.">SPque</expan> q.E.I.<emph.end type="italics"/></s> </p> <p type="main"> <s>Eodem modo demon&longs;tratur quod corpus, hac vi centripeta in <lb/>centrifugam ver&longs;a, movebitur in Hyperbola conjugata. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA XIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Latus rectum Parabolæ ad verticem quemvis pertinens, e&longs;t quadru­<lb/>plum di&longs;tantiæ verticis illius ab umbilico figuræ.<emph.end type="italics"/> Patet ex Conicis. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA XIV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Perpendiculum quod ab umbilico Parabolæ ad tangentem ejus demitti­<lb/>tur, medium e&longs;t proportionale inter di&longs;tantias umbilici a puncto con­<lb/>tactus & a vertice principali figuræ.<emph.end type="italics"/></s> </p> <p type="main"> <s>Sit enim <emph type="italics"/>AQP<emph.end type="italics"/> Parabola, <emph type="italics"/>S<emph.end type="italics"/> umbilicus ejus, <emph type="italics"/>A<emph.end type="italics"/> vertex principa­<lb/>lis <emph type="italics"/>P<emph.end type="italics"/> punctum <lb/> <arrow.to.target n="fig21"></arrow.to.target><lb/>contactus, <emph type="italics"/>PO<emph.end type="italics"/><lb/>ordinatim ap­<lb/>plicata ad dia­<lb/>metrum prin­<lb/>cipalem, <emph type="italics"/>PM<emph.end type="italics"/><lb/>tangens dia­<lb/>metro princi­<lb/>pali occurrens <lb/>in <emph type="italics"/>M,<emph.end type="italics"/> & <emph type="italics"/>SN,<emph.end type="italics"/><lb/>linea perpen­<lb/>dicularis ab umbilico in tangentem. </s> <s>Jungatur <emph type="italics"/>AN,<emph.end type="italics"/> & ob æquales <lb/><emph type="italics"/>MS<emph.end type="italics"/> & <emph type="italics"/>SP, MN<emph.end type="italics"/> & <emph type="italics"/>NP, MA<emph.end type="italics"/> & <emph type="italics"/>AO,<emph.end type="italics"/> parallelæ erunt rectæ <lb/><emph type="italics"/>AN<emph.end type="italics"/> & <emph type="italics"/>OP,<emph.end type="italics"/> & inde triangulum <emph type="italics"/>SAN<emph.end type="italics"/> rectangulum erit ad <emph type="italics"/>A<emph.end type="italics"/> & <lb/>&longs;imile triangulis æqualibus <emph type="italics"/>SNM, SNP:<emph.end type="italics"/> Ergo <emph type="italics"/>PS<emph.end type="italics"/> e&longs;t ad <emph type="italics"/>SN,<emph.end type="italics"/><lb/>ut <emph type="italics"/>SN<emph.end type="italics"/> ad <emph type="italics"/>SA. q.E.D.<emph.end type="italics"/></s> </p> <figure id="fig21"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. <emph type="italics"/><expan abbr="PSq.">PSque</expan><emph.end type="italics"/> e&longs;t ad <emph type="italics"/><expan abbr="SNq.">SNque</expan><emph.end type="italics"/> ut <emph type="italics"/>PS<emph.end type="italics"/> ad <emph type="italics"/>SA.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et ob datam <emph type="italics"/>SA,<emph.end type="italics"/> e&longs;t <emph type="italics"/><expan abbr="SNq.">SNque</expan><emph.end type="italics"/> ut <emph type="italics"/>PS.<emph.end type="italics"/> <pb pagenum="52"/> <arrow.to.target n="note28"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note28"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Et concur&longs;us tangentis cuju&longs;vis <emph type="italics"/>PM<emph.end type="italics"/> cum recta <emph type="italics"/>SN,<emph.end type="italics"/><lb/>quæ ab umbilico in ip&longs;am perpendicularis e&longs;t, incidit in rectam <emph type="italics"/>AN,<emph.end type="italics"/><lb/>quæ Parabolam tangit in vertice principali. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO. XIII. PROBLEMA VIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Moveatur corpus in perimetro Parabolæ: requiritur Lex vis centri­<lb/>petæ tendentis ad umbilicum hujus figuræ.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Maneat con&longs;tructio Lemmatis, &longs;itque <emph type="italics"/>P<emph.end type="italics"/> corpus in perimetro Pa­<lb/>rabolæ, & a loco <emph type="italics"/>Q<emph.end type="italics"/> in quem corpus proxime movetur, age ip&longs;i <emph type="italics"/>SP<emph.end type="italics"/><lb/>parallelam <emph type="italics"/>QR<emph.end type="italics"/> & perpendicularem <emph type="italics"/>QT,<emph.end type="italics"/> necnon <emph type="italics"/>Qv<emph.end type="italics"/> tangenti pa­<lb/>rallelam & occurrentem tum diametro <emph type="italics"/>YPG<emph.end type="italics"/> in <emph type="italics"/>v,<emph.end type="italics"/> tum di&longs;tantiæ <lb/><emph type="italics"/>SP<emph.end type="italics"/> in <emph type="italics"/>x.<emph.end type="italics"/> Jam ob &longs;imilia triangula <emph type="italics"/>Pxv, SPM<emph.end type="italics"/> & æqualia unius <lb/>latera <emph type="italics"/>SM, SP,<emph.end type="italics"/> æqualia &longs;unt alterius latera <emph type="italics"/>Px<emph.end type="italics"/> &longs;eu <emph type="italics"/>QR<emph.end type="italics"/> & <emph type="italics"/>Pv.<emph.end type="italics"/><lb/>Sed, ex Conicis, quadratum ordinatæ <emph type="italics"/>Qv<emph.end type="italics"/> æquale e&longs;t rectangulo &longs;ub <lb/>latere recto & &longs;egmento diametri <emph type="italics"/>Pv,<emph.end type="italics"/> id e&longs;t (per Lem. </s> <s>XIII.) rectangu­<lb/>lo 4 <emph type="italics"/>PSXPv,<emph.end type="italics"/> &longs;eu 4 <emph type="italics"/>PSXQR<emph.end type="italics"/>; & punctis <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>Q<emph.end type="italics"/> coeuntibus, ra­<lb/>tio <emph type="italics"/>Qv<emph.end type="italics"/> ad <emph type="italics"/>Qx<emph.end type="italics"/> per (per Corol. </s> <s>2 Lem. </s> <s>VII.) fit ratio æqualitatis. </s> <s>Er­<lb/>go <emph type="italics"/>Qxquad.<emph.end type="italics"/> eo <lb/> <arrow.to.target n="fig22"></arrow.to.target><lb/>in ca&longs;u, æquale <lb/>e&longs;t rectangu­<lb/>lo 4 <emph type="italics"/>PSXQR.<emph.end type="italics"/><lb/>E&longs;t autem (ob <lb/>&longs;imilia trian­<lb/>gula <emph type="italics"/>QxT, <lb/>SPN) <expan abbr="Qxq.">Qxque</expan><emph.end type="italics"/><lb/>ad <emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/> ut <lb/><emph type="italics"/><expan abbr="PSq.">PSque</expan><emph.end type="italics"/> ad <emph type="italics"/><expan abbr="SNq.">SNque</expan><emph.end type="italics"/><lb/>hoc e&longs;t (per <lb/>Corol. </s> <s>1. Lem. </s> <s>XIV.) ut <emph type="italics"/>PS<emph.end type="italics"/> ad <emph type="italics"/>SA,<emph.end type="italics"/> id e&longs;t ut 4 <emph type="italics"/>PSXQR<emph.end type="italics"/><lb/>ad 4<emph type="italics"/>SAXQR,<emph.end type="italics"/> & inde (per Prop. </s> <s>IX. Lib. </s> <s>v. Elem.) <emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/> & <lb/>4<emph type="italics"/>SAXQR<emph.end type="italics"/> æquantur. </s> <s>Ducantur hæc æqualia in (<emph type="italics"/>SPq./QR<emph.end type="italics"/>), & fiet <lb/>(<emph type="italics"/>SPq.XQTq./QR<emph.end type="italics"/>) æquale <emph type="italics"/>SPq.X4SA:<emph.end type="italics"/> & propterea (per Corol. </s> <s>1 & 5 <lb/>Prop. </s> <s>VI.) vis centripeta e&longs;t reciproce ut <emph type="italics"/>SPq.X4SA,<emph.end type="italics"/> id e&longs;t, ob da­<lb/>tam 4<emph type="italics"/>SA,<emph.end type="italics"/> reciproce in duplicata ratione di&longs;tantiæ <emph type="italics"/>SP. q.E.I.<emph.end type="italics"/></s> </p> <pb pagenum="53"/> <figure id="fig22"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Ex tribus novi&longs;&longs;imis Propo&longs;itionibus con&longs;equens e&longs;t, quod </s> </p> <p type="main"> <s> <arrow.to.target n="note29"></arrow.to.target><lb/>&longs;i corpus quodvis <emph type="italics"/>P,<emph.end type="italics"/> &longs;ecundum lineam quamvis rectam <emph type="italics"/>PR,<emph.end type="italics"/> qua­<lb/>cunque cum velocitate exeat de loco <emph type="italics"/>P,<emph.end type="italics"/> & vi centripeta quæ &longs;it re­<lb/>ciproce proportionalis quadrato di&longs;tantiæ locorum a centro, &longs;imul <lb/>agitetur; movebitur hoc corpus in aliqua &longs;ectionum Conicarum <lb/>umbilicum habente in centro virium; & contra. </s> <s>Nam datis umbi­<lb/>lico & puncto contactus & po&longs;itione tangentis, de&longs;cribi pote&longs;t &longs;ectio <lb/>Conica quæ curvaturam datam ad punctum illud habebit. </s> <s>Datur <lb/>autem curvatura ex data vi centripeta: & Orbes duo &longs;e mutuo tan­<lb/>gentes, eadem vi centripeta de&longs;cribi non po&longs;&longs;unt. </s> </p> <p type="margin"> <s><margin.target id="note29"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Si velocitas, quacum corpus exit de loco &longs;uo <emph type="italics"/>P,<emph.end type="italics"/> ea <lb/>&longs;it, qua lineola <emph type="italics"/>PR<emph.end type="italics"/> in minima aliqua temporis particula de&longs;cribi <lb/>po&longs;&longs;it, & vis centripeta potis &longs;it eodem tempore corpus idem mo­<lb/>vere per &longs;patium <emph type="italics"/>QR:<emph.end type="italics"/> movebitur hoc corpus in Conica aliqua &longs;e­<lb/>ctione, cujus latus rectum principale e&longs;t quantitas illa (<emph type="italics"/>QTq./QR<emph.end type="italics"/>) quæ <lb/>ultimo fit ubi lineolæ <emph type="italics"/>PR, QR<emph.end type="italics"/> in infinitum diminuuntur. </s> <s>Circu­<lb/>lum in his Corollariis refero ad Ellip&longs;in, & ca&longs;um excipio ubi cor­<lb/>pus recta de&longs;cendit ad centrum. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XIV. THEOREMA VI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si corpora plura revolvantur circa centrum commune, & vis centri­<lb/>peta &longs;it reciproce in duplicata ratione di&longs;tantiæ locorum a centro; <lb/>dico quod Orbium Latera recta principalia &longs;unt in duplicata ratio­<lb/>one arearum quas corpora, radiis ad centrum ductis, eodem tempore <lb/>de&longs;cribunt.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam, per Corol. </s> <s>2. Prop. </s> <s>XIII, Latus rectum <emph type="italics"/>L<emph.end type="italics"/> æquale e&longs;t quan­<lb/>titati (<emph type="italics"/>QTq./QR<emph.end type="italics"/>) quæ ultimo fit ubi coeunt puncta <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> Sed linea <lb/>minima <emph type="italics"/>QR,<emph.end type="italics"/> dato tempore, e&longs;t ut vis centripeta generans, hoc <lb/>e&longs;t (per Hypothe&longs;in) reciproce ut <emph type="italics"/><expan abbr="SPq.">SPque</expan><emph.end type="italics"/> Ergo (<emph type="italics"/>QTq./QR<emph.end type="italics"/>) e&longs;t ut <lb/><emph type="italics"/><expan abbr="QTq.XSPq.">QTq.XSPque</expan><emph.end type="italics"/> hoc e&longs;t, latus rectum <emph type="italics"/>L<emph.end type="italics"/> in duplicata ratione areæ <lb/><emph type="italics"/>QTXSP. q.E.D.<emph.end type="italics"/> <pb pagenum="54"/> <arrow.to.target n="note30"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note30"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Hinc Ellip&longs;eos area tota, eique proportionale rectangu­<lb/>lum &longs;ub axibus, e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione <lb/>lateris recti & ratione temporis periodici. </s> <s>Namque area tota e&longs;t <lb/>ut area <emph type="italics"/>QTXSP,<emph.end type="italics"/> quæ dato tempore de&longs;cribitur, ducta in &c. </s> <s>ducta in tempus periodicum. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XV. THEOREMA VII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod Tempora periodica in Ellip&longs;ibus &longs;unt in ratione <lb/>&longs;e&longs;quiplicata majorum axium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Namque axis minor e&longs;t medius proportionalis inter axem majo­<lb/>rem & latus rectum, atque adeo rectangulum &longs;ub axibus e&longs;t in ra­<lb/>tione compo&longs;ita ex &longs;ubduplicata ratione lateris recti & &longs;e&longs;quiplicata <lb/>ratione axis majoris. </s> <s>Sed hoc rectangulum, per Corollarium Prop. </s> <s><lb/>XIV. e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione lateris recti <lb/>& ratione periodici temporis. </s> <s>Dematur utrobique &longs;ubduplicata <lb/>ratio lateris recti, & manebit &longs;e&longs;quiplicata ratio majoris axis æqua­<lb/>lis rationi periodici temporis. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Sunt igitur tempora periodica in Ellip&longs;ibus eadem ac in <lb/>Circulis, quorum diametri æquantur majoribus axibus Ellip&longs;eon. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVI. THEOREMA VIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, & actis ad corpora lineis rectis, quæ ibidem tangant Or­<lb/>bitas, demi&longs;&longs;i&longs;que ab umbilico communi ad has tangentes perpendi­<lb/>cularibus: dico quod Velocitates corporum &longs;unt in ratione compo&longs;i­<lb/>ta ex ratione perpendiculorum inver&longs;e & &longs;ubduplicata ratione la­<lb/>terum rectorum principalium directe.<emph.end type="italics"/></s> </p> <p type="main"> <s>Ab umbilico <emph type="italics"/>S<emph.end type="italics"/> ad tangentem <emph type="italics"/>PR<emph.end type="italics"/> demitte perpendiculum <emph type="italics"/>SY<emph.end type="italics"/><lb/>& velocitas corporis <emph type="italics"/>P<emph.end type="italics"/> erit reciproce in &longs;ubduplicata ratione quan­<lb/>titatis (<emph type="italics"/>SYq/L<emph.end type="italics"/>). Nam velocitas illa e&longs;t ut arcus quam minimus <emph type="italics"/>PQ<emph.end type="italics"/><lb/>in data temporis particula de&longs;criptus, hoc e&longs;t (per Lem. </s> <s>VII.) ut <lb/>tangens <emph type="italics"/>PR,<emph.end type="italics"/> id e&longs;t (ob proportionales <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>QT<emph.end type="italics"/> & <emph type="italics"/>SP<emph.end type="italics"/> ad <emph type="italics"/>SY<emph.end type="italics"/>) ut <lb/>(<emph type="italics"/>SPXQT/SY<emph.end type="italics"/>), &longs;ive ut <emph type="italics"/>SY<emph.end type="italics"/> reciproce & <emph type="italics"/>SPXQT<emph.end type="italics"/> directe; e&longs;tque <pb pagenum="55"/><emph type="italics"/>SPXQT<emph.end type="italics"/> ut area dato tempore de&longs;cripta, id e&longs;t, per Prop. </s> <s>XIV. </s> </p> <p type="main"> <s> <arrow.to.target n="note31"></arrow.to.target><lb/>in &longs;ubduplicata ratione lateris recti. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note31"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Latera recta principalia &longs;unt in ratione compo&longs;ita ex <lb/>duplicata ratione perpendiculorum & duplicata ratione veloci­<lb/>tatum. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Velocitates corporum in maximis & minimis ab umbi­<lb/>lico communi di&longs;tantiis, &longs;unt in ratione compo&longs;ita ex ratione di­<lb/>&longs;tantiarum inver&longs;e & &longs;ubduplicata ratione laterum rectorum princi­<lb/>palium directe. </s> <s>Nam perpendicula jam &longs;unt ip&longs;æ di&longs;tantiæ. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Ideoque velocitas in Conica &longs;ectione, in maxima vel <lb/>minima ab umbilico di&longs;tantia, e&longs;t ad velocitatem in Circulo in ea­<lb/>dem à centro di&longs;tantia, in &longs;ubduplicata ratione lateris recti princi­<lb/>palis ad duplam illam di&longs;tantiam. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Corporum in Ellip&longs;ibus gyrantium velocitates in medi­<lb/>ocribus di&longs;tantus ab umbilico communi &longs;unt eædem quæ corporum <lb/>gyrantium in Circulis ad ea&longs;dem di&longs;tantias; hoc e&longs;t (per Corol 6. <lb/>Prop. </s> <s>IV.) reciproce in &longs;ubduplicata ratione di&longs;tantiarum. </s> <s>Nam <lb/>perpendicula jam &longs;unt &longs;emi-axes minores; & hi &longs;unt ut mediæ <lb/>proportionales inter di&longs;tantias & latera recta. </s> <s>Componatur hæc <lb/>ratio inver&longs;e cum &longs;ubduplicata ratione laterum rectorum directe, & <lb/>fiet ratio &longs;ubduplicata di&longs;tantiarum inver&longs;e. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. In eadem figura, vel etiam in figuris diver&longs;is, quarum <pb pagenum="56"/> <arrow.to.target n="note32"></arrow.to.target><lb/>latera recta principalia &longs;unt æqualia, velocitas corporis e&longs;t reciproce <lb/>ut perpendiculum demi&longs;&longs;um ab umbilico ad tangentem. </s> </p> <p type="margin"> <s><margin.target id="note32"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 6. In Parabola, velocitas e&longs;t reciproce in &longs;ubduplicata ra­<lb/>tione di&longs;tantiæ corporis ab umbilico figuræ; in Ellip&longs;i magis varia­<lb/><gap/>ur, in Hyperbola minus, quam in hac ratione. </s> <s>Nam (per Corol. </s> <s><lb/>2. Lem. </s> <s>XIV.) perpendiculum demi&longs;&longs;um ab umbilico ad tangentem <lb/>Parabolæ e&longs;t in &longs;ubduplicata ratione di&longs;tantiæ. </s> <s>In Hyperbola per­<lb/>pendiculum minus variatur, in Ellip&longs;i magis. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 7. In Parabola, velocitas corporis ad quamvis ab umbili­<lb/>co di&longs;tantiam, e&longs;t ad velocitatem corporis revolventis in Circulo <lb/>ad eandem a centro di&longs;tantiam, in &longs;ubduplicata ratione numeri bi­<lb/>narii ad unitatem; in Ellip&longs;i minor e&longs;t, in Hyperbola major quam <lb/>in hac ratione. </s> <s>Nam per hujus Corollarium &longs;ecundum, velocitas <lb/>in vertice Parabolæ e&longs;t in hac ratione, & per Corollaria &longs;exta hu­<lb/>jus & Propo&longs;itionis quartæ, &longs;ervatur eadem proportio in omnibus <lb/>di&longs;tantiis. </s> <s>Hinc etiam in Parabola velocitas ubique æqualis e&longs;t ve­<lb/>locitati corporis revolventis in Circulo ad dimidiam di&longs;tantiam, in <lb/>Ellip&longs;i minor e&longs;t, in Hyperbola major. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 8. Velocitas gyrantis in Sectione quavis Conica e&longs;t ad ve­<lb/>locitatem gyrantis in Circulo in di&longs;tantia dimidii lateris recti princi­<lb/>palis Sectionis, ut di&longs;tantia illa ad perpendiculum ab umbilico in <lb/>tangentem Sectionis demi&longs;&longs;um. </s> <s>Patet per Corollarium quintum. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 9. Unde cum (per Corol. </s> <s>6. Prop. </s> <s>IV.) velocitas gyrantis <lb/>in hoc Circulo &longs;it ad velocitatem gyrantis in Circulo quovis alio, <lb/>reciproce in &longs;ubduplicata ratione di&longs;tantiarum; fiet ex æquo velo­<lb/>citas gyrantis in Conica &longs;ectione ad velocitatem gyrantis in Circulo <lb/>in eadem di&longs;tantia, ut media proportionalis inter di&longs;tantiam illam <lb/>communem & &longs;emi&longs;&longs;em principalis lateris recti &longs;ectionis, ad per­<lb/>pendiculum ab umbilico communi in tangentem &longs;ectionis de­<lb/>mi&longs;&longs;um. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVII. PROBLEMA. IX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod vis centripeta &longs;it reciproce proportionalis quadrato di&longs;tan­<lb/>&longs;tantiæ locorum a centro, & quod vis illius quantitas ab&longs;oluta &longs;it <lb/>cognita; requiritur Linea quam corpus de&longs;cribit, de loco dato, cum <lb/>data velocitate, &longs;ecundum datam rectam egrediens.<emph.end type="italics"/></s> </p> <p type="main"> <s>Vis centripeta tendens ad punctum <emph type="italics"/>S<emph.end type="italics"/> ea &longs;it qua corpus <emph type="italics"/>p<emph.end type="italics"/> in or­<lb/>bita quavis data <emph type="italics"/>pq<emph.end type="italics"/> gyretur, & cogno&longs;catur hujus velocitas in loco <emph type="italics"/>p.<emph.end type="italics"/> <pb pagenum="57"/>De loco <emph type="italics"/>P,<emph.end type="italics"/> &longs;ecundum lineam <emph type="italics"/>PR,<emph.end type="italics"/> exeat corpus <emph type="italics"/>P,<emph.end type="italics"/> cum data velo­<lb/> <arrow.to.target n="note33"></arrow.to.target><lb/>citate, & mox inde, cogente vi centripeta, deflectat illud in Coni­<lb/>&longs;ectionem <emph type="italics"/><expan abbr="Pq.">Pque</expan><emph.end type="italics"/> Hanc igitur recta <emph type="italics"/>PR<emph.end type="italics"/> tanget in <emph type="italics"/>P.<emph.end type="italics"/> Tangat itidem <lb/>recta aliqua <emph type="italics"/>pr<emph.end type="italics"/> Orbitam <emph type="italics"/>pq<emph.end type="italics"/> in <emph type="italics"/>p,<emph.end type="italics"/> & &longs;i ab <emph type="italics"/>S<emph.end type="italics"/> ad eas tangentes demitti <lb/>intelligantur perpendicula, erit (per Corol. </s> <s>1. Prop. </s> <s>XVI.) latus re­<lb/>ctum principale Coni&longs;ectionis ad latus rectum principale Orbitæ, in <lb/>ratione compo&longs;ita ex duplicata ratione perpendiculorum & dupli­<lb/>cata ratione velocitatum, atque adeo datur. </s> <s>Sit i&longs;tud <emph type="italics"/>L.<emph.end type="italics"/> Da­<lb/>tur præterea Coni&longs;e­<lb/> <arrow.to.target n="fig23"></arrow.to.target><lb/>ctionis umbilicus <emph type="italics"/>S.<emph.end type="italics"/><lb/>Anguli <emph type="italics"/>RPS<emph.end type="italics"/> com­<lb/>plementum ad du­<lb/>os rectos fiat angu­<lb/>lus <emph type="italics"/>RPH,<emph.end type="italics"/> & dabi­<lb/>tur po&longs;itione linea <lb/><emph type="italics"/>PH,<emph.end type="italics"/> in qua umbilicus <lb/>alter <emph type="italics"/>H<emph.end type="italics"/> locatur. </s> <s>De­<lb/>mi&longs;&longs;o ad <emph type="italics"/>PH<emph.end type="italics"/> perpen­<lb/>diculo <emph type="italics"/>SK,<emph.end type="italics"/> erigi intelligatur &longs;emiaxis conjugatus <emph type="italics"/>BC,<emph.end type="italics"/> & erit <lb/><emph type="italics"/>SPq.-2KPH+PHq.=SHq.=4CHq.=4BHq-4BCq.= <lb/>―SP+PH: quad. -LX―SP+PH=SPq.+2SPH+PHq. <lb/>-LX―SP+PH.<emph.end type="italics"/> Addantur utrobique 2<emph type="italics"/>KPH-SPq-PHq <lb/>+LX―SP+PH,<emph.end type="italics"/> & fiet <emph type="italics"/>LX―SP+PH=2SPH+2KPH,<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>SP+PH,<emph.end type="italics"/> ad <emph type="italics"/>PH,<emph.end type="italics"/> ut 2<emph type="italics"/>SP+2KP<emph.end type="italics"/> ad <emph type="italics"/>L.<emph.end type="italics"/> Unde datur <emph type="italics"/>PH<emph.end type="italics"/><lb/>tam longitudine quam po&longs;itione. </s> <s>Nimirum &longs;i ea fit corporis &c. </s> <s>in <emph type="italics"/>P<emph.end type="italics"/><lb/>velocitas, ut latus rectum <emph type="italics"/>L<emph.end type="italics"/> minus fuerit quam 2 <emph type="italics"/>SP+2KP,<emph.end type="italics"/><lb/>jacebit <emph type="italics"/>PH<emph.end type="italics"/> ad eandem partem tangentis <emph type="italics"/>PR<emph.end type="italics"/> cum linea <emph type="italics"/>PS,<emph.end type="italics"/><lb/>adeoque figura erit Ellip&longs;is, & ex datis umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/> & axe <lb/>principali <emph type="italics"/>SP+PH,<emph.end type="italics"/> dabitur: Sin tanta &longs;it corporis velocitas ut <lb/>latus rectum <emph type="italics"/>L<emph.end type="italics"/> æquale fuerit 2 <emph type="italics"/>SP+2KP,<emph.end type="italics"/> longitudo <emph type="italics"/>PH<emph.end type="italics"/> infi­<lb/>nita erit, & propterea figura erit Parabola axem habens <emph type="italics"/>SH<emph.end type="italics"/> paral­<lb/>lelum lineæ <emph type="italics"/>PK,<emph.end type="italics"/> & inde dabitur. </s> <s>Quod &longs;i corpus majori adhuc <lb/>cum velocitate de loco &longs;uo <emph type="italics"/>P<emph.end type="italics"/> exeat, capienda erit longitudo <emph type="italics"/>PH<emph.end type="italics"/><lb/>ad alteram partem tangentis, adeoque tangente inter umbilicos per­<lb/>gente, figura erit Hyperbola axem habens principalem æqualem dif­<lb/>ferentiæ linearum <emph type="italics"/>SP<emph.end type="italics"/> & <emph type="italics"/>PH,<emph.end type="italics"/> & inde dabitur. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note33"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig23"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc in omni Coni&longs;ectione ex dato vertice principali <emph type="italics"/>D,<emph.end type="italics"/><lb/>latere recto <emph type="italics"/>L,<emph.end type="italics"/> & umbilico <emph type="italics"/>S,<emph.end type="italics"/> datur umbilicus alter <emph type="italics"/>H<emph.end type="italics"/> capiendo <emph type="italics"/>DH,<emph.end type="italics"/><lb/>ad <emph type="italics"/>DS<emph.end type="italics"/> ut e&longs;t latus rectum ad differentiam inter latus rectum & <lb/>4 <emph type="italics"/>DS.<emph.end type="italics"/> Nam proportio <emph type="italics"/>SP+PH<emph.end type="italics"/> ad <emph type="italics"/>PH<emph.end type="italics"/> ut 2 <emph type="italics"/>SP+2KP<emph.end type="italics"/> ad <emph type="italics"/>L,<emph.end type="italics"/> <pb pagenum="58"/> <arrow.to.target n="note34"></arrow.to.target><lb/>in ca&longs;u hujus Corollarii, &longs;it <emph type="italics"/>DS+DH<emph.end type="italics"/> ad <emph type="italics"/>DH<emph.end type="italics"/> ut 4 <emph type="italics"/>DS<emph.end type="italics"/> ad <emph type="italics"/>L,<emph.end type="italics"/> & <lb/>divi&longs;im <emph type="italics"/>DS<emph.end type="italics"/> ad <emph type="italics"/>DH<emph.end type="italics"/> ut 4 <emph type="italics"/>DS-L<emph.end type="italics"/> ad <emph type="italics"/>L.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note34"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Unde &longs;i datur corporis velocitas in vertice principali <emph type="italics"/>D,<emph.end type="italics"/><lb/>invenietur Orbita expedite, capiendo &longs;cilicet latus rectum ejus, ad <lb/>duplam di&longs;tantiam <emph type="italics"/>DS,<emph.end type="italics"/> in duplicata ratione velocitatis hujus datæ <lb/>ad velocitatem corporis in Circulo, ad di&longs;tantiam <emph type="italics"/>DS,<emph.end type="italics"/> gyrantis (per <lb/>Corol. </s> <s>3. Prop. </s> <s>XVI.) dein <emph type="italics"/>DH<emph.end type="italics"/> ad <emph type="italics"/>DS<emph.end type="italics"/> ut latus rectum ad differen­<lb/>tiam inter latus rectum & 4 <emph type="italics"/>DS.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Hinc etiam &longs;i corpus moveatur in Sectione quacunque <lb/>Conica, & ex Orbe &longs;uo impul&longs;u quocunque exturbetur; cogno&longs;ci <lb/>pote&longs;t Orbis in quo po&longs;tea cur&longs;um &longs;uum peraget. </s> <s>Nam componen­<lb/>do proprium corporis motum cum motu illo quem impul&longs;us &longs;olus <lb/>generaret, habebitur motus quocum corpus de dato impul&longs;us loco, <lb/>&longs;ecundum rectam po&longs;itione datam, exibit. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Et &longs;i corpus illud vi aliqua extrin&longs;ecus impre&longs;&longs;a conti­<lb/>nuo perturbetur, innote&longs;cet cur&longs;us quam proxime, colligendo mu­<lb/>tationes quas vis illa in punctis quibu&longs;dam inducit, & ex &longs;eriei ana­<lb/>logia mutationes continuas in locis intermediis æ&longs;timando. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Si corpus <emph type="italics"/>P<emph.end type="italics"/> vi centripeta ad <lb/> <arrow.to.target n="fig24"></arrow.to.target><lb/>punctum quodcunque datum <emph type="italics"/>R<emph.end type="italics"/><lb/>tendente moveatur in perimetro <lb/>datæ cuju&longs;cunque Sectionis co­<lb/>nicæ cujus centrum &longs;it <emph type="italics"/>C,<emph.end type="italics"/> & re­<lb/>quiratur Lex vis centripetæ: du­<lb/>catur <emph type="italics"/>CG<emph.end type="italics"/> radio <emph type="italics"/>RP<emph.end type="italics"/> paralle­<lb/>la, & Orbis tangenti <emph type="italics"/>PG<emph.end type="italics"/> oc­<lb/>currens in <emph type="italics"/>G<emph.end type="italics"/>; & vis illa (per <lb/>Corol. </s> <s>1 & Schol. </s> <s>Prop. </s> <s>X, & Corol. </s> <s>3 Prop. </s> <s>VII.) erit ut <lb/>(<emph type="italics"/>CG cub./RP quad.<emph.end type="italics"/>) <pb pagenum="59"/> <arrow.to.target n="note35"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note35"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig24"></figure> <p type="main"> <s><emph type="center"/>SECTIO IV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Inventione Orbium Ellipticorum, Parabolicorum & Hyperbolico­<lb/>rum ex umbilico dato.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>LEMMA XV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si ab Ellip&longs;eos vel Hyperbolæ cuju&longs;vis umbilicis duobus<emph.end type="italics"/> S, H, <emph type="italics"/>ad <lb/>punctum quodvis tertium<emph.end type="italics"/> V <emph type="italics"/>inflectantur rectæ duæ<emph.end type="italics"/> SV, HV, <lb/><emph type="italics"/>quarum una<emph.end type="italics"/> HV <emph type="italics"/>æqualis &longs;it axi principali figuræ, altera<emph.end type="italics"/> SV <emph type="italics"/>a <lb/>perpendiculo<emph.end type="italics"/> TR <emph type="italics"/>in &longs;e demi&longs;&longs;o bi-<emph.end type="italics"/><lb/> <arrow.to.target n="fig25"></arrow.to.target><lb/><emph type="italics"/>&longs;ecetur in<emph.end type="italics"/> T; <emph type="italics"/>perpendiculum illud<emph.end type="italics"/><lb/>TR <emph type="italics"/>&longs;ectionem Conicam alicubi tan­<lb/>get: & contra, &longs;i tangit, erit<emph.end type="italics"/> HV <lb/><emph type="italics"/>æqualis axi principali figuræ.<emph.end type="italics"/></s> </p> <figure id="fig25"></figure> <p type="main"> <s>Secet enim perpendiculum <emph type="italics"/>TR<emph.end type="italics"/> re­<lb/>ctam <emph type="italics"/>HV<emph.end type="italics"/> productam, &longs;i opus fuerit, <lb/>in <emph type="italics"/>R<emph.end type="italics"/>; & jungatur <emph type="italics"/>SR.<emph.end type="italics"/> Ob æquales <lb/><emph type="italics"/>TS, TV,<emph.end type="italics"/> æquales erunt & rectæ <emph type="italics"/>SR, VR<emph.end type="italics"/> & anguli <emph type="italics"/>TRS, TRV.<emph.end type="italics"/><lb/>Unde punctum <emph type="italics"/>R<emph.end type="italics"/> erit ad Sectionem Conicam, & perpendiculum <lb/><emph type="italics"/>TR<emph.end type="italics"/> tanget eandem: & contra. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVIII. PROBLEMA X.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Datis umbilico & axibus principalibus de&longs;cribere Trajectorias Ellipti­<lb/>cas & Hyperbolicas, quæ tran&longs;ibunt per puncta data, & rectas po­<lb/>&longs;itione datas contingent.<emph.end type="italics"/></s> </p> <p type="main"> <s>Sit <emph type="italics"/>S<emph.end type="italics"/> communis umbilicus figurarum; <emph type="italics"/>AB<emph.end type="italics"/> longitudo axis prin­<lb/>cipalis Trajectoriæ cuju&longs;vis; <emph type="italics"/>P<emph.end type="italics"/> punctum per quod Trajectoria de­<lb/>bet tran&longs;ire; & <emph type="italics"/>TR<emph.end type="italics"/> recta quam debet tangere. </s> <s>Centro <emph type="italics"/>P<emph.end type="italics"/> inter­<lb/>vallo <emph type="italics"/>AB-SP,<emph.end type="italics"/> &longs;i orbita &longs;it Ellip&longs;is, vel <emph type="italics"/>AB+SP,<emph.end type="italics"/> &longs;i ea &longs;it Hy­<lb/>perbola, de&longs;cribatur circulus <emph type="italics"/>HG.<emph.end type="italics"/> Ad tangentem <emph type="italics"/>TR<emph.end type="italics"/> demittatur <lb/>perpendiculum <emph type="italics"/>ST,<emph.end type="italics"/> & producatur idem ad <emph type="italics"/>V,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>TV<emph.end type="italics"/> æqualis <lb/><emph type="italics"/>ST<emph.end type="italics"/>; centroque <emph type="italics"/>V<emph.end type="italics"/> & intervallo <emph type="italics"/>AB<emph.end type="italics"/> de&longs;cribatur circulus <emph type="italics"/>FH.<emph.end type="italics"/> Hac <pb pagenum="60"/> <arrow.to.target n="note36"></arrow.to.target><lb/>methodo &longs;ive dentur duo puncta <emph type="italics"/>P, p,<emph.end type="italics"/> &longs;ive duæ tangentes <emph type="italics"/>TR, <lb/>tr,<emph.end type="italics"/> &longs;ive punctum <emph type="italics"/>P<emph.end type="italics"/> & tangens <lb/> <arrow.to.target n="fig26"></arrow.to.target><lb/><emph type="italics"/>TR,<emph.end type="italics"/> de&longs;cribendi &longs;unt circuli duo. </s> <s><lb/>Sit <emph type="italics"/>H<emph.end type="italics"/> eorum inter&longs;ectio com­<lb/>munis, & umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/> axe illo <lb/>dato de&longs;cribatur Trajectoria. </s> <s><lb/>Dico factum. </s> <s>Nam Trajecto­<lb/>ctoria de&longs;cripta (eo quod <emph type="italics"/>PH <lb/>+SP<emph.end type="italics"/> in Ellip&longs;i, & <emph type="italics"/>PH-SP<emph.end type="italics"/><lb/>in Hyperbola æquatur axi) <lb/>tran&longs;ibit per punctum <emph type="italics"/>P,<emph.end type="italics"/> & <lb/>(per Lemma &longs;uperius) tanget <lb/>rectam <emph type="italics"/>TR.<emph.end type="italics"/> Et eodem argu­<lb/>mento vel tran&longs;ibit eadem per <lb/>puncta duo <emph type="italics"/>P, p,<emph.end type="italics"/> vel tanget re­<lb/>ctas duas <emph type="italics"/>TR, tr. </s> <s>q.E.F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note36"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig26"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO XIX. PROBLEMA XI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Circa datum umbilicum Trajectoriam Parabolicam de&longs;cribere, quæ <lb/>tran&longs;ibit per puncta data, & rectas po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Sit <emph type="italics"/>S<emph.end type="italics"/> umbilicus, <emph type="italics"/>P<emph.end type="italics"/> punctum & <emph type="italics"/>TR<emph.end type="italics"/> tangens Trajectoriæ de&longs;cri­<lb/>bendæ. </s> <s>Centro <emph type="italics"/>P,<emph.end type="italics"/> intervallo <emph type="italics"/>PS<emph.end type="italics"/> de&longs;cribe cir­<lb/> <arrow.to.target n="fig27"></arrow.to.target><lb/>culum <emph type="italics"/>FG.<emph.end type="italics"/> Ab umbilico ad tangentem demit­<lb/>te perpendicularem <emph type="italics"/>ST,<emph.end type="italics"/> & produc eam ad <emph type="italics"/>V,<emph.end type="italics"/><lb/>ut &longs;it <emph type="italics"/>TV<emph.end type="italics"/> æqualis <emph type="italics"/>ST.<emph.end type="italics"/> Eodem modo de&longs;cri­<lb/>bendus e&longs;t alter circulus <emph type="italics"/>fg,<emph.end type="italics"/> &longs;i datur alterum <lb/>punctum <emph type="italics"/>p<emph.end type="italics"/>; vel inveniendum alterum punctum <lb/><emph type="italics"/>v,<emph.end type="italics"/> &longs;i datur altera tangens <emph type="italics"/>tr<emph.end type="italics"/>; dein ducenda re­<lb/><gap/>a <emph type="italics"/>IF<emph.end type="italics"/> quæ tangat duos circulos <emph type="italics"/>FG, fg<emph.end type="italics"/> &longs;i <lb/>dantur duo puncta <emph type="italics"/>P, p,<emph.end type="italics"/> vel tran&longs;eat per duo <lb/>puncta <emph type="italics"/>V, v,<emph.end type="italics"/> &longs;i dantur duæ tangentes <emph type="italics"/>TR, tr,<emph.end type="italics"/> vel <lb/>tangat circulum <emph type="italics"/>FG<emph.end type="italics"/> & tran&longs;eat per punctum <emph type="italics"/>V,<emph.end type="italics"/><lb/>&longs;i datur punctum <emph type="italics"/>P<emph.end type="italics"/> & tangens <emph type="italics"/>TR.<emph.end type="italics"/> Ad <emph type="italics"/>FI<emph.end type="italics"/> demitte perpendicula­<lb/>rem <emph type="italics"/>SI,<emph.end type="italics"/> eamque bi&longs;eca in <emph type="italics"/>K<emph.end type="italics"/>; & axe <emph type="italics"/>SK,<emph.end type="italics"/> vertice principali <emph type="italics"/>K<emph.end type="italics"/> de­<lb/>&longs;cribatur Parabola. </s> <s>Dico factum. </s> <s>Nam Parabola, ob æquales <lb/><emph type="italics"/>SK<emph.end type="italics"/> & <emph type="italics"/>IK, SP<emph.end type="italics"/> & <emph type="italics"/>FP,<emph.end type="italics"/> tran&longs;ibit per punctum <emph type="italics"/>P<emph.end type="italics"/>; & (per Lem­<lb/>matis XIV. Corol. </s> <s>3.) ob æquales <emph type="italics"/>ST<emph.end type="italics"/> & <emph type="italics"/>TV<emph.end type="italics"/> & angulum rectum <lb/><emph type="italics"/>STR,<emph.end type="italics"/> tanget rectam <emph type="italics"/>TR. q.E.F.<emph.end type="italics"/> <pb pagenum="61"/> <arrow.to.target n="note37"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note37"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig27"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO XX. PROBLEMA XII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Circa datum umbilicum Trajectoriam quamvis &longs;pecie datam de&longs;cribe­<lb/>re, quæ per data puncta tran&longs;ibit & rectas tanget pofitione datas.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Dato umbilico <emph type="italics"/>S,<emph.end type="italics"/> de&longs;cribenda &longs;it Trajectoria <emph type="italics"/>ABC<emph.end type="italics"/> per <lb/>puncta duo <emph type="italics"/>B, C.<emph.end type="italics"/> Quoniam Trajectoria datur &longs;pecie, dabitur ra­<lb/>tio axis principalis ad di&longs;tantiam <lb/> <arrow.to.target n="fig28"></arrow.to.target><lb/>umbilicorum. </s> <s>In ea ratione cape <lb/><emph type="italics"/>KB<emph.end type="italics"/> ad <emph type="italics"/>BS,<emph.end type="italics"/> & <emph type="italics"/>LC<emph.end type="italics"/> ad <emph type="italics"/>CS.<emph.end type="italics"/> Cen­<lb/>tris <emph type="italics"/>B, C,<emph.end type="italics"/> intervallis <emph type="italics"/>BK, CL,<emph.end type="italics"/> de­<lb/>&longs;cribe circulos duos, & ad rectam <lb/><emph type="italics"/>KL,<emph.end type="italics"/> quæ tangat eo&longs;dem in <emph type="italics"/>K<emph.end type="italics"/> & <lb/><emph type="italics"/>L,<emph.end type="italics"/> demitte perpendiculum <emph type="italics"/>SG,<emph.end type="italics"/> idemque &longs;eca in <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>a,<emph.end type="italics"/> ita ut &longs;it <lb/><emph type="italics"/>GA<emph.end type="italics"/> ad <emph type="italics"/>AS<emph.end type="italics"/> & <emph type="italics"/>Ga<emph.end type="italics"/> ad <emph type="italics"/>aS,<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>KB<emph.end type="italics"/> ad <emph type="italics"/>BS,<emph.end type="italics"/> & axe &c. <emph type="italics"/>Aa,<emph.end type="italics"/> verticibus <lb/><emph type="italics"/>A, a,<emph.end type="italics"/> de&longs;cribatur Trajectoria. </s> <s>Dico factum. </s> <s>Sit enim <emph type="italics"/>H<emph.end type="italics"/> umbilicus <lb/>alter Figuræ de&longs;criptæ, & cum &longs;it <emph type="italics"/>GA<emph.end type="italics"/> ad <emph type="italics"/>AS<emph.end type="italics"/> ut <emph type="italics"/>Ga<emph.end type="italics"/> ad <emph type="italics"/>aS,<emph.end type="italics"/> erit di­<lb/>vi&longs;im <emph type="italics"/>Ga-GA<emph.end type="italics"/> &longs;eu <emph type="italics"/>Aa<emph.end type="italics"/> ad <emph type="italics"/>aS-AS<emph.end type="italics"/> &longs;eu <emph type="italics"/>SH<emph.end type="italics"/> in eadem &c. </s> <s>ratione, <lb/>adeoque in ratione quam habet axis principalis Figuræ de&longs;cribendæ <lb/>ad di&longs;tantiam umbilicorum ejus; & propterea Figura de&longs;cripta e&longs;t <lb/>eju&longs;dem &longs;peciei cum de&longs;cribenda. </s> <s>Cumque &longs;int <emph type="italics"/>KB<emph.end type="italics"/> ad <emph type="italics"/>BS<emph.end type="italics"/> & <emph type="italics"/>LC<emph.end type="italics"/><lb/>ad <emph type="italics"/>CS<emph.end type="italics"/> in eadem ratione, tran&longs;ibit hæc Figura per puncta <emph type="italics"/>B, C,<emph.end type="italics"/> ut <lb/>ex Conicis manife&longs;tum e&longs;t. </s> </p> <figure id="fig28"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Dato umbilico <emph type="italics"/>S,<emph.end type="italics"/> de&longs;cribenda &longs;it Trajectoria quæ rectas <lb/>duas <emph type="italics"/>TR, tr<emph.end type="italics"/> alicubi contingat. </s> <s>Ab umbilico in tangentes demitte <lb/>perpendicula <emph type="italics"/>ST, St<emph.end type="italics"/> & produc ea­<lb/> <arrow.to.target n="fig29"></arrow.to.target><lb/>dem ad <emph type="italics"/>V, v,<emph.end type="italics"/> ut &longs;int <emph type="italics"/>TV, tv<emph.end type="italics"/> æ­<lb/>quales <emph type="italics"/>TS, tS.<emph.end type="italics"/> Bi&longs;eca <emph type="italics"/>Vv<emph.end type="italics"/> in <emph type="italics"/>O,<emph.end type="italics"/><lb/>& erige perpendiculum infinitum <lb/><emph type="italics"/>OH,<emph.end type="italics"/> rectamque <emph type="italics"/>VS<emph.end type="italics"/> infinite pro­<lb/>ductam &longs;eca in <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>k<emph.end type="italics"/> ita, ut &longs;it <lb/><emph type="italics"/>VK<emph.end type="italics"/> ad <emph type="italics"/>KS<emph.end type="italics"/> & <emph type="italics"/>Vk<emph.end type="italics"/> ad <emph type="italics"/>kS<emph.end type="italics"/> ut e&longs;t <lb/>Trajectoriæ de&longs;cribendæ axis prin­<lb/>cipalis ad umbilicorum di&longs;tantiam. </s> <s><lb/>Super diametro <emph type="italics"/>Kk<emph.end type="italics"/> de&longs;cribatur <lb/>circulus &longs;ecans <emph type="italics"/>OH<emph.end type="italics"/> in <emph type="italics"/>H<emph.end type="italics"/>; & umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/> axe principali ip&longs;am <lb/><emph type="italics"/>VH<emph.end type="italics"/> æquante, de&longs;cribatur Trajectoria. </s> <s>Dico factum. </s> <s>Nam bi&longs;eca <lb/><emph type="italics"/>Kk<emph.end type="italics"/> in <emph type="italics"/>X,<emph.end type="italics"/> & junge <emph type="italics"/>HX, HS, HV, Hv.<emph.end type="italics"/> Quoniam e&longs;t <emph type="italics"/>VK<emph.end type="italics"/> ad <emph type="italics"/>KS<emph.end type="italics"/><lb/>ut <emph type="italics"/>Vk<emph.end type="italics"/> ad <emph type="italics"/>kS<emph.end type="italics"/>; & compofite ut <emph type="italics"/>VK+Vk<emph.end type="italics"/> ad <emph type="italics"/>KS+kS<emph.end type="italics"/>; divi&longs;imque <pb pagenum="62"/> <arrow.to.target n="note38"></arrow.to.target><lb/>ut <emph type="italics"/>Vk-VK<emph.end type="italics"/> ad <emph type="italics"/>kS-KS,<emph.end type="italics"/> id e&longs;t ut 2 <emph type="italics"/>VX<emph.end type="italics"/> ad 2 <emph type="italics"/>KX<emph.end type="italics"/> & 2 <emph type="italics"/>KX<emph.end type="italics"/> ad <lb/>2 <emph type="italics"/>SX,<emph.end type="italics"/> adeoque ut <emph type="italics"/>VX<emph.end type="italics"/> ad <emph type="italics"/>HX<emph.end type="italics"/> & <emph type="italics"/>HX<emph.end type="italics"/> ad <emph type="italics"/>SX,<emph.end type="italics"/> &longs;imilia erunt tri­<lb/>angula <emph type="italics"/>VXH, HXS,<emph.end type="italics"/> & propterea <emph type="italics"/>VH<emph.end type="italics"/> erit ad <emph type="italics"/>SH<emph.end type="italics"/> ut <emph type="italics"/>VX<emph.end type="italics"/> ad <emph type="italics"/>XH,<emph.end type="italics"/><lb/>adeoque ut <emph type="italics"/>VK<emph.end type="italics"/> ad <emph type="italics"/>KS.<emph.end type="italics"/> Habet igitur Trajectoriæ de&longs;criptæ axis <lb/>principalis <emph type="italics"/>VH<emph.end type="italics"/> eam rationem ad ip&longs;ius umbilicorum di&longs;tantiam <emph type="italics"/>SH,<emph.end type="italics"/><lb/>quam habet Trajectoriæ de&longs;cribendæ axis principalis ad ip&longs;ius um­<lb/>bilicorum di&longs;tantiam, & propterea eju&longs;dem e&longs;t &longs;peciei. </s> <s>In&longs;uper cum <lb/><emph type="italics"/>VH, vH<emph.end type="italics"/> æquentur axi principali, & <emph type="italics"/>VS, vS<emph.end type="italics"/> a rectis <emph type="italics"/>TR, tr<emph.end type="italics"/><lb/>perpendiculariter bi&longs;ecentur, liquet, ex Lemmate XV, rectas illas <lb/>Trajectoriam de&longs;criptam tangere. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note38"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig29"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 3. Dato umbilico <emph type="italics"/>S<emph.end type="italics"/> de&longs;cribenda &longs;it Trajectoria quæ rect­<lb/>am <emph type="italics"/>TR<emph.end type="italics"/> tanget in puncto dato <emph type="italics"/>R.<emph.end type="italics"/> In rectam <emph type="italics"/>TR<emph.end type="italics"/> demitte perpen­<lb/>dicularem <emph type="italics"/>ST,<emph.end type="italics"/> & produc eandem ad <emph type="italics"/>V,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>TV<emph.end type="italics"/> æqualis <emph type="italics"/>ST.<emph.end type="italics"/> Junge <lb/><emph type="italics"/>VR,<emph.end type="italics"/> & rectam <emph type="italics"/>VS<emph.end type="italics"/> infinite productam &longs;eca in <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>k,<emph.end type="italics"/> ita ut &longs;it <lb/><emph type="italics"/>VK<emph.end type="italics"/> ad <emph type="italics"/>SK<emph.end type="italics"/> & <emph type="italics"/>Vk<emph.end type="italics"/> ad <emph type="italics"/>Sk<emph.end type="italics"/> ut Ellip&longs;eos de&longs;cribendæ axis principalis <lb/>ad di&longs;tantiam umbilicorum; circuloque &longs;uper diametro <emph type="italics"/>Kk<emph.end type="italics"/> de­<lb/>&longs;cripto, &longs;ecetur producta recta <emph type="italics"/>VR<emph.end type="italics"/> in <emph type="italics"/>H,<emph.end type="italics"/> & umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/> axe <lb/>principali rectam <emph type="italics"/>VH<emph.end type="italics"/> æquante, de&longs;cribatur Trajectoria. </s> <s>Dico fa­<lb/>ctum. </s> <s>Namque <emph type="italics"/>VH<emph.end type="italics"/> e&longs;&longs;e ad <lb/> <arrow.to.target n="fig30"></arrow.to.target><lb/><emph type="italics"/>SH<emph.end type="italics"/> ut <emph type="italics"/>VK<emph.end type="italics"/> ad <emph type="italics"/>SK,<emph.end type="italics"/> atque adeo <lb/>ut axis principalis Trajectoriæ <lb/>de&longs;cribendæ ad di&longs;tantiam um­<lb/>bilicorum ejus, patet ex demon­<lb/>&longs;lratis in Ca&longs;u &longs;ecundo, & prop­<lb/>terea Trajectoriam de&longs;criptam <lb/>eju&longs;dem e&longs;&longs;e &longs;peciei cum de&longs;cri­<lb/>benda; rectam vero <emph type="italics"/>TR<emph.end type="italics"/> qua an­<lb/>gulus <emph type="italics"/>VRS<emph.end type="italics"/> bi&longs;ecatur, tangere Trajectoriam in puncto <emph type="italics"/>R,<emph.end type="italics"/> patet ex <lb/>Conicis. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s> </p> <figure id="fig30"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 4. Circa umbilicum <emph type="italics"/>S<emph.end type="italics"/> de&longs;cribenda jam &longs;it Trajectoria <emph type="italics"/>APB,<emph.end type="italics"/><lb/>quæ tangat rectam <emph type="italics"/>TR,<emph.end type="italics"/> tran&longs;eatque per punctum quodvis <emph type="italics"/>P<emph.end type="italics"/> extra <lb/>tangentem datum, quæque &longs;imilis &longs;it Figuræ <emph type="italics"/>apb,<emph.end type="italics"/> axe principali <lb/><emph type="italics"/>ab<emph.end type="italics"/> & umbilicis <emph type="italics"/>s, h<emph.end type="italics"/> de&longs;criptæ. </s> <s>In tangentem <emph type="italics"/>TR<emph.end type="italics"/> demitte per­<lb/>pendiculum <emph type="italics"/>ST,<emph.end type="italics"/> & produc idem ad <emph type="italics"/>V,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>TV<emph.end type="italics"/> æqualis <emph type="italics"/>ST.<emph.end type="italics"/> An­<lb/>gulis autem <emph type="italics"/>VSP, SVP<emph.end type="italics"/> fac angulos <emph type="italics"/>hsq, shq<emph.end type="italics"/> æquales; cen­<lb/>troque <emph type="italics"/>q<emph.end type="italics"/> & intervallo quod &longs;it ad <emph type="italics"/>ab<emph.end type="italics"/> ut <emph type="italics"/>SP<emph.end type="italics"/> ad <emph type="italics"/>VS<emph.end type="italics"/> de&longs;cribe circu­<lb/>lum &longs;ecantem Figuram <emph type="italics"/>apb<emph.end type="italics"/> in <emph type="italics"/>p.<emph.end type="italics"/> Junge <emph type="italics"/>sp<emph.end type="italics"/> & age <emph type="italics"/>SH<emph.end type="italics"/> quæ &longs;it ad <lb/><emph type="italics"/>sh<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>SP<emph.end type="italics"/> ad <emph type="italics"/>sp,<emph.end type="italics"/> quæque angulum <emph type="italics"/>PSH<emph.end type="italics"/> angulo <emph type="italics"/>psh<emph.end type="italics"/> & angulum <lb/><emph type="italics"/>VSH<emph.end type="italics"/> angulo <emph type="italics"/>psq<emph.end type="italics"/> æquales con&longs;tituat. </s> <s>Denique umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/><lb/>& axe principali <emph type="italics"/>AB<emph.end type="italics"/> di&longs;tantiam <emph type="italics"/>VH<emph.end type="italics"/> æquante, de&longs;cribatur &longs;ectio <lb/>Conica. </s> <s>Dico factum. </s> <s>Nam &longs;i agatur <emph type="italics"/>sv<emph.end type="italics"/> quæ &longs;it ad <emph type="italics"/>sp<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>sh<emph.end type="italics"/> <pb pagenum="63"/>ad <emph type="italics"/>sq,<emph.end type="italics"/> quæque con&longs;tituat angulum <emph type="italics"/>vsp<emph.end type="italics"/> angulo <emph type="italics"/>hsq<emph.end type="italics"/> & angulum <lb/> <arrow.to.target n="note39"></arrow.to.target><lb/><emph type="italics"/>vsh<emph.end type="italics"/> angulo <emph type="italics"/>psq<emph.end type="italics"/> æquales, triangula <emph type="italics"/>svh, spq<emph.end type="italics"/> erunt &longs;imilia, & prop­<lb/>terea <emph type="italics"/>vh<emph.end type="italics"/> erit ad <emph type="italics"/>pq<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>sh<emph.end type="italics"/> ad <emph type="italics"/>sq,<emph.end type="italics"/> id e&longs;t (ob &longs;imilia triangula <lb/> <arrow.to.target n="fig31"></arrow.to.target><lb/><emph type="italics"/>VSP, hsq<emph.end type="italics"/>) ut e&longs;t <emph type="italics"/>VS<emph.end type="italics"/> ad <emph type="italics"/>SP<emph.end type="italics"/> &longs;eu <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/><expan abbr="pq.">pque</expan><emph.end type="italics"/> Æquantur ergo <lb/><emph type="italics"/>vh<emph.end type="italics"/> & <emph type="italics"/>ab.<emph.end type="italics"/> Porro ob &longs;imilia triangula <emph type="italics"/>VSH. vsh,<emph.end type="italics"/> e&longs;t <emph type="italics"/>VH<emph.end type="italics"/> ad <lb/><emph type="italics"/>SH<emph.end type="italics"/> ut <emph type="italics"/>vh<emph.end type="italics"/> ad <emph type="italics"/>sh,<emph.end type="italics"/> id e&longs;t, axis Conicæ &longs;ectionis jam de&longs;criptæ ad <lb/>illius umbilicorum intervallum, ut axis <emph type="italics"/>ab<emph.end type="italics"/> ad umbilicorum inter­<lb/>vallum <emph type="italics"/>sh<emph.end type="italics"/>; & propterea Figura jam de&longs;eripta &longs;imilis e&longs;t Figuræ <lb/><emph type="italics"/>apb.<emph.end type="italics"/> Tran&longs;it autem hæc Figura per punctum <emph type="italics"/>P,<emph.end type="italics"/> eo quod trian­<lb/>gulum <emph type="italics"/>PSH<emph.end type="italics"/> &longs;imile &longs;it triangulo <emph type="italics"/>psh<emph.end type="italics"/>; & quia <emph type="italics"/>VH<emph.end type="italics"/> æquatur ip&longs;ius <lb/>axi & <emph type="italics"/>VS<emph.end type="italics"/> bi&longs;ecatur perpendiculariter a recta <emph type="italics"/>TR,<emph.end type="italics"/> tangit eadem <lb/>rectam <emph type="italics"/>TR. q.E.F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note39"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig31"></figure> <p type="main"> <s><emph type="center"/>LEMMA XVI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>A datis tribus punctis ad quartum non datum inflectere tres rectas <lb/>quarum differentiæ vel dantur vel nullæ &longs;unt.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Sunto puncta illa data <emph type="italics"/>A, B, C<emph.end type="italics"/> & punctum quartum <emph type="italics"/>Z,<emph.end type="italics"/><lb/>quod invenire oportet; Ob datam differentiam linearum <emph type="italics"/>AZ, BZ,<emph.end type="italics"/><lb/>locabitur punctum <emph type="italics"/>Z<emph.end type="italics"/> in Hyperbola cujus umbilici &longs;unt <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>B,<emph.end type="italics"/> & <lb/>principalis axis differentia illa data. </s> <s>Sit axis ille <emph type="italics"/>MN.<emph.end type="italics"/> Cape <emph type="italics"/>PM.<emph.end type="italics"/> <pb pagenum="64"/> <arrow.to.target n="note40"></arrow.to.target><lb/>ad <emph type="italics"/>MA<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>MN<emph.end type="italics"/> ad <emph type="italics"/>AB,<emph.end type="italics"/> & erecta <emph type="italics"/>PR<emph.end type="italics"/> perpendiculari ad <emph type="italics"/>AB,<emph.end type="italics"/><lb/>demi&longs;&longs;aque <emph type="italics"/>ZR<emph.end type="italics"/> perpendiculari ad <emph type="italics"/>PR<emph.end type="italics"/>; erit, ex natura hujus Hy­<lb/>perbolæ, <emph type="italics"/>ZR<emph.end type="italics"/> ad <emph type="italics"/>AZ<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>MN<emph.end type="italics"/> ad <emph type="italics"/>AB.<emph.end type="italics"/> Simili di&longs;cur&longs;u punctum <lb/><emph type="italics"/>Z<emph.end type="italics"/> locabitur in alia Hyperbola, cujus umbilici &longs;unt <emph type="italics"/>A, C<emph.end type="italics"/> & princi­<lb/>palis axis differentia inter <emph type="italics"/>AZ<emph.end type="italics"/> & <emph type="italics"/>CZ,<emph.end type="italics"/> ducique pote&longs;t <emph type="italics"/>QS<emph.end type="italics"/> ip&longs;i <emph type="italics"/>AC<emph.end type="italics"/><lb/>perpendicularis, ad quam &longs;i ab Hyperbolæ hujus puncto quovis <emph type="italics"/>Z<emph.end type="italics"/><lb/>demittatur normalis <emph type="italics"/>ZS,<emph.end type="italics"/> hæc fuerit ad <emph type="italics"/>AZ<emph.end type="italics"/> ut e&longs;t differentia inter <lb/><emph type="italics"/>AZ<emph.end type="italics"/> & <emph type="italics"/>CZ<emph.end type="italics"/> ad <emph type="italics"/>AC.<emph.end type="italics"/> Dantur ergo rationes ip&longs;arum <emph type="italics"/>ZR<emph.end type="italics"/> & <emph type="italics"/>ZS<emph.end type="italics"/><lb/>ad <emph type="italics"/>AZ,<emph.end type="italics"/> & idcirco datur earun­<lb/> <arrow.to.target n="fig32"></arrow.to.target><lb/>dem <emph type="italics"/>ZR<emph.end type="italics"/> & <emph type="italics"/>ZS<emph.end type="italics"/> ratio ad invicem; <lb/>ideoque &longs;i rectæ <emph type="italics"/>RP, SQ<emph.end type="italics"/> concur­<lb/>rant in <emph type="italics"/>T,<emph.end type="italics"/> & agatur <emph type="italics"/>TZ,<emph.end type="italics"/> figura <lb/><emph type="italics"/>TRZS,<emph.end type="italics"/> dabitur &longs;pecie, & recta <lb/><emph type="italics"/>TZ<emph.end type="italics"/> in qua punctum <emph type="italics"/>Z<emph.end type="italics"/> alicubi lo­<lb/>catur, dabitur po&longs;itione. </s> <s>Eadem <lb/>methodo per Hyperbolam ter­<lb/>tiam, cujus umbilici &longs;unt <emph type="italics"/>B<emph.end type="italics"/> & <emph type="italics"/>C<emph.end type="italics"/><lb/>& axis principalis differentia re­<lb/>ctarum <emph type="italics"/>BZ, CZ,<emph.end type="italics"/> inveniri pote&longs;t <lb/>alia recta in qua <expan abbr="pũctum">punctum</expan> <emph type="italics"/>Z<emph.end type="italics"/> locatur. </s> <s><lb/>Habitis autem duobus Locis recti­<lb/>lineis, habetur punctum quæ&longs;itum <emph type="italics"/>Z<emph.end type="italics"/> in eorum inter&longs;ectione. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note40"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig32"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Si duæ ex tribus lineis, puta <emph type="italics"/>AZ<emph.end type="italics"/> & <emph type="italics"/>BZ<emph.end type="italics"/> æquantur, pun­<lb/>ctum <emph type="italics"/>Z<emph.end type="italics"/> locabitur in perpendiculo bi&longs;ecante di&longs;tantiam <emph type="italics"/>AB,<emph.end type="italics"/> & lo­<lb/>cus alius rectilineus invenietur ut &longs;upra. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 3. Si omnes tres æquantur, locabitur punctum <emph type="italics"/>Z<emph.end type="italics"/> in centro <lb/>Circuli per puncta <emph type="italics"/>A, B, C<emph.end type="italics"/> tran&longs;euntis. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="main"> <s>Solvitur etiam hoc Lemma problematicum per Librum Tactio­<lb/>num <emph type="italics"/>Apollonii<emph.end type="italics"/> a <emph type="italics"/>Vieta<emph.end type="italics"/> re&longs;titutum. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXI. PROBLEMA XIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam circa datum umbilicum de&longs;cribere, quæ tran&longs;ibit per <lb/>puncta data & rectas po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Detur umbilicus <emph type="italics"/>S,<emph.end type="italics"/> punctum <emph type="italics"/>P,<emph.end type="italics"/> & tangens <emph type="italics"/>TR,<emph.end type="italics"/> & invenien­<lb/>dus &longs;it umbilicus alter <emph type="italics"/>H.<emph.end type="italics"/> Ad tangentem demitte perpendiculum <lb/><emph type="italics"/>ST,<emph.end type="italics"/> & produc idem ad <emph type="italics"/>Y,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>TY<emph.end type="italics"/> æqualis <emph type="italics"/>ST,<emph.end type="italics"/> & erit <emph type="italics"/>YH<emph.end type="italics"/> æ­<lb/>qualis axi principali. </s> <s>Junge <emph type="italics"/>SP, HP,<emph.end type="italics"/> & erit <emph type="italics"/>SP<emph.end type="italics"/> differentia inter <lb/><emph type="italics"/>HP<emph.end type="italics"/> & axem principalem. </s> <s>Hoc modo &longs;i dentur plures tangen- <pb pagenum="65"/>tes <emph type="italics"/>TR,<emph.end type="italics"/> vel plura puncta <emph type="italics"/>P,<emph.end type="italics"/> devenietur &longs;emper ad lineas totidem <lb/> <arrow.to.target n="note41"></arrow.to.target><lb/><emph type="italics"/>YH,<emph.end type="italics"/> vel <emph type="italics"/>PH,<emph.end type="italics"/> a dictis punctis <emph type="italics"/>Y<emph.end type="italics"/> vel <lb/> <arrow.to.target n="fig33"></arrow.to.target><lb/><emph type="italics"/>P<emph.end type="italics"/> ad umbilicum <emph type="italics"/>H<emph.end type="italics"/> ductas, quæ vel <lb/>æquantur axibus, vel datis longitu­<lb/>dinibus <emph type="italics"/>SP<emph.end type="italics"/> differunt ab ii&longs;dem, at­<lb/>que adeo quæ vel æquantur &longs;ibi invi­<lb/>cem, vel datas habent differentias; & <lb/>inde, per Lemma &longs;uperius, datur umbi­<lb/>licus ille alter <emph type="italics"/>H.<emph.end type="italics"/> Habitis autem um­<lb/>bilicis una cum axis longitudine (quæ <lb/>vel e&longs;t <emph type="italics"/>YH<emph.end type="italics"/>; vel, &longs;i Trajectoria Ellip&longs;is e&longs;t, <emph type="italics"/>PH+SP<emph.end type="italics"/>; &longs;in Hy­<lb/>perbola, <emph type="italics"/>PH-SP<emph.end type="italics"/>) habetur Trajectoria. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note41"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig33"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Ca&longs;us ubi dantur tria puncta &longs;ic &longs;olvitur expeditius. </s> <s>Dentur <lb/>puncta <emph type="italics"/>B, C, D.<emph.end type="italics"/> Junctas <emph type="italics"/>BC, CD<emph.end type="italics"/> produc ad <emph type="italics"/>E, F,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>EB<emph.end type="italics"/> ad <lb/><emph type="italics"/>EC<emph.end type="italics"/> ut <emph type="italics"/>SB<emph.end type="italics"/> ad <emph type="italics"/>SC,<emph.end type="italics"/> & <emph type="italics"/>FC<emph.end type="italics"/> ad <emph type="italics"/>FD<emph.end type="italics"/> ut <emph type="italics"/>SC<emph.end type="italics"/> ad <emph type="italics"/>SD.<emph.end type="italics"/> Ad <emph type="italics"/>EF<emph.end type="italics"/> ductam <lb/>& productam demitte normales <emph type="italics"/>SG, BH,<emph.end type="italics"/> inque <emph type="italics"/>GS<emph.end type="italics"/> infinite <lb/>producta cape <emph type="italics"/>GA<emph.end type="italics"/> ad <emph type="italics"/>AS<emph.end type="italics"/> & <emph type="italics"/>Ga<emph.end type="italics"/> ad <emph type="italics"/>aS<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>HB<emph.end type="italics"/> ad <emph type="italics"/>BS<emph.end type="italics"/>; & erit <lb/><emph type="italics"/>A<emph.end type="italics"/> vertex, & <emph type="italics"/>Aa<emph.end type="italics"/> axis principalis Trajectoriæ: quæ, perinde ut <emph type="italics"/>GA<emph.end type="italics"/><lb/>major, æqualis, vel minor fuerit quam <emph type="italics"/>AS,<emph.end type="italics"/> erit Ellip&longs;is, Parabola <lb/>vel Hyperbola; pun­<lb/> <arrow.to.target n="fig34"></arrow.to.target><lb/>cto <emph type="italics"/>a<emph.end type="italics"/> in primo ca&longs;u <lb/>cadente ad eandem <lb/>partem lineæ <emph type="italics"/>GF<emph.end type="italics"/><lb/>cum puncto <emph type="italics"/>A<emph.end type="italics"/>; in <lb/>&longs;ecundo ca&longs;u abeunte <lb/>in infinitum; in tertio <lb/>cadente ad contrari­<lb/>am partem lineæ <emph type="italics"/>GF.<emph.end type="italics"/><lb/>Nam &longs;i demittantur <lb/>ad <emph type="italics"/>GF<emph.end type="italics"/> perpendicula <lb/><emph type="italics"/>CI, DK<emph.end type="italics"/>; erit <emph type="italics"/>IC<emph.end type="italics"/> ad <emph type="italics"/>HB<emph.end type="italics"/> ut <emph type="italics"/>EC<emph.end type="italics"/> ad <emph type="italics"/>EB,<emph.end type="italics"/> hoc e&longs;t, ut <emph type="italics"/>SC<emph.end type="italics"/> ad <emph type="italics"/>SB<emph.end type="italics"/>; & vi­<lb/>ci&longs;&longs;im <emph type="italics"/>IC<emph.end type="italics"/> ad <emph type="italics"/>SC<emph.end type="italics"/> ut <emph type="italics"/>HB<emph.end type="italics"/> ad <emph type="italics"/>SB<emph.end type="italics"/> &longs;ive ut <emph type="italics"/>GA<emph.end type="italics"/> ad <emph type="italics"/>SA.<emph.end type="italics"/> Et &longs;imili argumento <lb/>probabitur e&longs;&longs;e <emph type="italics"/>KD<emph.end type="italics"/> ad <emph type="italics"/>SD<emph.end type="italics"/> in eadem ratione. </s> <s>Jacent ergo puncta <emph type="italics"/>B, <lb/>C, D<emph.end type="italics"/> in Coni&longs;ectione circa umbilicum <emph type="italics"/>S<emph.end type="italics"/> ita de&longs;cripta, ut rectæ omnes <lb/>ab umbilico <emph type="italics"/>S<emph.end type="italics"/> ad &longs;ingula Sectionis puncta ductæ, &longs;int ad perpendicula <lb/>a punctis ii&longs;dem ad rectam <emph type="italics"/>GF<emph.end type="italics"/> demi&longs;&longs;a in data illa ratione. </s> </p> <figure id="fig34"></figure> <p type="main"> <s>Methodo haud multum di&longs;&longs;imili hujus problematis &longs;olutionem <lb/>tradit Clari&longs;&longs;imus Geometra <emph type="italics"/>de la Hire,<emph.end type="italics"/> Conicorum &longs;uorum Lib. </s> <s><lb/>VIII. Prop. </s> <s>XXV. <pb pagenum="66"/> <arrow.to.target n="note42"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note42"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>SECTIO V.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Inventio Orbium ubi umbilicus neuter datur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>LEMMA XVII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si a datæ Conicæ Sectionis puncto quovis<emph.end type="italics"/> P, <emph type="italics"/>ad Trapezii alicujus<emph.end type="italics"/><lb/>ABDC, <emph type="italics"/>in Conica illa &longs;ectione in&longs;cripti, latera quatuor infinite <lb/>producta<emph.end type="italics"/> AB, CD, AC, DB, <emph type="italics"/>totidem rectæ<emph.end type="italics"/> PQ, PR, PS, PT <lb/><emph type="italics"/>in datis angulis ducantur, &longs;ingulæ ad &longs;ingula: rectangulum duc­<lb/>tarum ad oppo&longs;ita duo latera<emph.end type="italics"/> PQXPR, <emph type="italics"/>erit ad rectangulum duc­<lb/>tarum ad alia duo latera oppo&longs;ita<emph.end type="italics"/> PSXPT <emph type="italics"/>in data ratione.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Ponamus primo lineas ad <lb/> <arrow.to.target n="fig35"></arrow.to.target><lb/>oppo&longs;ita latera ductas parallelas e&longs;­<lb/>&longs;e alterutri reliquorum laterum, <lb/>puta <emph type="italics"/>PQ<emph.end type="italics"/> & <emph type="italics"/>PR<emph.end type="italics"/> lateri <emph type="italics"/>AC,<emph.end type="italics"/> & <emph type="italics"/>PS<emph.end type="italics"/><lb/>ac <emph type="italics"/>PT<emph.end type="italics"/> lateri <emph type="italics"/>AB.<emph.end type="italics"/> Sintque in&longs;uper <lb/>latera duo ex oppo&longs;itis, puta <emph type="italics"/>AC<emph.end type="italics"/><lb/>& <emph type="italics"/>BD,<emph.end type="italics"/> &longs;ibi invicem paralle­<lb/>la. </s> <s>Et recta quæ bi&longs;ecat paralle­<lb/>la illa latera erit una ex diametris <lb/>Conicæ &longs;ectionis, & bi&longs;ecabit eti­<lb/>am <emph type="italics"/><expan abbr="Rq.">Rque</expan><emph.end type="italics"/> Sit <emph type="italics"/>O<emph.end type="italics"/> punctum in quo <lb/><emph type="italics"/>RQ<emph.end type="italics"/> bi&longs;ecatur, & erit <emph type="italics"/>PO<emph.end type="italics"/> ordinatim applicata ad diametrum illam. </s> <s><lb/>Produc <emph type="italics"/>PO<emph.end type="italics"/> ad <emph type="italics"/>K<emph.end type="italics"/> ut &longs;it <emph type="italics"/>OK<emph.end type="italics"/> æqualis <emph type="italics"/>PO,<emph.end type="italics"/> & erit <emph type="italics"/>OK<emph.end type="italics"/> ordinatim <lb/>applicata ad contrarias partes diametri. </s> <s>Cum igitur puncta <emph type="italics"/>A, B, <lb/>P<emph.end type="italics"/> & <emph type="italics"/>K<emph.end type="italics"/> &longs;int ad Conicam &longs;ectionem, & <emph type="italics"/>PK<emph.end type="italics"/> &longs;ecet <emph type="italics"/>AB<emph.end type="italics"/> in dato an­<lb/>gulo, erit (per Prop.17 & 18 Lib. </s> <s>III Conicorum <emph type="italics"/>Apollonii<emph.end type="italics"/>) rectangu­<lb/>lum <emph type="italics"/>PQK<emph.end type="italics"/> ad rectangulum <emph type="italics"/>AQB<emph.end type="italics"/> in data ratione. </s> <s>Sed <emph type="italics"/>QK<emph.end type="italics"/> & <emph type="italics"/>PR<emph.end type="italics"/><lb/>æquales &longs;unt, utpote æqualium <emph type="italics"/>OK, OP,<emph.end type="italics"/> & <emph type="italics"/>OQ, OR<emph.end type="italics"/> differentiæ, <lb/>& inde etiam rectangula <emph type="italics"/>PQK<emph.end type="italics"/> & <emph type="italics"/>PQXPR<emph.end type="italics"/> æqualia &longs;unt; at­<lb/>que adeo rectangulum <emph type="italics"/>PQXPR<emph.end type="italics"/> e&longs;t ad rectangulum <emph type="italics"/>AQB,<emph.end type="italics"/> hoc <lb/>e&longs;t ad rectangulum <emph type="italics"/>PSXPT<emph.end type="italics"/> in data ratione. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <pb pagenum="67"/> <figure id="fig35"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Ponamus jam Trapezii latera oppo&longs;ita <emph type="italics"/>AC<emph.end type="italics"/> & <emph type="italics"/>BD<emph.end type="italics"/> non <lb/> <arrow.to.target n="note43"></arrow.to.target><lb/>e&longs;&longs;e parallela. </s> <s>Age <emph type="italics"/>Bd<emph.end type="italics"/> parallelam <emph type="italics"/>AC<emph.end type="italics"/> & occurrentem tum rectæ <lb/><emph type="italics"/>ST<emph.end type="italics"/> in <emph type="italics"/>t,<emph.end type="italics"/> tum Conicæ &longs;ectioni in <emph type="italics"/>d.<emph.end type="italics"/> Junge <emph type="italics"/>Cd<emph.end type="italics"/> &longs;ecantem <emph type="italics"/>PQ<emph.end type="italics"/> in <emph type="italics"/>r,<emph.end type="italics"/><lb/>& ip&longs;i <emph type="italics"/>PQ<emph.end type="italics"/> parallelam age <emph type="italics"/>DM<emph.end type="italics"/><lb/> <arrow.to.target n="fig36"></arrow.to.target><lb/>&longs;ecantem <emph type="italics"/>Cd<emph.end type="italics"/> in <emph type="italics"/>M<emph.end type="italics"/> & <emph type="italics"/>AB<emph.end type="italics"/> in <emph type="italics"/>N.<emph.end type="italics"/><lb/>Jam ob &longs;imilia triangula <emph type="italics"/>BTt, <lb/>DBN<emph.end type="italics"/>; e&longs;t <emph type="italics"/>Bt<emph.end type="italics"/> &longs;eu <emph type="italics"/>PQ<emph.end type="italics"/> ad <emph type="italics"/>Tt<emph.end type="italics"/> ut <lb/><emph type="italics"/>DN<emph.end type="italics"/> ad <emph type="italics"/>NB.<emph.end type="italics"/> Sic & <emph type="italics"/>Rr<emph.end type="italics"/> e&longs;t ad <lb/><emph type="italics"/>AQ<emph.end type="italics"/> &longs;eu <emph type="italics"/>PS<emph.end type="italics"/> ut <emph type="italics"/>DM<emph.end type="italics"/> ad <emph type="italics"/>AN.<emph.end type="italics"/><lb/>Ergo, ducendo antecedentes in <lb/>antecedentes & con&longs;equentes in <lb/>con&longs;equentes, ut rectangulum <emph type="italics"/>PQ<emph.end type="italics"/><lb/>in <emph type="italics"/>Rr<emph.end type="italics"/> e&longs;t ad rectangulum <emph type="italics"/>PS<emph.end type="italics"/> in <lb/><emph type="italics"/>Tt,<emph.end type="italics"/> ita rectangulum <emph type="italics"/>NDM<emph.end type="italics"/> e&longs;t <lb/>ad rectangulum <emph type="italics"/>ANB,<emph.end type="italics"/> & (per Ca&longs;.1) ita rectangulum <emph type="italics"/>PQ<emph.end type="italics"/> in <emph type="italics"/>Pr<emph.end type="italics"/> e&longs;t <lb/>ad rectangulum <emph type="italics"/>PS<emph.end type="italics"/> in <emph type="italics"/>Pt,<emph.end type="italics"/> ac divi&longs;im ita rectangulum <emph type="italics"/>PQXPR<emph.end type="italics"/><lb/>e&longs;t ad rectangulum <emph type="italics"/>PSXPT. q.E.D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note43"></margin.target>LIBIR <lb/>PRIMUS.</s> </p> <figure id="fig36"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 3. Ponamus denique lineas <lb/> <arrow.to.target n="fig37"></arrow.to.target><lb/>quatuor <emph type="italics"/>PQ, PR, PS, PT<emph.end type="italics"/> non <lb/>e&longs;&longs;e parallelas lateribus <emph type="italics"/>AC, AB,<emph.end type="italics"/><lb/>&longs;ed ad ea utcunque inclinatas. </s> <s>Ea­<lb/>rum vice age <emph type="italics"/>Pq, Pr<emph.end type="italics"/> parallelas <lb/>ip&longs;i <emph type="italics"/>AC<emph.end type="italics"/>; & <emph type="italics"/>Ps, Pt<emph.end type="italics"/> parallelas <lb/>ip&longs;i <emph type="italics"/>AB<emph.end type="italics"/>; & propter datos angu­<lb/>los triangulorum <emph type="italics"/>PQq, PRr, <lb/>PSs, PTt,<emph.end type="italics"/> dabuntur rationes <lb/><emph type="italics"/>PQ<emph.end type="italics"/> ad <emph type="italics"/>Pq, PR<emph.end type="italics"/> ad <emph type="italics"/>Pr, PS<emph.end type="italics"/><lb/>ad <emph type="italics"/>Ps,<emph.end type="italics"/> & <emph type="italics"/>PT<emph.end type="italics"/> ad <emph type="italics"/>Pt<emph.end type="italics"/>; atque adeo rationes compo&longs;itæ <emph type="italics"/>PQXPR<emph.end type="italics"/><lb/>ad <emph type="italics"/>PqXPr,<emph.end type="italics"/> & <emph type="italics"/>PSXPT<emph.end type="italics"/> ad <emph type="italics"/>PsXPt.<emph.end type="italics"/> Sed, per &longs;uperius de­<lb/>mon&longs;trata, ratio <emph type="italics"/>PqXPr<emph.end type="italics"/> ad <emph type="italics"/>PsXPt<emph.end type="italics"/> data e&longs;t: Ergo & ratio <lb/><emph type="italics"/>PQXPR<emph.end type="italics"/> ad <emph type="italics"/>PSXPT. q.E.D.<emph.end type="italics"/></s> </p> <figure id="fig37"></figure> <p type="main"> <s><emph type="center"/>LEMMA XVIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, &longs;i rectangulum ductarum ad oppo&longs;ita duo latera Tra­<lb/>pezii<emph.end type="italics"/> PQXPR <emph type="italics"/>&longs;it adrectangulum ductarum ad reliqua duo late­<lb/>ra<emph.end type="italics"/> PSXPT <emph type="italics"/>in data ratione; punctum<emph.end type="italics"/> P, <emph type="italics"/>a quo lineæ ducuntur, <lb/>tanget Conicam &longs;ectionem circa Trapezium de&longs;criptam.<emph.end type="italics"/></s> </p> <pb pagenum="68"/> <p type="main"> <s>Per puncta <emph type="italics"/>A, B, C, D<emph.end type="italics"/> & aliquod infinitorum punctorum <emph type="italics"/>P,<emph.end type="italics"/> pu­<lb/> <arrow.to.target n="note44"></arrow.to.target><lb/>ta <emph type="italics"/>p,<emph.end type="italics"/> concipe Conicam &longs;ectionem de&longs;cribi: dico punctum <emph type="italics"/>P<emph.end type="italics"/> hanc <lb/>&longs;emper tangere. </s> <s>Si negas, <lb/> <arrow.to.target n="fig38"></arrow.to.target><lb/>junge <emph type="italics"/>AP<emph.end type="italics"/> &longs;ecantem hanc <lb/>Conicam &longs;ectionem alibi <lb/>quam in <emph type="italics"/>P,<emph.end type="italics"/> &longs;i fieri pote&longs;t, <lb/>puta in <emph type="italics"/>b.<emph.end type="italics"/> Ergo &longs;i ab his <lb/>punctis <emph type="italics"/>p<emph.end type="italics"/> & <emph type="italics"/>b<emph.end type="italics"/> ducantur in <lb/>datis angulis ad latera Tra­<lb/>pezii rectæ <emph type="italics"/>pq, pr, ps, pt<emph.end type="italics"/><lb/>& <emph type="italics"/>bk, br, b&longs;, bd<emph.end type="italics"/>; erit <lb/>ut <emph type="italics"/>bkXb<emph.end type="italics"/>r ad <emph type="italics"/>b&longs;Xbd<emph.end type="italics"/> ita <lb/>(per Lem. </s> <s>XVII) <emph type="italics"/>pqXpr<emph.end type="italics"/><lb/>ad <emph type="italics"/>psXpt,<emph.end type="italics"/> & ita (per <lb/>Hypoth.) <emph type="italics"/>PQXPR<emph.end type="italics"/> ad <lb/><emph type="italics"/>PSXPT.<emph.end type="italics"/> E&longs;t & prop­<lb/>ter &longs;imilitudinem Trapeziorum <emph type="italics"/>bkA&longs;, PQAS,<emph.end type="italics"/> ut <emph type="italics"/>bk<emph.end type="italics"/> ad <emph type="italics"/>b&longs;<emph.end type="italics"/> ita <lb/><emph type="italics"/>PQ<emph.end type="italics"/> ad <emph type="italics"/>PS.<emph.end type="italics"/> Quare, applicando terminos prioris proportionis ad <lb/>terminos corre&longs;pondentes hujus, erit <emph type="italics"/>b<emph.end type="italics"/>r ad <emph type="italics"/>bd<emph.end type="italics"/> ut <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>PT.<emph.end type="italics"/> Er­<lb/>go Trapezia æquiangula <emph type="italics"/>Dr bd, DRPT<emph.end type="italics"/> &longs;imilia &longs;unt, & eorum <lb/>diagonales <emph type="italics"/>Db, DP<emph.end type="italics"/> propterea coincidunt. </s> <s>Incidit itaque <emph type="italics"/>b<emph.end type="italics"/> in <lb/>inter&longs;ectionem rectarum <emph type="italics"/>AP, DP<emph.end type="italics"/> adeoque coincidit cum puncto <lb/><emph type="italics"/>P.<emph.end type="italics"/> Quare punctum <emph type="italics"/>P,<emph.end type="italics"/> ubicunque &longs;umatur, incidit in a&longs;&longs;ignatam <lb/>Conicam &longs;ectionem. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note44"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig38"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Hinc &longs;i rectæ tres <emph type="italics"/>PQ, PR, PS<emph.end type="italics"/> a puncto communi <emph type="italics"/>P<emph.end type="italics"/><lb/>ad alias totidem po&longs;itione datas rectas <emph type="italics"/>AB, CD, AC,<emph.end type="italics"/> &longs;ingulæ ad <lb/>&longs;ingulas, in datis angulis ducantur, &longs;itque rectangulum &longs;ub duabus <lb/>ductis <emph type="italics"/>PQXPR<emph.end type="italics"/> ad quadratum tertiæ <emph type="italics"/>PS quad.<emph.end type="italics"/> in data ratione: <lb/>punctum <emph type="italics"/>P,<emph.end type="italics"/> a quibus rectæ ducuntur, locabitur in &longs;ectione Conica <lb/>quæ tangit lineas <emph type="italics"/>AB, CD<emph.end type="italics"/> in <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>C<emph.end type="italics"/>; & contra. </s> <s>Nam coeat linea <lb/><emph type="italics"/>BD<emph.end type="italics"/> cum linea <emph type="italics"/>AC<emph.end type="italics"/> manente po&longs;itione trium <emph type="italics"/>AB, CD, AC<emph.end type="italics"/>; de­<lb/>in coeat etiam linea <emph type="italics"/>PT<emph.end type="italics"/> cum linea <emph type="italics"/>PS:<emph.end type="italics"/> & rectangulum <emph type="italics"/>PSXPT<emph.end type="italics"/><lb/>evadet <emph type="italics"/>PS quad.<emph.end type="italics"/> rectæque <emph type="italics"/>AB, CD<emph.end type="italics"/> quæ curvam in punctis <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>B, <lb/>C<emph.end type="italics"/> & <emph type="italics"/>D<emph.end type="italics"/> &longs;ecabant, jam Curvam in punctis illis coeuntibus non am­<lb/>plius &longs;ecare po&longs;&longs;unt &longs;ed tantum tangent. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Nomen Conicæ &longs;ectionis in hoc Lemmate late &longs;umitur, ita ut <lb/>&longs;ectio tam Rectilinea per verticem Coni tran&longs;iens, quam Circularis <lb/>ba&longs;i parallela includatur. </s> <s>Nam &longs;i punctum <emph type="italics"/>p<emph.end type="italics"/> incidit in rectam, qua <lb/>quævis ex punctis quatuor <emph type="italics"/>A, B, C, D<emph.end type="italics"/> junguntur, Conica &longs;ectio <pb pagenum="69"/>vertetur in geminas Rectas, quarum una e&longs;t recta illa in quam pun­<lb/> <arrow.to.target n="note45"></arrow.to.target><lb/>ctum <emph type="italics"/>p<emph.end type="italics"/> incidit, & altera e&longs;t recta qua alia duo ex punctis quatuor jun­<lb/>guntur. </s> <s>Si Trapezii anguli duo oppo&longs;iti &longs;imul &longs;umpti æquentur <lb/>duobus rectis, & lineæ quatuor <emph type="italics"/>PQ, PR, PS, PT<emph.end type="italics"/> ducantur ad <lb/>latera ejus vel perpendiculariter vel in angulis quibu&longs;vis æqualibus, <lb/>&longs;itque rectangulum &longs;ub duabus ductis <emph type="italics"/>PQXPR<emph.end type="italics"/> æquale rectangu­<lb/>lo &longs;ub duabus aliis <emph type="italics"/>PSXPT,<emph.end type="italics"/> Sectio conica evadet Circulus. </s> <s>Idem <lb/>fiet &longs;i lineæ quatuor ducantur in angulis quibu&longs;vis & rectangulum <lb/>&longs;ub duabus ductis <emph type="italics"/>PQXPR<emph.end type="italics"/> &longs;it ad rectangulum &longs;ub aliis duabus <lb/><emph type="italics"/>PSXPT<emph.end type="italics"/> ut rectangulum &longs;ub &longs;inubus angulorum <emph type="italics"/>S, T,<emph.end type="italics"/> in quibus <lb/>duæ ultimæ <emph type="italics"/>PS, PT<emph.end type="italics"/> ducuntur, ad rectangulum &longs;ub &longs;inubus angu­<lb/>lorum <emph type="italics"/>Q, R,<emph.end type="italics"/> in quibus duæ primæ <emph type="italics"/>PQ, PR<emph.end type="italics"/> ducuntur. </s> <s>Cæteris <lb/>in ca&longs;ibus Locus puncti <emph type="italics"/>P<emph.end type="italics"/> erit aliqua trium figurarum quæ vulgo <lb/>nominantur Sectiones Conicæ. </s> <s>Vice autem Trapezii <emph type="italics"/>ABCD<emph.end type="italics"/> &longs;ub­<lb/>&longs;titui pote&longs;t Quadrilaterum cujus latera duo oppo&longs;ita &longs;e mutuo in­<lb/>&longs;tar diagonalium decu&longs;&longs;ant. </s> <s>Sed & e punctis quatuor <emph type="italics"/>A, B, C, D<emph.end type="italics"/><lb/>po&longs;&longs;unt unum vel duo abire ad infinitum, eoque pacto latera fi­<lb/>guræ quæ ad puncta illa convergunt, evadere parallela: quo in <lb/>ca&longs;u Sectio Conica tran&longs;ibit per cætera puncta, & in plagas paralle­<lb/>larum abibit in infinitum. </s> </p> <p type="margin"> <s><margin.target id="note45"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>LEMMA XIX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Invenire <expan abbr="punctũ">punctum</expan><emph.end type="italics"/> P, <emph type="italics"/>a quo &longs;irectæ<emph.end type="italics"/><lb/> <arrow.to.target n="fig39"></arrow.to.target><lb/><emph type="italics"/>quatuor<emph.end type="italics"/> PQ, PR, PS, PT, <lb/><emph type="italics"/>ad alias totidem po&longs;itione da <lb/>tas rectas<emph.end type="italics"/> AB, CD, AC, BD, <lb/><emph type="italics"/>&longs;ingulæ ad &longs;ingulas in datis <lb/>angulis ducantur, <expan abbr="rectangulũ">rectangulum</expan> <lb/>&longs;ub duabus ductis,<emph.end type="italics"/> PQXPR, <lb/><emph type="italics"/>&longs;it ad rectangulum &longs;ub aliis <lb/>duabus,<emph.end type="italics"/> PSXPT, <emph type="italics"/>in data ra­<lb/>tione.<emph.end type="italics"/></s> </p> <figure id="fig39"></figure> <p type="main"> <s>Lineæ <emph type="italics"/>AB, CD,<emph.end type="italics"/> ad quas rectæ duæ <emph type="italics"/>PQ, PR,<emph.end type="italics"/> unum rectan­<lb/>gulorum continentes ducuntur, conveniant cum aliis duabus po&longs;i­<lb/>tione datis lineis in punctis <emph type="italics"/>A, B, C, D.<emph.end type="italics"/> Ab eorum aliquo <emph type="italics"/>A<emph.end type="italics"/> age <lb/>rectam quamlibet <emph type="italics"/>AH,<emph.end type="italics"/> in qua velis punctum <emph type="italics"/>P<emph.end type="italics"/> reperiri. </s> <s>Secet ea <lb/>lineas oppo&longs;itas <emph type="italics"/>BD, CD,<emph.end type="italics"/> nimirum <emph type="italics"/>BD<emph.end type="italics"/> in <emph type="italics"/>H<emph.end type="italics"/> & <emph type="italics"/>CD<emph.end type="italics"/> in <emph type="italics"/>I,<emph.end type="italics"/> & ob <lb/>datos omnes angulos figuræ, dabuntur rationes <emph type="italics"/>PQ<emph.end type="italics"/> ad <emph type="italics"/>PA<emph.end type="italics"/> & <emph type="italics"/>PA<emph.end type="italics"/> <pb pagenum="70"/> <arrow.to.target n="note46"></arrow.to.target><lb/>ad <emph type="italics"/>PS,<emph.end type="italics"/> adeoque ratio <emph type="italics"/>PQ<emph.end type="italics"/> ad <lb/> <arrow.to.target n="fig40"></arrow.to.target><lb/><emph type="italics"/>PS.<emph.end type="italics"/> Auferendo hanca data ra­<lb/>tione <emph type="italics"/>PQXPR<emph.end type="italics"/> ad <emph type="italics"/>PSXPT,<emph.end type="italics"/><lb/>dabitur ratio <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>PT,<emph.end type="italics"/> & <lb/>addendo datas rationes <emph type="italics"/>PI<emph.end type="italics"/> ad <lb/><emph type="italics"/>PR,<emph.end type="italics"/> & <emph type="italics"/>PT<emph.end type="italics"/> ad <emph type="italics"/>PH<emph.end type="italics"/> dabitur <lb/>ratio <emph type="italics"/>PI<emph.end type="italics"/> ad <emph type="italics"/>PH<emph.end type="italics"/> atque adeo <lb/>punctum <emph type="italics"/>P. q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note46"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig40"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc etiam ad Loci <lb/>punctorum infinitorum <emph type="italics"/>P<emph.end type="italics"/> pun­<lb/>ctum quodvis <emph type="italics"/>D<emph.end type="italics"/> tangens duci <lb/>pote&longs;t. </s> <s>Nam chorda <emph type="italics"/>PD<emph.end type="italics"/> ubi <lb/>puncta <emph type="italics"/>P<emph.end type="italics"/> ac <emph type="italics"/>D<emph.end type="italics"/> conveniunt, hoc <lb/>e&longs;t, ubi <emph type="italics"/>AH<emph.end type="italics"/> ducitur per punctum <emph type="italics"/>D,<emph.end type="italics"/> tangens evadit. </s> <s>Quo in ca&longs;u, <lb/>ultima ratio evane&longs;centium <emph type="italics"/>IP<emph.end type="italics"/> & <emph type="italics"/>PH<emph.end type="italics"/> invenietur ut &longs;upra. </s> <s>Ip&longs;i <lb/>igitur <emph type="italics"/>AD<emph.end type="italics"/> due parallelam <emph type="italics"/>CF,<emph.end type="italics"/> occurrentem <emph type="italics"/>BD<emph.end type="italics"/> in <emph type="italics"/>F,<emph.end type="italics"/> & in ea ul­<lb/>tima ratione &longs;ectam in <emph type="italics"/>E,<emph.end type="italics"/> & <emph type="italics"/>DE<emph.end type="italics"/> tangens erit, propterea quod <emph type="italics"/>CF<emph.end type="italics"/><lb/>& evane&longs;cens <emph type="italics"/>IH<emph.end type="italics"/> parallelæ &longs;unt, & in <emph type="italics"/>E<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> fimiliter &longs;ectæ. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Hinc etiam Locus punctorum omnium <emph type="italics"/>P<emph.end type="italics"/> definiri pote&longs;t. </s> <s><lb/>Per quodvis punctorum <emph type="italics"/>A, B, C, D,<emph.end type="italics"/> puta <emph type="italics"/>A,<emph.end type="italics"/> duc Loci tangentem <lb/><emph type="italics"/>AE<emph.end type="italics"/> & per aliud quodvis punctum <emph type="italics"/>B<emph.end type="italics"/> duc tangenti parallelam <emph type="italics"/>BF<emph.end type="italics"/><lb/>occurrentem Loco in <emph type="italics"/>F.<emph.end type="italics"/> Invenie­<lb/> <arrow.to.target n="fig41"></arrow.to.target><lb/>tur autem punctum <emph type="italics"/>F<emph.end type="italics"/> per Lem. </s> <s>XIX. </s> <s><lb/>Bi&longs;eca <emph type="italics"/>BF<emph.end type="italics"/> in <emph type="italics"/>G,<emph.end type="italics"/> & acta indefinita <lb/><emph type="italics"/>AG<emph.end type="italics"/> erit po&longs;itio diametri ad quam <lb/><emph type="italics"/>BG<emph.end type="italics"/> & <emph type="italics"/>FG<emph.end type="italics"/> ordinatim applicantur. </s> <s><lb/>Hæc <emph type="italics"/>AG<emph.end type="italics"/> occurrat Loco in <emph type="italics"/>H,<emph.end type="italics"/> & <lb/>erit <emph type="italics"/>AH<emph.end type="italics"/> diameter &longs;ive latus tran&longs;­<lb/>ver&longs;um, ad quod latus rectum erit <lb/>ut <emph type="italics"/><expan abbr="BGq.">BGque</expan><emph.end type="italics"/> ad <emph type="italics"/>AGH.<emph.end type="italics"/> Si <emph type="italics"/>AG<emph.end type="italics"/> nullibi <lb/>occurrit Loco, linea <emph type="italics"/>AH<emph.end type="italics"/> exi&longs;tente <lb/>infinita, Locus erit Parabola & la­<lb/>rum rectum ejus ad diametrum <emph type="italics"/>AG<emph.end type="italics"/><lb/>pertinens erit (<emph type="italics"/>BGq./AG<emph.end type="italics"/>) Sin ea alicubi occurrit, Locus Hyperbola erit <lb/>ubi puncta <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>H<emph.end type="italics"/> &longs;ita &longs;unt ad ea&longs;dem partes ip&longs;ius <emph type="italics"/>G:<emph.end type="italics"/> & Ellip&longs;is, <lb/>ubi <emph type="italics"/>G<emph.end type="italics"/> intermedium e&longs;t, ni&longs;i forte angulus <emph type="italics"/>AGB<emph.end type="italics"/> rectus &longs;it & in&longs;uper <lb/><emph type="italics"/>BG quad.<emph.end type="italics"/> æquale rectangulo <emph type="italics"/>AGH,<emph.end type="italics"/> quo in ca&longs;u Circulus habebitur. </s> </p> <figure id="fig41"></figure> <p type="main"> <s>Atque ita Problematis Veterum de quatuor lineis ab <emph type="italics"/>Euclide<emph.end type="italics"/> incæp­<lb/>ti & ab <emph type="italics"/>Apollonio<emph.end type="italics"/> continuati non calculus, &longs;ed compo&longs;itio Geometri­<lb/>ca, qualem Veteres quærebant, in hoc Corollario exhibetur. <pb pagenum="71"/> <arrow.to.target n="note47"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note47"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>LEMMA XX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si Parallelogr ammum quodvis<emph.end type="italics"/> ASPQ <emph type="italics"/>angulis duobus oppo&longs;itis<emph.end type="italics"/> A <emph type="italics"/>&<emph.end type="italics"/><lb/>P <emph type="italics"/>tangit &longs;ectionem quamvis Conicam in punctis<emph.end type="italics"/> A <emph type="italics"/>&<emph.end type="italics"/> P; <emph type="italics"/>&, lateri­<lb/>bus unius angulorum illorum infinite productis<emph.end type="italics"/> AQ, AS, <emph type="italics"/>occurrit <lb/>eidem &longs;ectioni Conicæ in<emph.end type="italics"/> B <emph type="italics"/>&<emph.end type="italics"/> C; <emph type="italics"/>a punctis autem occur&longs;uum<emph.end type="italics"/> B <emph type="italics"/>&<emph.end type="italics"/><lb/>C <emph type="italics"/>ad quintum quodvis &longs;ectionis Conicæ punctum<emph.end type="italics"/> D <emph type="italics"/>agantur rec­<lb/>tæ duæ<emph.end type="italics"/> BD, CD <emph type="italics"/>occurrentes alteris duobus infinite productis pa­<lb/>rallelogrammi lateribus<emph.end type="italics"/> PS, PQ <emph type="italics"/>in<emph.end type="italics"/> T <emph type="italics"/>&<emph.end type="italics"/> R: <emph type="italics"/>erunt &longs;emper ab&longs;ci&longs;&longs;æ <lb/>laterum partes<emph.end type="italics"/> PR <emph type="italics"/>&<emph.end type="italics"/> PT <emph type="italics"/>adinvicem in data ratione. </s> <s>Et contra, &longs;i <lb/>partes illæ ab&longs;ci&longs;&longs;æ &longs;unt ad invicem in data ratione, punctum<emph.end type="italics"/> D <emph type="italics"/>tan­<lb/>get Sectionem Conicam per puncta quatuor<emph.end type="italics"/> A, B, C, P <emph type="italics"/>tran&longs;euntem.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Jungantur <emph type="italics"/>BP, CP<emph.end type="italics"/> & a puncto <emph type="italics"/>D<emph.end type="italics"/> agantur rectæ duæ <lb/><emph type="italics"/>DG, DE,<emph.end type="italics"/> quarum prior <lb/> <arrow.to.target n="fig42"></arrow.to.target><lb/><emph type="italics"/>DG<emph.end type="italics"/> ip&longs;i <emph type="italics"/>AB<emph.end type="italics"/> parallela &longs;it & <lb/>occurrat <emph type="italics"/>PB, PQ, CA<emph.end type="italics"/> in <lb/><emph type="italics"/>H, I, G<emph.end type="italics"/>; altera <emph type="italics"/>DE<emph.end type="italics"/> paral­<lb/>lela &longs;it ipfi <emph type="italics"/>AC<emph.end type="italics"/> & occurrat <lb/><emph type="italics"/>PC, PS, AB<emph.end type="italics"/> in <emph type="italics"/>F, K, E:<emph.end type="italics"/><lb/>& erit (per Lemma XVII.) re­<lb/>ctangulum <emph type="italics"/>DEXDF<emph.end type="italics"/> ad re­<lb/>ctangulum <emph type="italics"/>DGXDH<emph.end type="italics"/> in ra­<lb/>tione data. </s> <s>Sed e&longs;t <emph type="italics"/>PQ<emph.end type="italics"/> ad <lb/><emph type="italics"/>DE<emph.end type="italics"/> (&longs;eu <emph type="italics"/>IQ<emph.end type="italics"/>) ut <emph type="italics"/>PB<emph.end type="italics"/> ad <emph type="italics"/>HB,<emph.end type="italics"/><lb/>adeoque ut <emph type="italics"/>PT<emph.end type="italics"/> ad <emph type="italics"/>DH<emph.end type="italics"/>; & <lb/>vici&longs;&longs;im <emph type="italics"/>PQ<emph.end type="italics"/> ad <emph type="italics"/>PT<emph.end type="italics"/> ut <emph type="italics"/>DE<emph.end type="italics"/> ad <emph type="italics"/>DH.<emph.end type="italics"/> E&longs;t & <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>DF<emph.end type="italics"/> ut <emph type="italics"/>RC<emph.end type="italics"/><lb/>ad <emph type="italics"/>DC,<emph.end type="italics"/> adeoque ut (<emph type="italics"/>IG<emph.end type="italics"/> vel) <emph type="italics"/>PS<emph.end type="italics"/> ad <emph type="italics"/>DG,<emph.end type="italics"/> & vici&longs;&longs;im <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>PS<emph.end type="italics"/><lb/>ut <emph type="italics"/>DF<emph.end type="italics"/> ad <emph type="italics"/>DG<emph.end type="italics"/>; & conjunctis rationibus fit rectangulum <emph type="italics"/>PQXPR<emph.end type="italics"/><lb/>ad rectangulum <emph type="italics"/>PSXPT<emph.end type="italics"/> ut rectangulum <emph type="italics"/>DEXDF<emph.end type="italics"/> ad rectan­<lb/>gulum <emph type="italics"/>DGXDH,<emph.end type="italics"/> atque adeo in data ratione. </s> <s>Sed dantur <emph type="italics"/>PQ<emph.end type="italics"/><lb/>& <emph type="italics"/>PS<emph.end type="italics"/> & propterea ratio <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>PT<emph.end type="italics"/> datur. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <figure id="fig42"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Quod &longs;i <emph type="italics"/>PR<emph.end type="italics"/> & <emph type="italics"/>PT<emph.end type="italics"/> ponantur in data ratione ad invi­<lb/>cem, tum &longs;imili ratiocinio regrediendo, &longs;equetur e&longs;&longs;e rectangulum <lb/><emph type="italics"/>DEXDF<emph.end type="italics"/> ad rectangulum <emph type="italics"/>DGXDH<emph.end type="italics"/> in ratione data, adeoque <lb/>punctum <emph type="italics"/>D<emph.end type="italics"/> (per Lemma XVIII.) contingere Conicam &longs;ectionem <lb/>tran&longs;euntem per puncta <emph type="italics"/>A, B, C, P. q.E.D.<emph.end type="italics"/> <pb pagenum="72"/> <arrow.to.target n="note48"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note48"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i agatur <emph type="italics"/>BC<emph.end type="italics"/> &longs;ecans <emph type="italics"/>PQ<emph.end type="italics"/> in <emph type="italics"/>r,<emph.end type="italics"/> & in <emph type="italics"/>PT<emph.end type="italics"/> capiatur <lb/><emph type="italics"/>Pt<emph.end type="italics"/> in ratione ad <emph type="italics"/>Pr<emph.end type="italics"/> quam habet <emph type="italics"/>PT<emph.end type="italics"/> ad <emph type="italics"/>PR:<emph.end type="italics"/> erit <emph type="italics"/>Bt<emph.end type="italics"/> tangens <lb/>Conicæ &longs;ectionis ad punctum <emph type="italics"/>B.<emph.end type="italics"/> Nam concipe punctum <emph type="italics"/>D<emph.end type="italics"/> coire <lb/>cum puncto <emph type="italics"/>B<emph.end type="italics"/> ita ut, chorda <emph type="italics"/>BD<emph.end type="italics"/> evane&longs;cente, <emph type="italics"/>BT<emph.end type="italics"/> tangens eva­<lb/>dat; & <emph type="italics"/>CD<emph.end type="italics"/> ac <emph type="italics"/>BT<emph.end type="italics"/> coincident cum <emph type="italics"/>CB<emph.end type="italics"/> & <emph type="italics"/>Bt.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et vice ver&longs;a &longs;i <lb/> <arrow.to.target n="fig43"></arrow.to.target><lb/><emph type="italics"/>Bt<emph.end type="italics"/> fit tangens, & ad quod­<lb/>vis Conicæ &longs;ectionis punc­<lb/>tum <emph type="italics"/>D<emph.end type="italics"/> conveniant <emph type="italics"/>BD, <lb/>CD<emph.end type="italics"/>; erit <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>PT<emph.end type="italics"/> ut <lb/>ut <emph type="italics"/>Pr<emph.end type="italics"/> ad <emph type="italics"/>Pt.<emph.end type="italics"/> Et contra, <lb/>&longs;i &longs;it <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>PT<emph.end type="italics"/> ut <emph type="italics"/>Pr<emph.end type="italics"/> ad <lb/><emph type="italics"/>Pt:<emph.end type="italics"/> convenient <emph type="italics"/>BD, CD<emph.end type="italics"/><lb/>ad Conicæ Sectionis punc­<lb/>um aliquod <emph type="italics"/>D.<emph.end type="italics"/></s> </p> <figure id="fig43"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Conica &longs;ectio <lb/>non &longs;ecat Conicam &longs;ectio­<lb/>nem in punctis pluribus quam quatuor. </s> <s>Nam, &longs;i fieri pote&longs;t, tran&longs;­<lb/>eant duæ Conicæ &longs;ectiones per quinque puncta <emph type="italics"/>A, B, C, P, O<emph.end type="italics"/>; ea&longs;­<lb/>que &longs;ecet recta <emph type="italics"/>BD<emph.end type="italics"/> in punctis <emph type="italics"/>D, d,<emph.end type="italics"/> & ip&longs;am <emph type="italics"/>PQ<emph.end type="italics"/> &longs;ecet recta <emph type="italics"/>Cd<emph.end type="italics"/><lb/>in r. </s> <s>Ergo <emph type="italics"/>PR<emph.end type="italics"/> e&longs;t ad <emph type="italics"/>PT<emph.end type="italics"/> ut <emph type="italics"/>P<emph.end type="italics"/>r ad <emph type="italics"/>PT<emph.end type="italics"/>; unde <emph type="italics"/>PR<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/>r &longs;ibi <lb/>invicem æquantur, contra Hypothe&longs;in. </s> </p> <p type="main"> <s><emph type="center"/>LEMMA XXI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si rectæ duæ mobiles & infinitæ<emph.end type="italics"/> BM, CM <emph type="italics"/>per data puncta<emph.end type="italics"/> B, C, <emph type="italics"/>ceu <lb/>polos ductæ, concur&longs;u &longs;uo<emph.end type="italics"/> M <emph type="italics"/>de&longs;cribant tertiam po&longs;itione da­<lb/>tam rectam<emph.end type="italics"/> MN; <emph type="italics"/>& aliæ duæ infinitæ rectæ<emph.end type="italics"/> BD, CD <emph type="italics"/>cum <lb/>prioribus duabus ad puncta illa data<emph.end type="italics"/> B, C <emph type="italics"/>datos angulos<emph.end type="italics"/><lb/>MBD, MCD <emph type="italics"/>efficientes ducantur; dico quod hæ duæ<emph.end type="italics"/> BD, <lb/>CD <emph type="italics"/>concur&longs;u &longs;uo<emph.end type="italics"/> D <emph type="italics"/>de&longs;cribent &longs;ectionem Conicam per puncta<emph.end type="italics"/><lb/>B, C <emph type="italics"/>tran&longs;euntem. </s> <s>Et vice ver&longs;a, &longs;i rectæ<emph.end type="italics"/> BD, CD <emph type="italics"/>concur&longs;u <lb/>&longs;uo<emph.end type="italics"/> D <emph type="italics"/>de&longs;cribant Sectionem Conicam per data puncta<emph.end type="italics"/> B, C, A <lb/><emph type="italics"/>tran&longs;euntem, & &longs;it angulus<emph.end type="italics"/> DBM <emph type="italics"/>&longs;emper æqualis angulo dato<emph.end type="italics"/><lb/>ABC, <emph type="italics"/>angulu&longs;que<emph.end type="italics"/> DCM <emph type="italics"/>&longs;emper æqualis angulo dato<emph.end type="italics"/> ACB: <lb/><emph type="italics"/>punctum<emph.end type="italics"/> M <emph type="italics"/>continget rectam po&longs;itione datam.<emph.end type="italics"/></s> </p> <pb pagenum="73"/> <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/>PRIMUS.</s> </p> <p type="main"> <s>Nam in recta <emph type="italics"/>MN<emph.end type="italics"/> detur punctum <emph type="italics"/>N,<emph.end type="italics"/> & ubi punctum mobile <lb/><emph type="italics"/>M<emph.end type="italics"/> incidit in immotum <emph type="italics"/>N,<emph.end type="italics"/> incidat punctum mobile <emph type="italics"/>D<emph.end type="italics"/> in immo­<lb/>tum <emph type="italics"/>P,<emph.end type="italics"/> Junge <emph type="italics"/>CN, BN,<emph.end type="italics"/><lb/> <arrow.to.target n="fig44"></arrow.to.target><lb/><emph type="italics"/>CP, BP,<emph.end type="italics"/> & a puncto <lb/><emph type="italics"/>P<emph.end type="italics"/> age rectas <emph type="italics"/>PT, PR<emph.end type="italics"/><lb/>occurrentes ip&longs;is <emph type="italics"/>BD, <lb/>CD<emph.end type="italics"/> in <emph type="italics"/>T<emph.end type="italics"/> & <emph type="italics"/>R,<emph.end type="italics"/> & fa­<lb/>cientes angulum <emph type="italics"/>BPT<emph.end type="italics"/><lb/>æqualem angulo dato <lb/><emph type="italics"/>BNM,<emph.end type="italics"/> & angulum <lb/><emph type="italics"/>CPR<emph.end type="italics"/> æqualem angu­<lb/>gulo dato <emph type="italics"/>CNM.<emph.end type="italics"/> Cum <lb/>ergo (ex Hypothe&longs;i) <lb/>æquales &longs;int anguli <lb/><emph type="italics"/>MBD, NBP,<emph.end type="italics"/> ut & <lb/>anguli <emph type="italics"/>MCD, NCP<emph.end type="italics"/>; <lb/>aufer communes <emph type="italics"/>NBD<emph.end type="italics"/><lb/>& <emph type="italics"/>NCD,<emph.end type="italics"/> & re&longs;tabunt <lb/>æquales <emph type="italics"/>NBM<emph.end type="italics"/> & <emph type="italics"/>PBT, <lb/>NCM<emph.end type="italics"/> & <emph type="italics"/>PCR:<emph.end type="italics"/> adeoque triangula <emph type="italics"/>NBM, PBT<emph.end type="italics"/> &longs;imilia &longs;unt, ut <lb/>& triangula <emph type="italics"/>NCM, PCR.<emph.end type="italics"/> Quare <emph type="italics"/>PT<emph.end type="italics"/> e&longs;t ad <emph type="italics"/>NM<emph.end type="italics"/> ut <emph type="italics"/>PB<emph.end type="italics"/> ad <lb/><emph type="italics"/>NB,<emph.end type="italics"/> & <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>NM<emph.end type="italics"/> ut <emph type="italics"/>PC<emph.end type="italics"/> ad <emph type="italics"/>NC.<emph.end type="italics"/> Sunt autem puncta <emph type="italics"/>B, C, N, P<emph.end type="italics"/><lb/>immobilia. </s> <s>Ergo <emph type="italics"/>PT<emph.end type="italics"/> & <emph type="italics"/>PR<emph.end type="italics"/> datam habent rationem ad <emph type="italics"/>NM,<emph.end type="italics"/> pro­<lb/>indeque datam rationem inter &longs;e; atque adeo, per Lemma xx, <lb/>punctum <emph type="italics"/>D<emph.end type="italics"/> (perpetuus rectarum mobilium <emph type="italics"/>BT<emph.end type="italics"/> & <emph type="italics"/>CR<emph.end type="italics"/> concur&longs;us) <lb/>contingit &longs;ectionem Conicam, per puncta <emph type="italics"/>B, C, P<emph.end type="italics"/> tran&longs;euntem. <lb/><emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <figure id="fig44"></figure> <p type="main"> <s>Et contra, &longs;i punctum mobile <emph type="italics"/>D<emph.end type="italics"/> contingat &longs;ectionem Conicam <lb/>tran&longs;euntem per data puncta <emph type="italics"/>B, C, A,<emph.end type="italics"/> & &longs;it angulus <emph type="italics"/>DBM<emph.end type="italics"/> &longs;emper <lb/>æqualis angulo dato <emph type="italics"/>ABC,<emph.end type="italics"/> & angulus <emph type="italics"/>DCM<emph.end type="italics"/> &longs;emper æqualis angu­<lb/>lo dato <emph type="italics"/>ACB,<emph.end type="italics"/> & ubi punctum <emph type="italics"/>D<emph.end type="italics"/> incidit &longs;ucce&longs;&longs;ive in duo quævis &longs;e­<lb/>ctionis puncta immobilia <emph type="italics"/>p, P,<emph.end type="italics"/> punctum mobile <emph type="italics"/>M<emph.end type="italics"/> incidat &longs;ucce&longs;&longs;ive <lb/>in puncta duo immobilia <emph type="italics"/>n, N:<emph.end type="italics"/> per eadem <emph type="italics"/>n, N<emph.end type="italics"/> agatur Recta <emph type="italics"/>n N,<emph.end type="italics"/><lb/>& hæc erit Locus perpetuus puncti illius mobilis <emph type="italics"/>M.<emph.end type="italics"/> Nam, &longs;i fieri <lb/>pote&longs;t, ver&longs;etur punctum <emph type="italics"/>M<emph.end type="italics"/> in linea aliqua Curva. </s> <s>Tanget ergo <lb/>punctum <emph type="italics"/>D<emph.end type="italics"/> &longs;ectionem Conicam per puncta quinque <emph type="italics"/>B, CA, p, P,<emph.end type="italics"/><lb/>tran&longs;euntem, ubi punctum <emph type="italics"/>M<emph.end type="italics"/> perpetuo tangit lineam Curvam. </s> <s>Sed <lb/>& ex jam demon&longs;tratis tanget etiam punctum <emph type="italics"/>D<emph.end type="italics"/> &longs;ectionem Coni­<lb/>cam per eadem quinque puncta <emph type="italics"/>B, C, A, p, P<emph.end type="italics"/> tran&longs;euntem, ubi pun-</s> </p> <pb pagenum="74"/> <p type="main"> <s> <arrow.to.target n="note50"></arrow.to.target><lb/>ctum <emph type="italics"/>M<emph.end type="italics"/> perpetuo tangit lineam Rectam. </s> <s>Ergo duæ &longs;ectiones Co­<lb/>nicæ tran&longs;ibunt per eadem quinque puncta, contra Corol. </s> <s>3. Lem. </s> <s><lb/>xx. </s> <s>Igitur punctum <emph type="italics"/>M<emph.end type="italics"/> ver&longs;ari in linea Curva ab&longs;urdum e&longs;t. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note50"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXII. PROBLEMA. XIV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam per data quinque puncta de&longs;cribere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Dentur puncta quinque <emph type="italics"/>A, B, C, P, D.<emph.end type="italics"/> Ab eorum aliquo <emph type="italics"/>A<emph.end type="italics"/> ad <lb/>alia duo quævis <emph type="italics"/>B, C,<emph.end type="italics"/> quæ poli nominentur, age rectas <emph type="italics"/>AB, AC,<emph.end type="italics"/><lb/> <arrow.to.target n="fig45"></arrow.to.target><lb/>hi&longs;que parallelas <emph type="italics"/>TPS, PRQ<emph.end type="italics"/> per punctum quartum <emph type="italics"/>P.<emph.end type="italics"/> De­<lb/>inde a polis duobus <emph type="italics"/>B, C<emph.end type="italics"/> age per punctum quintum <emph type="italics"/>D<emph.end type="italics"/> infini­<lb/>tas duas <emph type="italics"/>BDT, CRD,<emph.end type="italics"/> novi&longs;&longs;ime ductis <emph type="italics"/>TPS, PRQ<emph.end type="italics"/> (prio­<lb/>rem priori & po&longs;teriorem po&longs;teriori) occurrentes in <emph type="italics"/>T<emph.end type="italics"/> & <emph type="italics"/>R.<emph.end type="italics"/> De­<lb/>nique de rectis <emph type="italics"/>PT, PR,<emph.end type="italics"/> acta recta <emph type="italics"/>tr<emph.end type="italics"/> ip&longs;i <emph type="italics"/>TR<emph.end type="italics"/> parallela, ab­<lb/>&longs;cinde qua&longs;vis <emph type="italics"/>Pt, Pr<emph.end type="italics"/> ip&longs;is <emph type="italics"/>PT, PR<emph.end type="italics"/> proportionales; & &longs;i per <lb/>earum terminos <emph type="italics"/>t, r<emph.end type="italics"/> & polos <emph type="italics"/>B, C<emph.end type="italics"/> actæ <emph type="italics"/>Bt, Cr<emph.end type="italics"/> concurrant in <lb/><emph type="italics"/>d,<emph.end type="italics"/> locabitur punctum illud <emph type="italics"/>d<emph.end type="italics"/> in Trajectoria quæ&longs;ita. </s> <s>Nam punc­<lb/>tum illud <emph type="italics"/>d<emph.end type="italics"/> (per Lemma xx) ver&longs;atur in Conica Sectione per <lb/>puncta quatuor <emph type="italics"/>A, B, C, P<emph.end type="italics"/> tran&longs;eunte; &, lineis <emph type="italics"/>Rr, Tt<emph.end type="italics"/> evane­<lb/>&longs;centibus, coit punctum <emph type="italics"/>d<emph.end type="italics"/> cum puncto <emph type="italics"/>D.<emph.end type="italics"/> Tran&longs;it ergo &longs;ectio Co­<lb/>nica per puncta quinque <emph type="italics"/>A, B, C, P, D. q.E.D.<emph.end type="italics"/></s> </p> <pb pagenum="75"/> <figure id="fig45"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/><lb/> <arrow.to.target n="note51"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note51"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s>E punctis datis junge tria quævis <emph type="italics"/>A, B, C<emph.end type="italics"/>; &, circum duo eorum <lb/><emph type="italics"/>B, C<emph.end type="italics"/> ceu polos, rotando angulos magnitudine datos <emph type="italics"/>ABC, <lb/>ACB,<emph.end type="italics"/> applicentur cru­<lb/> <arrow.to.target n="fig46"></arrow.to.target><lb/>ra <emph type="italics"/>BA, CA<emph.end type="italics"/> primo ad <lb/>punctum <emph type="italics"/>D,<emph.end type="italics"/> deinde <lb/>ad punctum <emph type="italics"/>P,<emph.end type="italics"/> & no­<lb/>tentur puncta <emph type="italics"/>M, N<emph.end type="italics"/> in <lb/>quibus altera crura <lb/><emph type="italics"/>BL, CL<emph.end type="italics"/> ca&longs;u utroque <lb/>&longs;e decu&longs;&longs;ant. </s> <s>Agatur <lb/>recta infinita <emph type="italics"/>MN,<emph.end type="italics"/> & <lb/>rotentur anguli illi mo­<lb/>biles circum polos &longs;uos <lb/><emph type="italics"/>B, C,<emph.end type="italics"/> ea lege ut cru­<lb/>rum <emph type="italics"/>BL, CL<emph.end type="italics"/> vel <lb/><emph type="italics"/>BM, CM<emph.end type="italics"/> inter&longs;ectio <lb/>quæ jam &longs;it <emph type="italics"/>m<emph.end type="italics"/> incidat <lb/>&longs;emper in rectam illam <lb/>infinitam <emph type="italics"/>MN<emph.end type="italics"/> & cru­<lb/>rum <emph type="italics"/>BA, CA,<emph.end type="italics"/> vel <emph type="italics"/>BD, CD<emph.end type="italics"/> inter&longs;ectio, quæ jam &longs;it <emph type="italics"/>d,<emph.end type="italics"/> Trajecto­<lb/>riam quæ&longs;itam <emph type="italics"/>PAD dB<emph.end type="italics"/> delineabit. </s> <s>Nam punctum <emph type="italics"/>d,<emph.end type="italics"/> per Lem. </s> <s><lb/>XXI, continget &longs;ectionem Conicam per puncta <emph type="italics"/>B, C<emph.end type="italics"/> tran&longs;euntem; & <lb/>ubi punctum <emph type="italics"/>m<emph.end type="italics"/> accedit ad puncta <emph type="italics"/>L, M, N,<emph.end type="italics"/> punctum <emph type="italics"/>d<emph.end type="italics"/> (per con­<lb/>&longs;tructionem) accedet ad puncta <emph type="italics"/>A, D, P.<emph.end type="italics"/> De&longs;cribetur itaque &longs;ec­<lb/>tio Conica tran&longs;iens per puncta quinque <emph type="italics"/>A, B, C, P, D. q.E.F.<emph.end type="italics"/></s> </p> <figure id="fig46"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc recta expedite duci pote&longs;t quæ Trajectoriam quæ­<lb/>&longs;itam, in puncto quovis dato <emph type="italics"/>B,<emph.end type="italics"/> continget. </s> <s>Accedat punctum <emph type="italics"/>d<emph.end type="italics"/> ad <lb/>punctum <emph type="italics"/>B,<emph.end type="italics"/> & recta <emph type="italics"/>Bd<emph.end type="italics"/> evadet tangens quæ&longs;ita. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Unde etiam Trajectoriarum Centra, Diametri & Latera <lb/>recta inveniri po&longs;&longs;unt, ut in Corollario &longs;ecundo Lemmatis XIX. </s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Con&longs;tructio prior evadet paulo &longs;implicior jungendo <emph type="italics"/>BP,<emph.end type="italics"/> & in ea, <lb/>&longs;i opus e&longs;t, producta capiendo <emph type="italics"/>Bp<emph.end type="italics"/> ad <emph type="italics"/>BP<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>PR<emph.end type="italics"/> ad <emph type="italics"/>PT<emph.end type="italics"/>; & <lb/>per <emph type="italics"/>p<emph.end type="italics"/> agendo rectam infinitam <emph type="italics"/>p<emph.end type="italics"/>d ip&longs;i <emph type="italics"/>SPT<emph.end type="italics"/> parallelam, inque ea <lb/>capiendo &longs;emper <emph type="italics"/>p<emph.end type="italics"/>d æqualem <emph type="italics"/>Pr<emph.end type="italics"/>; & agendo rectas <emph type="italics"/>Bd, Cr<emph.end type="italics"/> con­<lb/>currentes in <emph type="italics"/>d.<emph.end type="italics"/> Nam cum &longs;int <emph type="italics"/>Pr<emph.end type="italics"/> ad <emph type="italics"/>Pt, PR<emph.end type="italics"/> ad <emph type="italics"/>PT, pB<emph.end type="italics"/> ad <emph type="italics"/>PB, <lb/>p<emph.end type="italics"/>d ad <emph type="italics"/>Pt<emph.end type="italics"/> in eadem ratione; erunt <emph type="italics"/>p<emph.end type="italics"/>d & <emph type="italics"/>Pr<emph.end type="italics"/> &longs;emper æqua- <pb pagenum="76"/>les. </s> <s>Hac methodo puncta Trajectoriæ inveniuntur expediti&longs;&longs;ime, </s> </p> <p type="main"> <s> <arrow.to.target n="note52"></arrow.to.target><lb/>ni&longs;i mavis Curvam, ut in con&longs;tructione &longs;ecunda, de&longs;eribere Me­<lb/>chanice. </s> </p> <p type="margin"> <s><margin.target id="note52"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIII. PROBLEMA XV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere quæ per data quatuor puncta tran&longs;ibit, & rec­<lb/>tam continget po&longs;itione datam.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Dentur tangens <emph type="italics"/>HB,<emph.end type="italics"/> punctum contactus <emph type="italics"/>B,<emph.end type="italics"/> & alia tria <lb/>puncta <emph type="italics"/>C, D, P.<emph.end type="italics"/> Junge <emph type="italics"/>BC,<emph.end type="italics"/> & agendo <emph type="italics"/>PS<emph.end type="italics"/> parallelam <emph type="italics"/>BH,<emph.end type="italics"/><lb/>& <emph type="italics"/>PQ<emph.end type="italics"/> parallelam <emph type="italics"/>BC,<emph.end type="italics"/> comple parallelogrammum <emph type="italics"/><expan abbr="BSPq.">BSPque</expan><emph.end type="italics"/><lb/> <arrow.to.target n="fig47"></arrow.to.target><lb/>Age <emph type="italics"/>BD<emph.end type="italics"/> &longs;ecantem <emph type="italics"/>SP<emph.end type="italics"/> in <emph type="italics"/>T,<emph.end type="italics"/> & <emph type="italics"/>CD<emph.end type="italics"/> &longs;ecantem <emph type="italics"/>PQ<emph.end type="italics"/> in <emph type="italics"/>R.<emph.end type="italics"/> De­<lb/>nique, agendo quamvis <emph type="italics"/>tr<emph.end type="italics"/> ip&longs;i <emph type="italics"/>TR<emph.end type="italics"/> parallelam, de <emph type="italics"/>PQ, PS<emph.end type="italics"/><lb/>ab&longs;cinde <emph type="italics"/>Pr, Pt<emph.end type="italics"/> ip&longs;is <emph type="italics"/>PR, PT<emph.end type="italics"/> proportionales re&longs;pective; & <lb/>actarum <emph type="italics"/>Cr, Bt<emph.end type="italics"/> concur&longs;us <emph type="italics"/>d<emph.end type="italics"/> (per Lem. </s> <s>xx) incidet &longs;emper in <lb/>Trajectoriam de&longs;cribendam. <pb pagenum="77"/> <arrow.to.target n="note53"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note53"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig47"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Revolvatur tum angulus magnitudine datus <emph type="italics"/>CBH<emph.end type="italics"/> circa polum <lb/><emph type="italics"/>B,<emph.end type="italics"/> tum radius quilibet rectilineus & utrinque productus <emph type="italics"/>DC<emph.end type="italics"/> cir­<lb/>ca polum <emph type="italics"/>C.<emph.end type="italics"/> Notentur puncta <emph type="italics"/>M, N<emph.end type="italics"/> in quibus anguli crus <emph type="italics"/>BC<emph.end type="italics"/><lb/>&longs;ecat radium illum ubi crus alterum <emph type="italics"/>BH<emph.end type="italics"/> concurrit cum eodem ra­<lb/>dio in punctis <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>D.<emph.end type="italics"/> Deinde ad actam infinitam <emph type="italics"/>MN<emph.end type="italics"/> con­<lb/> <arrow.to.target n="fig48"></arrow.to.target><lb/>currant perpetuo radius ille <emph type="italics"/>CP<emph.end type="italics"/> vel <emph type="italics"/>CD<emph.end type="italics"/> & anguli crus <emph type="italics"/>BC,<emph.end type="italics"/> & <lb/>cruris alterius <emph type="italics"/>BH<emph.end type="italics"/> concur&longs;us cum radio delineabit Trajectoriam <lb/>quæ&longs;itam. </s> </p> <figure id="fig48"></figure> <p type="main"> <s>Nam &longs;i in con&longs;tructionibus Problematis &longs;uperioris accedat punc­<lb/>tum <emph type="italics"/>A<emph.end type="italics"/> ad punctum <emph type="italics"/>B,<emph.end type="italics"/> lineæ <emph type="italics"/>CA<emph.end type="italics"/> & <emph type="italics"/>CB<emph.end type="italics"/> coincident, & linea <emph type="italics"/>AB<emph.end type="italics"/> in <lb/>ultimo &longs;uo &longs;itu fiet tangens <emph type="italics"/>BH,<emph.end type="italics"/> atque adeo con&longs;tructiones ibi po­<lb/>&longs;itæ evadent eædem cum con&longs;tructionibus hic de&longs;criptis. </s> <s>Delinea­<lb/>bit igitur cruris <emph type="italics"/>BH<emph.end type="italics"/> concur&longs;us cum radio &longs;ectionem Conicam per <lb/>puncta <emph type="italics"/>C, D, P<emph.end type="italics"/> tran&longs;euntem, & rectam <emph type="italics"/>BH<emph.end type="italics"/> tangentem in puncto <lb/><emph type="italics"/>B. q.E.F.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Dentur puncta quatuor <emph type="italics"/>B, C, D, P<emph.end type="italics"/> extra tangentem <lb/><emph type="italics"/>HI<emph.end type="italics"/> &longs;ita. </s> <s>Junge bina lineis <emph type="italics"/>BD, CP<emph.end type="italics"/> concurrentibus in <emph type="italics"/>G,<emph.end type="italics"/> tangen- <pb pagenum="78"/> <arrow.to.target n="note54"></arrow.to.target><lb/>tique occurrentibus in <emph type="italics"/>H<emph.end type="italics"/> & <emph type="italics"/>I.<emph.end type="italics"/> Secetur tangens in <emph type="italics"/>A,<emph.end type="italics"/> ita ut &longs;it <lb/><emph type="italics"/>HA<emph.end type="italics"/> ad <emph type="italics"/>AI,<emph.end type="italics"/> ut e&longs;t rectan­<lb/> <arrow.to.target n="fig49"></arrow.to.target><lb/>gulum &longs;ub media proportio­<lb/>nali inter <emph type="italics"/>CG<emph.end type="italics"/> & <emph type="italics"/>GP<emph.end type="italics"/> & me­<lb/>dia proportionali inter <emph type="italics"/>BH<emph.end type="italics"/> & <lb/><emph type="italics"/>HD,<emph.end type="italics"/> ad rectangulum &longs;ub me­<lb/>dia proportionali inter <emph type="italics"/>DG<emph.end type="italics"/> & <lb/><emph type="italics"/>GB<emph.end type="italics"/> & media proportionali in­<lb/>ter <emph type="italics"/>PI<emph.end type="italics"/> & <emph type="italics"/>IC<emph.end type="italics"/>; & erit <emph type="italics"/>A<emph.end type="italics"/> punc­<lb/>tum contactus. </s> <s>Nam &longs;i rectæ <lb/><emph type="italics"/>PI<emph.end type="italics"/> parallela <emph type="italics"/>HX<emph.end type="italics"/> Trajecto­<lb/>riam &longs;ecet in punctis quibu&longs;­<lb/>vis <emph type="italics"/>X<emph.end type="italics"/> & <emph type="italics"/>Y:<emph.end type="italics"/> erit (ex Conicis) <lb/>punctum <emph type="italics"/>A<emph.end type="italics"/> ita locandum, ut fuerit <emph type="italics"/>HA quad.<emph.end type="italics"/> ad <emph type="italics"/>AI quad.<emph.end type="italics"/> in ra­<lb/>tione compo&longs;ita ex ratione rectanguli <emph type="italics"/>XHY<emph.end type="italics"/> ad rectangulum <emph type="italics"/>BHD<emph.end type="italics"/><lb/>&longs;eu rectanguli <emph type="italics"/>CGP<emph.end type="italics"/> ad rectangulum <emph type="italics"/>DGB<emph.end type="italics"/> & ex ratione rectan­<lb/>guli <emph type="italics"/>BHD<emph.end type="italics"/> ad rectangulum <emph type="italics"/>PIC.<emph.end type="italics"/> Invento autem contactus <lb/>puncto <emph type="italics"/>A,<emph.end type="italics"/> de&longs;cribetur Trajectoria ut in ca&longs;u primo. <emph type="italics"/>q.E.F.<emph.end type="italics"/><lb/>Capi autem pote&longs;t punctum <emph type="italics"/>A<emph.end type="italics"/> vel inter puncta <emph type="italics"/>H<emph.end type="italics"/> & <emph type="italics"/>I,<emph.end type="italics"/> vel extra; <lb/>& perinde Trajectoria dupliciter de&longs;cribi. </s> </p> <p type="margin"> <s><margin.target id="note54"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig49"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIV. PROBLEMA XVI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere quæ tran&longs;ibit per data tria puncta & rectas <lb/>duas po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Dentur tangentes <emph type="italics"/>HI, KL<emph.end type="italics"/> & <lb/> <arrow.to.target n="fig50"></arrow.to.target><lb/>puncta <emph type="italics"/>B, C, D.<emph.end type="italics"/> Per punctorum <lb/>duo quævis <emph type="italics"/>B, D<emph.end type="italics"/> age rectam in­<lb/>finitam <emph type="italics"/>BD<emph.end type="italics"/> tangentibus occur­<lb/>rentem in punctis <emph type="italics"/>H, K.<emph.end type="italics"/> Deinde <lb/>etiam per alia duo quævis <emph type="italics"/>C, D<emph.end type="italics"/><lb/>age infinitam <emph type="italics"/>CD<emph.end type="italics"/> tangentibus oc­<lb/>currentem in punctis <emph type="italics"/>I, L.<emph.end type="italics"/> Actas <lb/>ita &longs;eca in <emph type="italics"/>R<emph.end type="italics"/> & <emph type="italics"/>S,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>HR<emph.end type="italics"/> ad <lb/><emph type="italics"/>KR<emph.end type="italics"/> ut e&longs;t media proportionalis <lb/>inter <emph type="italics"/>BH<emph.end type="italics"/> & <emph type="italics"/>HD<emph.end type="italics"/> ad mediam <lb/>proportionalem inter <emph type="italics"/>BK<emph.end type="italics"/> & <emph type="italics"/>KD<emph.end type="italics"/>; <lb/>& <emph type="italics"/>IS<emph.end type="italics"/> ad <emph type="italics"/>LS<emph.end type="italics"/> ut e&longs;t media pro­<lb/>portionalis inter <emph type="italics"/>CI<emph.end type="italics"/> & <emph type="italics"/>ID<emph.end type="italics"/> ad me­<lb/>diam proportionalem inter <emph type="italics"/>CL<emph.end type="italics"/> <pb pagenum="79"/>& <emph type="italics"/>LD.<emph.end type="italics"/> Seca autem pro lubitu vel inter puncta <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>H,<emph.end type="italics"/><lb/> <arrow.to.target n="note55"></arrow.to.target><lb/><emph type="italics"/>I<emph.end type="italics"/> & <emph type="italics"/>L,<emph.end type="italics"/> vel extra eadem: dein age <emph type="italics"/>RS<emph.end type="italics"/> &longs;ecantem tangentes in <emph type="italics"/>A<emph.end type="italics"/><lb/>& <emph type="italics"/>P,<emph.end type="italics"/> & erunt <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> puncta contactuum. </s> <s>Nam &longs;i <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/><lb/>&longs;upponantur e&longs;&longs;e puncta contactuum alicubi in tangentibus &longs;i­<lb/>ta; & per punctorum <emph type="italics"/>H, I, K, L<emph.end type="italics"/> quodvis <emph type="italics"/>I,<emph.end type="italics"/> in tangente al­<lb/>terutra <emph type="italics"/>HI<emph.end type="italics"/> &longs;itum, agatur recta <emph type="italics"/>IY<emph.end type="italics"/> tangenti alteri <emph type="italics"/>KL<emph.end type="italics"/> paral­<lb/>lela, quæ occurrat curvæ in <emph type="italics"/>X<emph.end type="italics"/> & <emph type="italics"/>Y,<emph.end type="italics"/> & in ea &longs;umatur <emph type="italics"/>IZ<emph.end type="italics"/> me­<lb/>dia proportionalis inter <emph type="italics"/>IX<emph.end type="italics"/> & <emph type="italics"/>IY:<emph.end type="italics"/> erit, ex Conicis, rectangulum <lb/><emph type="italics"/>XIY<emph.end type="italics"/> &longs;eu <emph type="italics"/>IZ quad.<emph.end type="italics"/> ad <emph type="italics"/>LP quad.<emph.end type="italics"/> ut rectangulum <emph type="italics"/>CID<emph.end type="italics"/> ad rectan­<lb/>gulum <emph type="italics"/>CLD,<emph.end type="italics"/> id e&longs;t (per con&longs;tructionem) ut <emph type="italics"/>SI quad.<emph.end type="italics"/> ad <lb/><emph type="italics"/>SL quad:<emph.end type="italics"/> atque adeo <emph type="italics"/>IZ<emph.end type="italics"/> ad <emph type="italics"/>LP<emph.end type="italics"/> ut <emph type="italics"/>SI<emph.end type="italics"/> ad <emph type="italics"/>SL.<emph.end type="italics"/> Jacent ergo punc­<lb/>ta <emph type="italics"/>S, P, Z<emph.end type="italics"/> in una recta. </s> <s>Porro tangentibus concurrentibus in <emph type="italics"/>G,<emph.end type="italics"/> e­<lb/>rit (ex Conicis) rectangulum <emph type="italics"/>XIY<emph.end type="italics"/> &longs;eu <emph type="italics"/>IZ quad.<emph.end type="italics"/> ad <emph type="italics"/>IA quad.<emph.end type="italics"/> ut <lb/><emph type="italics"/>GP quad<emph.end type="italics"/> ad <emph type="italics"/>GA quad:<emph.end type="italics"/> adeoque <emph type="italics"/>IZ<emph.end type="italics"/> & <emph type="italics"/>IA<emph.end type="italics"/> ut <emph type="italics"/>GP<emph.end type="italics"/> ad <emph type="italics"/>GA.<emph.end type="italics"/> Jacent <lb/>ergo puncta <emph type="italics"/>P, Z<emph.end type="italics"/> & <emph type="italics"/>A<emph.end type="italics"/> in una recta, adeoque puncta <emph type="italics"/>S, P<emph.end type="italics"/> & <emph type="italics"/>A<emph.end type="italics"/><lb/>&longs;unt in una recta. </s> <s>Et eodem argumento probabitur quod puncta <lb/><emph type="italics"/>R, P<emph.end type="italics"/> & <emph type="italics"/>A<emph.end type="italics"/> &longs;unt in una recta. </s> <s>Jacent igitur puncta contactuum <emph type="italics"/>A<emph.end type="italics"/><lb/>& <emph type="italics"/>P<emph.end type="italics"/> in recta <emph type="italics"/>RS.<emph.end type="italics"/> Hi&longs;ce autem inventis, Trajectoria de&longs;eribetur <lb/>ut in ca&longs;u primo Problematis &longs;uperioris. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note55"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig50"></figure> <p type="main"> <s><emph type="center"/>LEMMA XXII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Figuras in alias eju&longs;dem generis figur as mutare.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Tran&longs;mutanda &longs;it figura quævis <emph type="italics"/>HGI.<emph.end type="italics"/> Ducantur pro lubitu <lb/>rectæ duæ parallelæ <emph type="italics"/>AO, BL<emph.end type="italics"/> tertiam quamvis po&longs;itione datam <lb/><emph type="italics"/>AB<emph.end type="italics"/> &longs;ecantes in <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>B,<emph.end type="italics"/><lb/> <arrow.to.target n="fig51"></arrow.to.target><lb/>& a figuræ puncto quo­<lb/>vis <emph type="italics"/>G,<emph.end type="italics"/> ad rectam <emph type="italics"/>AB<emph.end type="italics"/><lb/>ducatur quævis <emph type="italics"/>GD,<emph.end type="italics"/><lb/>ip&longs;i <emph type="italics"/>OA<emph.end type="italics"/> parallela. </s> <s>De­<lb/>inde a puncto aliquo <emph type="italics"/>O,<emph.end type="italics"/><lb/>in linea <emph type="italics"/>OA<emph.end type="italics"/> dato, ad <lb/>punctum <emph type="italics"/>D<emph.end type="italics"/> ducatur <lb/>recta <emph type="italics"/>OD,<emph.end type="italics"/> ip&longs;i <emph type="italics"/>BL<emph.end type="italics"/> oc­<lb/>currens in <emph type="italics"/>d,<emph.end type="italics"/> & a puncto <lb/>occur&longs;us erigatur recta <lb/><emph type="italics"/>dg<emph.end type="italics"/> datum quemvis angulum cum recta <emph type="italics"/>BL<emph.end type="italics"/> continens, atque eam <lb/>habens rationem ad <emph type="italics"/>Od<emph.end type="italics"/> quam habet <emph type="italics"/>DG<emph.end type="italics"/> ad <emph type="italics"/>OD<emph.end type="italics"/>; & erit <emph type="italics"/>g<emph.end type="italics"/> punc­<lb/>tum in figura nova <emph type="italics"/>hgi<emph.end type="italics"/> puncto <emph type="italics"/>G<emph.end type="italics"/> re&longs;pondens. </s> <s>Eadem ratione <lb/>puncta &longs;ingula figuræ primæ dabunt puncta totidem figura novæ. <pb pagenum="80"/> <arrow.to.target n="note56"></arrow.to.target><lb/>Concipe igitur punctum <emph type="italics"/>G<emph.end type="italics"/> motu continuo percurrere puncta om­<lb/>nia figuræ primæ, & punctum <emph type="italics"/>g<emph.end type="italics"/> motu itidem continuo percurret <lb/>puncta omnia figuræ novæ & eandem de&longs;cribet. </s> <s>Di&longs;tinctionis gra­<lb/>tia nominemus <emph type="italics"/>DG<emph.end type="italics"/> ordinatam primam, <emph type="italics"/>dg<emph.end type="italics"/> ordinatam novam; <lb/><emph type="italics"/>AD<emph.end type="italics"/> ab&longs;ci&longs;&longs;am primam, <emph type="italics"/>ad<emph.end type="italics"/> ab&longs;ci&longs;&longs;am novam; <emph type="italics"/>O<emph.end type="italics"/> polum, <emph type="italics"/>OD<emph.end type="italics"/> ra­<lb/>dium ab&longs;cidentem, <emph type="italics"/>OA<emph.end type="italics"/> radium ordinatum primum, & <emph type="italics"/>Oa<emph.end type="italics"/> (qno <lb/>parallelogrammum <emph type="italics"/>OABa<emph.end type="italics"/> completur) radium ordinatum novum. </s> </p> <p type="margin"> <s><margin.target id="note56"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig51"></figure> <p type="main"> <s>Dico jam quod, &longs;i punctum <emph type="italics"/>G<emph.end type="italics"/> tangit rectam Lineam po&longs;itione da­<lb/>tam, punctum <emph type="italics"/>g<emph.end type="italics"/> tanget etiam Lineam rectam po&longs;itione datam. </s> <s>Si <lb/>punctum <emph type="italics"/>G<emph.end type="italics"/> tangit Conicam &longs;ectionem, punctum <emph type="italics"/>g<emph.end type="italics"/> tanget etiam <lb/>Conicam &longs;ectionem. </s> <s>Conicis &longs;ectionibus hic Circulum annumero. </s> <s><lb/>Porro &longs;i punctum <emph type="italics"/>G<emph.end type="italics"/> tan­<lb/> <arrow.to.target n="fig52"></arrow.to.target><lb/>git Lineam tertii ordinis <lb/>Analytici, punctum <emph type="italics"/>g<emph.end type="italics"/><lb/>tanget Lineam tertii iti­<lb/>dem ordinis; & &longs;ic de <lb/>curvis lineis &longs;uperiorum <lb/>ordinum. </s> <s>Lineæ duæ e­<lb/>runt eju&longs;dem &longs;emper or­<lb/>dinis Analytici quas pun­<lb/>cta <emph type="italics"/>G, g<emph.end type="italics"/> tangunt. </s> <s>Et­<lb/>enim ut e&longs;t <emph type="italics"/>ad<emph.end type="italics"/> ad <emph type="italics"/>OA<emph.end type="italics"/><lb/>ita &longs;unt <emph type="italics"/>Od<emph.end type="italics"/> ad <emph type="italics"/>OD, dg<emph.end type="italics"/> ad <emph type="italics"/>DG,<emph.end type="italics"/> & <emph type="italics"/>AB<emph.end type="italics"/> ad <emph type="italics"/>AD<emph.end type="italics"/>; adeoque <emph type="italics"/>AD<emph.end type="italics"/><lb/>æqualis e&longs;t (<emph type="italics"/>OAXAB/ad<emph.end type="italics"/>), & <emph type="italics"/>DG<emph.end type="italics"/> æqualis e&longs;t (<emph type="italics"/>OAXdg/ad<emph.end type="italics"/>). Jam &longs;i punc­<lb/>tum <emph type="italics"/>G<emph.end type="italics"/> tangit rectam Lineam, atque adeo in æquatione quavis, <lb/>qua relatio inter ab&longs;ci&longs;&longs;am <emph type="italics"/>AD<emph.end type="italics"/> & ordinatam <emph type="italics"/>DG<emph.end type="italics"/> habetur, in­<lb/>determinatæ illæ <emph type="italics"/>AD<emph.end type="italics"/> & <emph type="italics"/>DG<emph.end type="italics"/> ad unicam tantum dimen&longs;ionem <lb/>a&longs;cendunt, &longs;cribendo in hac æquatione (<emph type="italics"/>OAXAB/ad<emph.end type="italics"/>) pro <emph type="italics"/>AD,<emph.end type="italics"/> & <lb/>(<emph type="italics"/>OAXdg/ad<emph.end type="italics"/>) pro <emph type="italics"/>DG,<emph.end type="italics"/> producetur æquatio nova, in qua ab&longs;ci&longs;&longs;a no­<lb/>va <emph type="italics"/>ad<emph.end type="italics"/> & ordinata nova <emph type="italics"/>dg<emph.end type="italics"/> ad unicam tantum dimen&longs;ionem a&longs;cen­<lb/>dent, atque adeo quæ de&longs;ignat Lineam rectam. </s> <s>Sin <emph type="italics"/>AD<emph.end type="italics"/> & <emph type="italics"/>DG<emph.end type="italics"/><lb/>(vel earum alterutra) a&longs;cendebant ad duas dimen&longs;iones in æquati­<lb/>one prima, a&longs;cendent itidem <emph type="italics"/>ad<emph.end type="italics"/> & <emph type="italics"/>dg<emph.end type="italics"/> ad duas in æquatione &longs;ecun­<lb/>da. </s> <s>Et &longs;ic de tribus vel pluribus dimen&longs;ionibus. </s> <s>Indeterminatæ <lb/><emph type="italics"/>ad, dg<emph.end type="italics"/> in æquatione &longs;ecunda & <emph type="italics"/>AD, DG<emph.end type="italics"/> in prima a&longs;cendent &longs;em­<lb/>per ad eundem dimen&longs;ionum numerum, & propterea Lineæ, quas <lb/>puncta <emph type="italics"/>G, g<emph.end type="italics"/> tangunt, &longs;unt eju&longs;dem ordinis Analytici. </s> </p> <pb pagenum="81"/> <figure id="fig52"></figure> <p type="main"> <s>Dico præterea quod &longs;i recta aliqua tangat lineam curvam in fi­<lb/> <arrow.to.target n="note57"></arrow.to.target><lb/>gura prima; hæc recta eodem modo cum curva in figuram novam <lb/>tran&longs;lata tanget lineam illam curvam in figura nova: & contra. </s> <s>Nam <lb/>&longs;i Curvæ puncta quævis duo accedunt ad invicem & coeunt in fi­<lb/>gura prima, puncta eadem tran&longs;lata accedent ad invicem & coibunt <lb/>in figura nova, atque adeo rectæ, quibus hæc puncta junguntur, &longs;i­<lb/>mul evadent curvarum tangentes in figura utraque. </s> <s>Componi po&longs;­<lb/>&longs;ent harum a&longs;&longs;ertionum Demon&longs;trationes more magis Geometrico. </s> <s><lb/>Sed brevitati con&longs;ulo. </s> </p> <p type="margin"> <s><margin.target id="note57"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s>Igitur &longs;i figura rectilinea in aliam tran&longs;mutanda e&longs;t, &longs;ufficit rec­<lb/>tarum a quibus conflatur inter&longs;ectiones transferre, & per ea&longs;dem <lb/>in figura nova lineas rectas ducere. </s> <s>Sin curvilineam tran&longs;mutare <lb/>oportet, transferenda &longs;unt puncta, tangentes & aliæ rectæ quarum <lb/>ope curva linea definitur. </s> <s>In&longs;ervit autem hoc Lemma &longs;olutioni <lb/>difficiliorum Problematum, tran&longs;mutando figuras propo&longs;itas in &longs;im­<lb/>pliciores. </s> <s>Nam rectæ quævis convergentes tran&longs;mutantur in pa­<lb/>rallelas, adhibendo pro radio ordinato primo, lineam quam­<lb/>vis rectam quæ per concur&longs;um convergentium tran&longs;it: id adeo quia <lb/>concur&longs;us ille hoc pacto abit in infinitum, lineæ autem parallelæ <lb/>&longs;unt quæ ad punctum infinite di&longs;tans tendunt. </s> <s>Po&longs;tquam autem <lb/>Problema &longs;olvitur in figura nova, &longs;i per inver&longs;as operationes tran&longs;­<lb/>mutetur hæc figura in figuram primam, habebitur &longs;olutio quæ&longs;ita. </s> </p> <p type="main"> <s>Utile e&longs;t etiam hoc Lemma in &longs;olutione Solidorum Problema­<lb/>tum. </s> <s>Nam quoties duæ &longs;ectiones Conicæ obvenerint, quarum in­<lb/>ter&longs;ectione Problema &longs;olvi pote&longs;t, tran&longs;mutare licet earum alter­<lb/>utram, &longs;i Hyperbola &longs;it vel Parabola, in Ellip&longs;in: deinde Ellip&longs;is <lb/>facile mutatur in Circulum. </s> <s>Recta item & &longs;ectio Conica, in con­<lb/>&longs;tructione Planorum Problematum, vertuntur in Rectam & Cir­<lb/>culum. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXV. PROBLEMA XVII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere qua per data duo puncta tran&longs;ibit & rectas <lb/>tres continget po&longs;itione datas.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Per concur&longs;um tangentium quarumvis duarum cum &longs;e invicem, & <lb/>concur&longs;um tangentis tertiæ cum recta illa, quæ per puncta duo data <lb/>tran&longs;it, age rectam infinitam; eaque adhibita pro radio ordinato pri­<lb/>mo, tran&longs;mutetur figura, per Lemma &longs;uperius, in figuram novam. </s> <s>In <pb pagenum="82"/> <arrow.to.target n="note58"></arrow.to.target><lb/>hac figura tangentes illæ duæ evadent &longs;ibi invicem parallelæ, & tan­<lb/>gens tertia fiet parallela rectæ per <lb/> <arrow.to.target n="fig53"></arrow.to.target><lb/>puncta duo data tran&longs;eunti. </s> <s>Sunto <lb/><emph type="italics"/>hi, kl<emph.end type="italics"/> tangentes illæ duæ parallelæ, <lb/><emph type="italics"/>ik<emph.end type="italics"/> tangens tertia, & <emph type="italics"/>hl<emph.end type="italics"/> recta huic <lb/>parallela tran&longs;iens per puncta illa <lb/><emph type="italics"/>a, b,<emph.end type="italics"/> per quæ Conica &longs;ectio in hac <lb/>figura nova tran&longs;ire debet, & pa­<lb/>rallelogrammum <emph type="italics"/>hikl<emph.end type="italics"/> complens. </s> <s><lb/>Secentur rectæ <emph type="italics"/>hi, ik, kl<emph.end type="italics"/> in <emph type="italics"/>c, d, e,<emph.end type="italics"/><lb/>ita ut &longs;it <emph type="italics"/>hc<emph.end type="italics"/> ad latus quadratum <lb/>rectanguli <emph type="italics"/>ahb, ic<emph.end type="italics"/> ad <emph type="italics"/>id,<emph.end type="italics"/> & <emph type="italics"/>ke<emph.end type="italics"/><lb/>ad <emph type="italics"/>kd<emph.end type="italics"/> ut e&longs;t &longs;umma rectarum <emph type="italics"/>hi<emph.end type="italics"/><lb/>& <emph type="italics"/>kl<emph.end type="italics"/> ad &longs;ummam trium linea­<lb/>rum quarum prima e&longs;t recta <emph type="italics"/>ik,<emph.end type="italics"/> & alteræ duæ &longs;unt latera quadrata <lb/>rectangulorum <emph type="italics"/>ahb<emph.end type="italics"/> & <emph type="italics"/>alb<emph.end type="italics"/> & erunt <emph type="italics"/>c, d, e<emph.end type="italics"/> puncta contactuum. </s> <s>Et­<lb/>enim, ex Conicis, &longs;unt <emph type="italics"/>hc<emph.end type="italics"/> quadratum ad rectangulum <emph type="italics"/>ahb,<emph.end type="italics"/> & <lb/><emph type="italics"/>ic<emph.end type="italics"/> quadratum ad <emph type="italics"/>id<emph.end type="italics"/> quadratum, & <emph type="italics"/>ke<emph.end type="italics"/> quadratum ad <emph type="italics"/>kd<emph.end type="italics"/> quadratum, <lb/>& <emph type="italics"/>el<emph.end type="italics"/> quadratum ad rectangulum <emph type="italics"/>alb<emph.end type="italics"/> in eadem ratione; & propter­<lb/>ea <emph type="italics"/>hc<emph.end type="italics"/> ad latus quadratum ip&longs;ius <emph type="italics"/>ahb, ic<emph.end type="italics"/> ad <emph type="italics"/>id, ke<emph.end type="italics"/> ad <emph type="italics"/>kd,<emph.end type="italics"/> & <emph type="italics"/>el<emph.end type="italics"/> ad <lb/>latus quadratum ip&longs;ius <emph type="italics"/>alb<emph.end type="italics"/> &longs;unt in &longs;ubduplicata illa ratione, & <lb/>compo&longs;ite, in data ratione omnium antecedentium <emph type="italics"/>hi<emph.end type="italics"/> & <emph type="italics"/>kl<emph.end type="italics"/> ad <lb/>omnes con&longs;equentes, quæ &longs;unt latus quadratum rectanguli <emph type="italics"/>ahb<emph.end type="italics"/> & <lb/>recta <emph type="italics"/>ik<emph.end type="italics"/> & latus quadratum rectanguli <emph type="italics"/>alb.<emph.end type="italics"/> Habentur igitur ex <lb/>data illa ratione puncta contactuum <emph type="italics"/>c, d, e,<emph.end type="italics"/> in figura nova. </s> <s>Per <lb/>inver&longs;as operationes Lemmatis novi&longs;&longs;imi transferantur hæc pun­<lb/>cta in figuram primam & ibi, per Probl. </s> <s>XIV, de&longs;cribetur <lb/>Trajectoria. <emph type="italics"/>q.E.F.<emph.end type="italics"/> Ceterum perinde ut puncta <emph type="italics"/>a, b<emph.end type="italics"/> ja­<lb/>cent vel inter puncta <emph type="italics"/>h, l,<emph.end type="italics"/> vel extra, debent puncta <emph type="italics"/>c, d, e<emph.end type="italics"/> vel <lb/>inter puncta <emph type="italics"/>h, i, k, l<emph.end type="italics"/> capi, vel extra. </s> <s>Si punctorum <emph type="italics"/>a, b<emph.end type="italics"/> al­<lb/>terutrum cadit inter puncta <emph type="italics"/>h, l,<emph.end type="italics"/> & alterum extra, Problema im­<lb/>po&longs;&longs;ibile e&longs;t. </s> </p> <p type="margin"> <s><margin.target id="note58"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig53"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVI. PROBLEMA XVIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere quæ tran&longs;ibit per punctum datum & rectas <lb/>quatuor po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Ab inter&longs;ectione communi duarum quarumlibet tangentium ad <lb/>inter&longs;ectionem communem reliquarum duarum agatur recta infini- <pb pagenum="83"/>ta, & eadem pro radio ordinato primo adhibita, tran&longs;mutetur fi­<lb/> <arrow.to.target n="note59"></arrow.to.target><lb/>gura (per Lem. </s> <s>XXII) in figuram novam, & tangentes binæ, quæ ad <lb/>radium ordinatum primum concurrebant, jam evadent parallelæ. </s> <s>Sun­<lb/>to illæ <emph type="italics"/>hi<emph.end type="italics"/> & <emph type="italics"/>kl, ik<emph.end type="italics"/> & <emph type="italics"/>hl<emph.end type="italics"/> continentes parallelogrammum <emph type="italics"/>hikl.<emph.end type="italics"/> Sit­<lb/>que <emph type="italics"/>p<emph.end type="italics"/> punctum in hac nova figura, puncto in figura prima dato <lb/>re&longs;pondens. </s> <s>Per figuræ centrum <emph type="italics"/>O<emph.end type="italics"/> agatur <emph type="italics"/>pq,<emph.end type="italics"/> & exi&longs;tente <emph type="italics"/>Oq<emph.end type="italics"/> æ­<lb/>quali <emph type="italics"/>Op,<emph.end type="italics"/> erit <emph type="italics"/>q<emph.end type="italics"/> punctum alterum per quod &longs;ectio Conica in hac <lb/>figura nova tran&longs;ire debet. </s> <s>Per Lemmatis XXII operationem in­<lb/>ver&longs;am transferatur hoc punctum in figuram primam, & ibi habe­<lb/>buntur puncta duo per quæ Trajectoria de&longs;cribenda e&longs;t. </s> <s>Per ea­<lb/>dem vero de&longs;cribi pote&longs;t Trajectoria illa per Prob. </s> <s>XVII. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note59"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>LEMMA XXIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si rectæ duæ po&longs;itione datæ<emph.end type="italics"/> AC, BD <emph type="italics"/>ad data puncta<emph.end type="italics"/> A, B, <emph type="italics"/>ter­<lb/>minentur, datamque habeant rationem ad invicem, & recta<emph.end type="italics"/><lb/>CD, <emph type="italics"/>qua puncta indeterminata<emph.end type="italics"/> C, D <emph type="italics"/>junguntur, &longs;ecetur in ra­<lb/>tione data in<emph.end type="italics"/> K: <emph type="italics"/>dico quod punctum<emph.end type="italics"/> K <emph type="italics"/>locabitur in recta po&longs;i­<lb/>tione data.<emph.end type="italics"/></s> </p> <p type="main"> <s>Concurrant enim rectæ <emph type="italics"/>AC,<emph.end type="italics"/><lb/> <arrow.to.target n="fig54"></arrow.to.target><lb/><emph type="italics"/>BD<emph.end type="italics"/> in <emph type="italics"/>E,<emph.end type="italics"/> & in <emph type="italics"/>BE<emph.end type="italics"/> capiatur <emph type="italics"/>BG<emph.end type="italics"/><lb/>ad <emph type="italics"/>AE<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>AC,<emph.end type="italics"/> &longs;it­<lb/>que <emph type="italics"/>FD<emph.end type="italics"/> &longs;emper æqualis datæ <lb/><emph type="italics"/>EG<emph.end type="italics"/>; & erit ex con&longs;tructione <lb/><emph type="italics"/>EC<emph.end type="italics"/> ad <emph type="italics"/>GD,<emph.end type="italics"/> hoc e&longs;t, ad <emph type="italics"/>EF<emph.end type="italics"/> ut <lb/><emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>BD,<emph.end type="italics"/> adeoque in ratione <lb/>data, & propterea dabitur &longs;pecie <lb/>triangulum <emph type="italics"/>EFC.<emph.end type="italics"/> Secetur <emph type="italics"/>CF<emph.end type="italics"/><lb/>in <emph type="italics"/>L<emph.end type="italics"/> ut &longs;it <emph type="italics"/>CL<emph.end type="italics"/> ad <emph type="italics"/>CF<emph.end type="italics"/> in ratio­<lb/>ne <emph type="italics"/>CK<emph.end type="italics"/> ad <emph type="italics"/>CD<emph.end type="italics"/>; &, ob datam il­<lb/>lam rationem, dabitur etiam &longs;pecie triangulum <emph type="italics"/>EFL<emph.end type="italics"/>; proindeque <lb/>punctum <emph type="italics"/>L<emph.end type="italics"/> locabitur in recta <emph type="italics"/>EL<emph.end type="italics"/> po&longs;itione data. </s> <s>Junge <emph type="italics"/>LK,<emph.end type="italics"/> & <lb/>&longs;imilia erunt triangula <emph type="italics"/>CLK, CFD<emph.end type="italics"/>; &, ob datam <emph type="italics"/>FD<emph.end type="italics"/> & datam <lb/>rationem <emph type="italics"/>LK<emph.end type="italics"/> ad <emph type="italics"/>FD,<emph.end type="italics"/> dabitur <emph type="italics"/>LK.<emph.end type="italics"/> Huic æqualis capiatur <emph type="italics"/>EH,<emph.end type="italics"/><lb/>& erit &longs;emper <emph type="italics"/>ELKH<emph.end type="italics"/> parallelogrammum. </s> <s>Locatur igitur punc­<lb/>tum <emph type="italics"/>K<emph.end type="italics"/> in parallelogrammi illius latere po&longs;itione dato <emph type="italics"/>HK. q.E.D.<emph.end type="italics"/> <pb pagenum="84"/> <arrow.to.target n="note60"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note60"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig54"></figure> <p type="main"> <s><emph type="center"/>LEMMA XXIV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si rectæ tres tangant quamcunque Coni&longs;ectionem, quarum duæ pa­<lb/>rallelæ &longs;int ac dentur po&longs;itione; dico quod Sectionis &longs;emidia­<lb/>meter hi&longs;ce duabus parallela, &longs;it media proportionalis inter ha­<lb/>rum &longs;egmenta, punctis contactuum & tangenti tertiæ inter­<lb/>jecta.<emph.end type="italics"/></s> </p> <p type="main"> <s>Sunto <emph type="italics"/>AF, GB<emph.end type="italics"/> pa­<lb/> <arrow.to.target n="fig55"></arrow.to.target><lb/>rallelæ duæ Coni&longs;ec­<lb/>tionem <emph type="italics"/>ADB<emph.end type="italics"/> tan­<lb/>gentes in <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>B; EF<emph.end type="italics"/><lb/>recta tertia Coni&longs;ec­<lb/>tionem tangens in <emph type="italics"/>I,<emph.end type="italics"/><lb/>& occurrens prioribus <lb/>tangentibus in <emph type="italics"/>F<emph.end type="italics"/> & <emph type="italics"/>G<emph.end type="italics"/>; <lb/>&longs;itque <emph type="italics"/>CD<emph.end type="italics"/> &longs;emidiame­<lb/>ter Figuræ tangenti­<lb/>bus parallela: Dico <lb/>quod <emph type="italics"/>AF, CD, BG<emph.end type="italics"/><lb/>&longs;unt continue proportionales. </s> </p> <figure id="fig55"></figure> <p type="main"> <s>Nam &longs;i diametri conjugatæ <emph type="italics"/>AB, DM<emph.end type="italics"/> tangenti <emph type="italics"/>FG<emph.end type="italics"/> occurrant <lb/>in <emph type="italics"/>E<emph.end type="italics"/> & <emph type="italics"/>H,<emph.end type="italics"/> &longs;eque mutuo &longs;ecent in <emph type="italics"/>C,<emph.end type="italics"/> & compleatur parallelogram­<lb/>mum <emph type="italics"/>IKCL<emph.end type="italics"/>; erit, ex natura Sectionum Conicarum, ut <emph type="italics"/>EC<emph.end type="italics"/> ad <lb/><emph type="italics"/>CA<emph.end type="italics"/> ita <emph type="italics"/>CA<emph.end type="italics"/> ad <emph type="italics"/>CL,<emph.end type="italics"/> & ita divi&longs;im <emph type="italics"/>EC-CA<emph.end type="italics"/> ad <emph type="italics"/>CA-CL,<emph.end type="italics"/> &longs;eu <lb/><emph type="italics"/>EA<emph.end type="italics"/> ad <emph type="italics"/>AL,<emph.end type="italics"/> & compo&longs;ite <emph type="italics"/>EA<emph.end type="italics"/> ad <emph type="italics"/>EA+AL<emph.end type="italics"/> &longs;eu <emph type="italics"/>EL<emph.end type="italics"/> ut <emph type="italics"/>EC<emph.end type="italics"/> ad <lb/><emph type="italics"/>EC+CA<emph.end type="italics"/> &longs;eu <emph type="italics"/>EB<emph.end type="italics"/>; adeoque (ob &longs;imilitudinem triangulorum <emph type="italics"/>EAF, <lb/>ELI, ECH, EBG) AF<emph.end type="italics"/> ad <emph type="italics"/>LI<emph.end type="italics"/> ut <emph type="italics"/>CH<emph.end type="italics"/> ad <emph type="italics"/>BG.<emph.end type="italics"/> E&longs;t itidem, <lb/>ex natura Sectionum Conicarum, <emph type="italics"/>LI<emph.end type="italics"/> (&longs;eu <emph type="italics"/>CK<emph.end type="italics"/>) ad <emph type="italics"/>CD<emph.end type="italics"/> ut <emph type="italics"/>CD<emph.end type="italics"/> ad <lb/><emph type="italics"/>CH<emph.end type="italics"/>; atque, adeo ex æquo perturbate, <emph type="italics"/>AF<emph.end type="italics"/> ad <emph type="italics"/>CD<emph.end type="italics"/> ut <emph type="italics"/>CD<emph.end type="italics"/> ad <emph type="italics"/>BG. <lb/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i tangentes duæ <emph type="italics"/>FG, PQ<emph.end type="italics"/> tangentibus parallelis <lb/><emph type="italics"/>AF, BG<emph.end type="italics"/> occurrant in <emph type="italics"/>F<emph.end type="italics"/> & <emph type="italics"/>G, P<emph.end type="italics"/> & <emph type="italics"/>Q,<emph.end type="italics"/> &longs;eque mutuo &longs;ecent in <emph type="italics"/>O<emph.end type="italics"/>; <lb/>erit (ex æquo perturbate) <emph type="italics"/>AF<emph.end type="italics"/> ad <emph type="italics"/>BQ<emph.end type="italics"/> ut <emph type="italics"/>AP<emph.end type="italics"/> ad <emph type="italics"/>BG,<emph.end type="italics"/> & divi&longs;im <lb/>ut <emph type="italics"/>FP<emph.end type="italics"/> ad <emph type="italics"/>GQ,<emph.end type="italics"/> atque adeo ut <emph type="italics"/>FO<emph.end type="italics"/> ad <emph type="italics"/>OG.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Unde etiam rectæ duæ <emph type="italics"/>PG, FQ<emph.end type="italics"/> per puncta <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>G, <lb/>F<emph.end type="italics"/> & <emph type="italics"/>Q<emph.end type="italics"/> ductæ, concurrent ad rectam <emph type="italics"/>ACB<emph.end type="italics"/> per centrum Figuræ & <lb/>puncta contactuum <emph type="italics"/>A, B<emph.end type="italics"/> tran&longs;euntem. <pb pagenum="85"/> <arrow.to.target n="note61"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note61"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>LEMMA XXV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si parallelogrammi latera quatuor infinite producta tangant Sectio­<lb/>nem quamcunque Conicam, & ab&longs;cindantur ad tangentem quamvis <lb/>quintam; &longs;umantur autem laterum quorumvis duorum contermi­<lb/>norum ab&longs;ci&longs;&longs;æ terminatæ ad angulos oppo&longs;itos parallelogrammi: <lb/>dico quod ab&longs;ci&longs;&longs;a alterutra &longs;it ad latus illud a quo est ab&longs;ci&longs;&longs;a, ut <lb/>pars lateris alterius contermini inter punctum contactus & latus <lb/>tertium, est ad ab&longs;ci&longs;&longs;arum alteram.<emph.end type="italics"/></s> </p> <p type="main"> <s>Tangant parallelogrammi <emph type="italics"/>MLIK<emph.end type="italics"/> latera quatuor <emph type="italics"/>ML, IK, KL, <lb/>MI<emph.end type="italics"/> &longs;ectionem Conicam in <emph type="italics"/>A, B, C, D,<emph.end type="italics"/> & &longs;ecet tangens quinta <emph type="italics"/>FQ<emph.end type="italics"/><lb/>hæc latera in <emph type="italics"/>F, Q, H<emph.end type="italics"/><lb/> <arrow.to.target n="fig56"></arrow.to.target><lb/>& <emph type="italics"/>E<emph.end type="italics"/>; &longs;umantur autem <lb/>laterum <emph type="italics"/>MI, KI<emph.end type="italics"/> ab­<lb/>&longs;ci&longs;&longs;æ <emph type="italics"/>ME, KQ,<emph.end type="italics"/> vel <lb/>laterum <emph type="italics"/>KL, ML<emph.end type="italics"/> ab­<lb/>&longs;ci&longs;&longs;æ <emph type="italics"/>KH, MF:<emph.end type="italics"/> di­<lb/>co quod &longs;it <emph type="italics"/>ME<emph.end type="italics"/> ad <lb/><emph type="italics"/>MI<emph.end type="italics"/> ut <emph type="italics"/>BK<emph.end type="italics"/> ad <emph type="italics"/>KQ<emph.end type="italics"/>; <lb/>& <emph type="italics"/>KH<emph.end type="italics"/> ad <emph type="italics"/>KL<emph.end type="italics"/> ut <lb/><emph type="italics"/>AM<emph.end type="italics"/> ad <emph type="italics"/>MF.<emph.end type="italics"/> Nam <lb/>per Corollarium &longs;e­<lb/>cundum Lemmatis &longs;uperioris, e&longs;t <emph type="italics"/>ME<emph.end type="italics"/> ad <emph type="italics"/>EI<emph.end type="italics"/> ut (<emph type="italics"/>AM<emph.end type="italics"/> &longs;eu) <emph type="italics"/>BK<emph.end type="italics"/> ad <lb/><emph type="italics"/>BQ,<emph.end type="italics"/> & componendo <emph type="italics"/>ME<emph.end type="italics"/> ad <emph type="italics"/>MI<emph.end type="italics"/> ut <emph type="italics"/>BK<emph.end type="italics"/> ad <emph type="italics"/><expan abbr="Kq.">Kque</expan> q.E.D.<emph.end type="italics"/><lb/>Item <emph type="italics"/>KH<emph.end type="italics"/> ad <emph type="italics"/>HL<emph.end type="italics"/> ut (<emph type="italics"/>BK<emph.end type="italics"/> &longs;eu) <emph type="italics"/>AM<emph.end type="italics"/> ad <emph type="italics"/>AF,<emph.end type="italics"/> & dividendo <emph type="italics"/>KH<emph.end type="italics"/> ad <lb/><emph type="italics"/>KL<emph.end type="italics"/> ut <emph type="italics"/>AM<emph.end type="italics"/> ad <emph type="italics"/>MF. q.E.D.<emph.end type="italics"/></s> </p> <figure id="fig56"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i datur parallelogramum <emph type="italics"/>IKLM,<emph.end type="italics"/> circa datam Sec­<lb/>tionem Conicam de&longs;eriptum, dabitur rectangulum <emph type="italics"/>KQXME,<emph.end type="italics"/> ut <lb/>& huic æquale rectangulum <emph type="italics"/>KHXMF.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et &longs;i &longs;exta ducatur tangens <emph type="italics"/>eq<emph.end type="italics"/> tangentibus <emph type="italics"/>KI, MI<emph.end type="italics"/><lb/>occurrens in <emph type="italics"/>q<emph.end type="italics"/> & <emph type="italics"/>e<emph.end type="italics"/>; rectangulum <emph type="italics"/>KQXME<emph.end type="italics"/> æquabitur rectan­<lb/>gulo <emph type="italics"/>KqXMe<emph.end type="italics"/>; eritque <emph type="italics"/>KQ<emph.end type="italics"/> ad <emph type="italics"/>Me<emph.end type="italics"/> ut <emph type="italics"/>Kq<emph.end type="italics"/> ad <emph type="italics"/>ME,<emph.end type="italics"/> & divi&longs;im ut <lb/><emph type="italics"/>Qq<emph.end type="italics"/> ad <emph type="italics"/>Ee.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Unde etiam &longs;i <emph type="italics"/>Eq, eQ<emph.end type="italics"/> jungantur & bi&longs;ecentur, & recta <lb/>per puncta bi&longs;ectionum agatur, tran&longs;ibit hæc per centrum Sectio­<lb/>nis Conicæ. </s> <s>Nam cum &longs;it <emph type="italics"/>Qq<emph.end type="italics"/> ad <emph type="italics"/>Ee<emph.end type="italics"/> ut <emph type="italics"/>KQ<emph.end type="italics"/> ad <emph type="italics"/>Me,<emph.end type="italics"/> tran&longs;ibit ea- <pb pagenum="86"/> <arrow.to.target n="note62"></arrow.to.target><lb/>dem recta per medium omnium <emph type="italics"/>Eq, eQ, MK<emph.end type="italics"/>; (per Lem. </s> <s>XXIII) <lb/>& medium rectæ <emph type="italics"/>MK<emph.end type="italics"/> e&longs;t centrum Sectionis. </s> </p> <p type="margin"> <s><margin.target id="note62"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVII. PROBLEMA XIX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere quæ rectas quinque po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Dentur pofitione tangentes <emph type="italics"/>ABG, BCF, GCD, FDE, EA.<emph.end type="italics"/><lb/>Figuræ quadrilateræ &longs;ub quatuor quibu&longs;vis contentæ <emph type="italics"/>ABFE<emph.end type="italics"/> dia­<lb/>gonales <emph type="italics"/>AF, BE<emph.end type="italics"/> bi&longs;eca, & (per Corol. </s> <s>3. Lem. </s> <s>XXV) recta <emph type="italics"/>MN<emph.end type="italics"/><lb/>per puncta bi&longs;ectionum acta tran&longs;ibit per centrum Trajectoriæ. </s> <s><lb/>Rur&longs;us Figuræ quadrilateræ <emph type="italics"/>BGDF,<emph.end type="italics"/> &longs;ub aliis quibu&longs;vis quatuor <lb/> <arrow.to.target n="fig57"></arrow.to.target><lb/>tangentibus contentæ, diagonales (ut ita dicam) <emph type="italics"/>BD, GF<emph.end type="italics"/> bi­<lb/>&longs;eca in <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>Q:<emph.end type="italics"/> & recta <emph type="italics"/>PQ<emph.end type="italics"/> per puncta bi&longs;ectionum acta tran&longs;­<lb/>ibit per centrum Trajectoriæ. </s> <s>Dabitur ergo centrum in concur&longs;u bi­<lb/>&longs;ecantium. </s> <s>Sit illud <emph type="italics"/>O.<emph.end type="italics"/> Tangenti cuivis <emph type="italics"/>BC<emph.end type="italics"/> parallelam age <emph type="italics"/>KL,<emph.end type="italics"/><lb/>ad eam di&longs;tantiam ut centrum <emph type="italics"/>O<emph.end type="italics"/> in medio inter parallelas locetur, <lb/>& acta <emph type="italics"/>KL<emph.end type="italics"/> tanget Trajectoriam de&longs;cribendam. </s> <s>Secet hæc tan- <pb pagenum="87"/>gentes alias qua&longs;vis duas <emph type="italics"/>GCD, FDE<emph.end type="italics"/> in <emph type="italics"/>L<emph.end type="italics"/> & <emph type="italics"/>K.<emph.end type="italics"/> Per harum <lb/> <arrow.to.target n="note63"></arrow.to.target><lb/>tangentium non parallelarum <emph type="italics"/>CL, FK<emph.end type="italics"/> cum parallelis <emph type="italics"/>CF, KL<emph.end type="italics"/><lb/>concur&longs;us <emph type="italics"/>C<emph.end type="italics"/> & <emph type="italics"/>K, F<emph.end type="italics"/> & <emph type="italics"/>L<emph.end type="italics"/> age <emph type="italics"/>CK, FL<emph.end type="italics"/> concurrentes in <emph type="italics"/>R,<emph.end type="italics"/> & rec­<lb/>ta <emph type="italics"/>OR<emph.end type="italics"/> ducta & producta &longs;ecabit tangentes parallelas <emph type="italics"/>CF, KL<emph.end type="italics"/> in <lb/>punctis contactuum. </s> <s>Patet hoc per Corol. </s> <s>2. Lem. </s> <s>XXIV. </s> <s>Ea­<lb/>dem methodo invenire licet alia contactuum puncta, & tum de­<lb/>mum per Probl. </s> <s>XIV. &c. </s> <s>Trajectoriam de&longs;cribere. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note63"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig57"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Problemata, ubi dantur Trajectoriarum vel centra vel A&longs;ymp­<lb/>toti, includuntnr in præcedentibus. </s> <s>Nam datis punctis & tangen­<lb/>tibus una cum centro, dantur alia totidem puncta aliæque tangen­<lb/>tes a centro ex altera ejus parte æqualiter di&longs;tantes. </s> <s>A&longs;ymptotos <lb/>autem pro tangente habenda e&longs;t, & ejus terminus infinite di&longs;tans <lb/>(&longs;i ita loqui fas &longs;it) pro puncto contactus. </s> <s>Concipe tangentis cu­<lb/>ju&longs;vis punctum contactus abire in infinitum, & tangens vertetur in <lb/>A&longs;ymptoton, atque con&longs;tructiones Problematis XIV & Ca&longs;us pri­<lb/>mi Problematis XV vertentur in con&longs;tructiones Problematum ubi <lb/>A&longs;ymptoti dantur. </s> </p> <p type="main"> <s>Po&longs;tquam Trajectoria de&longs;cripta e&longs;t, invenire licet axes & umbi­<lb/>licos ejus hac methodo. </s> <s>In con&longs;tructione & figura Lemmatis XXI, <lb/>fac ut angulorum mobi­<lb/> <arrow.to.target n="fig58"></arrow.to.target><lb/>lium <emph type="italics"/>PBN, PCN<emph.end type="italics"/> cru­<lb/>ra <emph type="italics"/>BP, CP,<emph.end type="italics"/> quorum <lb/>concur&longs;u Trajectoria de­<lb/>&longs;cribebatur, &longs;int &longs;ibi invi­<lb/>cem parallela, eumque <lb/>&longs;ervantia &longs;itum revolvan­<lb/>tur circa polos &longs;uos <emph type="italics"/>B, C<emph.end type="italics"/><lb/>in figura illa. </s> <s>Interea ve­<lb/>ro de&longs;cribant altera an­<lb/>gulorum illorum crura <lb/><emph type="italics"/>CN, BN,<emph.end type="italics"/> concur&longs;u <lb/>&longs;uo <emph type="italics"/>K<emph.end type="italics"/> vel <emph type="italics"/>k,<emph.end type="italics"/> Circulum <lb/><emph type="italics"/>IBKGC.<emph.end type="italics"/> Sit Circuli <lb/>hujus centrum <emph type="italics"/>O.<emph.end type="italics"/> Ab <lb/>hoc centro ad Regulam <lb/><emph type="italics"/>MN,<emph.end type="italics"/> ad quam altera illa crura <emph type="italics"/>CN, BN<emph.end type="italics"/> interea concurrebant <pb pagenum="88"/> <arrow.to.target n="note64"></arrow.to.target><lb/>dum Trajectoria de&longs;cribebatur, demitte normalem <emph type="italics"/>OH<emph.end type="italics"/> Circulo oc­<lb/>currentem in <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>L.<emph.end type="italics"/> Et ubi crura illa altera <emph type="italics"/>CK, BK<emph.end type="italics"/> concur­<lb/>runt ad punctum illud <emph type="italics"/>K<emph.end type="italics"/> quod Regulæ propius e&longs;t, crura prima <lb/><emph type="italics"/>CP, BP<emph.end type="italics"/> parallela erunt axi majori, & perpendicularia minori; <lb/>& contrarium eveniet &longs;i crura eadem concurrunt ad punctum remo­<lb/>tius <emph type="italics"/>L.<emph.end type="italics"/> Unde &longs;i detur Trajectoriæ centrum, dabuntur axes. </s> <s>Hi&longs;ce <lb/>autem datis, umbilici &longs;unt in promptu. </s> </p> <p type="margin"> <s><margin.target id="note64"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig58"></figure> <p type="main"> <s>Axium vero quadrata &longs;unt ad invicem ut <emph type="italics"/>KH<emph.end type="italics"/> ad <emph type="italics"/>LH,<emph.end type="italics"/> & inde <lb/>facile e&longs;t Trajectoriam <lb/> <arrow.to.target n="fig59"></arrow.to.target><lb/>&longs;pecie datam per data <lb/>quatuor puncta de&longs;cri­<lb/>bere. </s> <s>Nam &longs;i duo ex <lb/>punctis datis con&longs;titu­<lb/>antur poli <emph type="italics"/>C, B,<emph.end type="italics"/> tertium <lb/>dabit angulos mobiles <lb/><emph type="italics"/>PCK, PBK<emph.end type="italics"/>; his au­<lb/>tem datis de&longs;cribi pote&longs;t <lb/>Circulus <emph type="italics"/>IBKGC.<emph.end type="italics"/><lb/>Tum ob datam &longs;pecie <lb/>Trajectoriam, dabitur <lb/>ratio <emph type="italics"/>OH<emph.end type="italics"/> ad <emph type="italics"/>OK,<emph.end type="italics"/> ad­<lb/>eoque ip&longs;a <emph type="italics"/>OH.<emph.end type="italics"/> Cen­<lb/>tro <emph type="italics"/>O<emph.end type="italics"/> & intervallo <emph type="italics"/>OH<emph.end type="italics"/><lb/>de&longs;cribe alium circulum, <lb/>& recta quæ tangit hunc circulum, & tran&longs;it per concur&longs;um crurum <lb/><emph type="italics"/>CK, BK,<emph.end type="italics"/> ubi crura prima <emph type="italics"/>CP, BP<emph.end type="italics"/> concurrunt ad quartum da­<lb/>tum punctum erit Regula illa <emph type="italics"/>MN<emph.end type="italics"/> cujus ope Trajectoria de&longs;cri­<lb/>betur. </s> <s>Unde etiam vici&longs;&longs;im Trapezium &longs;pecie datum (&longs;i ca&longs;us qui­<lb/>dam impo&longs;&longs;ibiles excipiantur) in data quavis Sectione Conica in­<lb/>&longs;cribi pote&longs;t. </s> </p> <figure id="fig59"></figure> <p type="main"> <s>Sunt & alia Le<gap/>mata quorum ope Trajectoriæ &longs;pecie datæ, <lb/>datis punctis & tangentibus, de&longs;cribi po&longs;&longs;unt. </s> <s>Ejus generis <lb/>e&longs;t quod, &longs;i recta linea per punctum quodvis po&longs;itione datum <lb/>ducatur, quæ datam Coni&longs;ectionem in punctis duobus inter&longs;e­<lb/>cet, & inter&longs;ectionum intervallum bi&longs;ecetur, punctum bi&longs;ectionis <lb/>tanget aliam Coni&longs;ectionem eju&longs;dem &longs;peciei cum priore, atque <lb/>axes habentem prioris axibus parallelos. </s> <s>Sed propero ad magis <lb/>utilia. <pb pagenum="89"/> <arrow.to.target n="note65"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note65"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>LEMMA XXVI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Trianguli &longs;pecie & magnitudine dati tres angulos ad rectas tot­<lb/>idem po&longs;itione datas, quæ non &longs;unt omnes parallelæ, &longs;ingulos ad <lb/>&longs;ingulas ponere.<emph.end type="italics"/></s> </p> <p type="main"> <s>Dantur po&longs;itione tres rectæ infinitæ <emph type="italics"/>AB, AC, BC,<emph.end type="italics"/> & opor­<lb/>tet triangulum <emph type="italics"/>DEF<emph.end type="italics"/> ita locare, ut angulus ejus <emph type="italics"/>D<emph.end type="italics"/> lineam <emph type="italics"/>AB,<emph.end type="italics"/><lb/>angulus <emph type="italics"/>E<emph.end type="italics"/> lineam <emph type="italics"/>AC,<emph.end type="italics"/><lb/> <arrow.to.target n="fig60"></arrow.to.target><lb/>& angulus <emph type="italics"/>F<emph.end type="italics"/> lineam <lb/><emph type="italics"/>BC<emph.end type="italics"/> tangat. </s> <s>Super <emph type="italics"/>DE, <lb/>DF<emph.end type="italics"/> & <emph type="italics"/>EF<emph.end type="italics"/> de&longs;cribe <lb/>tria circulorum &longs;eg­<lb/>menta <emph type="italics"/>DRE, DGF, <lb/>EMF,<emph.end type="italics"/> quæ capiant <lb/>angulos angulis <emph type="italics"/>BAC, <lb/>ABC, ACB<emph.end type="italics"/> æquales <lb/>re&longs;pective. </s> <s>De&longs;criban­<lb/>tur autem hæc &longs;egmen­<lb/>ta ad eas partes linea­<lb/>rum <emph type="italics"/>DE, DF, EF<emph.end type="italics"/> ut <lb/>literæ <emph type="italics"/>DRED<emph.end type="italics"/> eodem <lb/>ordine cum literis <lb/><emph type="italics"/>BACB,<emph.end type="italics"/> literæ <emph type="italics"/>DGFD<emph.end type="italics"/><lb/>eodem cum literis <lb/><emph type="italics"/>ABCA,<emph.end type="italics"/> & literæ <lb/><emph type="italics"/>EMFE<emph.end type="italics"/> eodem cum <lb/>literis <emph type="italics"/>ACBA<emph.end type="italics"/> in orbem <lb/>redeant; deinde com­<lb/>pleantur hæc &longs;egmenta <lb/>in circulos integros. </s> <s>Se­<lb/>cent circuli duo prio­<lb/>res &longs;e mutuo in <emph type="italics"/>G,<emph.end type="italics"/> &longs;int­<lb/>que centra eorum <emph type="italics"/>P<emph.end type="italics"/> & <lb/><emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> Junctis <emph type="italics"/>GP, PQ,<emph.end type="italics"/><lb/>cape <emph type="italics"/>Ga<emph.end type="italics"/> ad <emph type="italics"/>AB<emph.end type="italics"/> ut e&longs;t <lb/><emph type="italics"/>GP<emph.end type="italics"/> ad <emph type="italics"/>PQ,<emph.end type="italics"/> & cen­<lb/>tro <emph type="italics"/>G,<emph.end type="italics"/> intervallo <emph type="italics"/>Ga<emph.end type="italics"/><lb/>de&longs;cribe circulum, qui &longs;ecet circulum primum <emph type="italics"/>DGE<emph.end type="italics"/> in <emph type="italics"/>a.<emph.end type="italics"/> Jungatur <lb/>tum <emph type="italics"/>aD<emph.end type="italics"/> &longs;ecans circulum &longs;ecundum <emph type="italics"/>DFG<emph.end type="italics"/> in <emph type="italics"/>b,<emph.end type="italics"/> tum <emph type="italics"/>aE<emph.end type="italics"/> &longs;ecans cir- <pb pagenum="90"/> <arrow.to.target n="note66"></arrow.to.target><lb/>culum tertium <emph type="italics"/>EMF<emph.end type="italics"/> in <emph type="italics"/>c.<emph.end type="italics"/> Et compleatur Figura <emph type="italics"/>ABC def<emph.end type="italics"/> &longs;imi­<lb/>lis & æqualis Figuræ <emph type="italics"/>abcDEF.<emph.end type="italics"/> Dico factum. </s> </p> <p type="margin"> <s><margin.target id="note66"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig60"></figure> <p type="main"> <s>Agatur enim <emph type="italics"/>Fc<emph.end type="italics"/> ip&longs;i <emph type="italics"/>aD<emph.end type="italics"/> occurrens in <emph type="italics"/>n,<emph.end type="italics"/> & jungantur <emph type="italics"/>aG, bG, <lb/>QG, QD, PD.<emph.end type="italics"/> Ex con&longs;tructione e&longs;t angulus <emph type="italics"/>EaD<emph.end type="italics"/> æqualis an­<lb/>gulo <emph type="italics"/>CAB,<emph.end type="italics"/> & angulus <lb/> <arrow.to.target n="fig61"></arrow.to.target><lb/><emph type="italics"/>acF<emph.end type="italics"/> æqualis angulo <lb/><emph type="italics"/>ACB,<emph.end type="italics"/> adeoque trian­<lb/>gulum <emph type="italics"/>anc<emph.end type="italics"/> triangulo <lb/><emph type="italics"/>ABC<emph.end type="italics"/> æquiangulum. </s> <s><lb/>Ergo angulus <emph type="italics"/>anc<emph.end type="italics"/> &longs;eu <lb/><emph type="italics"/>FnD<emph.end type="italics"/> angulo <emph type="italics"/>ABC,<emph.end type="italics"/><lb/>adeoque angulo <emph type="italics"/>FbD<emph.end type="italics"/><lb/>æqualis e&longs;t; & propter­<lb/>ea punctum <emph type="italics"/>n<emph.end type="italics"/> incidit in <lb/>punctum <emph type="italics"/>b.<emph.end type="italics"/> Porro an­<lb/>gulus <emph type="italics"/>GPQ,<emph.end type="italics"/> qui di­<lb/>midius e&longs;t anguli ad <lb/>centrum <emph type="italics"/>GPD,<emph.end type="italics"/> æqua­<lb/>lis e&longs;t angulo ad cir­<lb/>cumferentiam <emph type="italics"/>GaD<emph.end type="italics"/>; <lb/>& angulus <emph type="italics"/>GQP,<emph.end type="italics"/> qui <lb/>dimidius e&longs;t anguli ad <lb/>centrum <emph type="italics"/>GQD,<emph.end type="italics"/> æ­<lb/>qualis e&longs;t complemen­<lb/>to ad duos rectos an­<lb/>guli ad circumferenti­<lb/>am <emph type="italics"/>GbD,<emph.end type="italics"/> adeoque æ­<lb/>qualis angulo <emph type="italics"/>Gba<emph.end type="italics"/>; <lb/>funtque ideo triangu­<lb/>la <emph type="italics"/>GPQ, Gab<emph.end type="italics"/> &longs;imi­<lb/>lia; & <emph type="italics"/>Ga<emph.end type="italics"/> e&longs;t ad <emph type="italics"/>ab<emph.end type="italics"/><lb/>ut <emph type="italics"/>GP<emph.end type="italics"/> ad <emph type="italics"/>PQ<emph.end type="italics"/>; id e&longs;t <lb/>(ex con&longs;tructione) ut <lb/><emph type="italics"/>Ga<emph.end type="italics"/> ad <emph type="italics"/>AB.<emph.end type="italics"/> Æquan­<lb/>tur itaque <emph type="italics"/>ab<emph.end type="italics"/> & <emph type="italics"/>AB<emph.end type="italics"/>; & propterea triangula <emph type="italics"/>abc, ABC,<emph.end type="italics"/> quæ mo­<lb/>do &longs;imilia e&longs;&longs;e probavimus, &longs;unt etiam æqualia. </s> <s>Unde, cum tan­<lb/>gant in&longs;uper trianguli <emph type="italics"/>DEF<emph.end type="italics"/> anguli <emph type="italics"/>D, E, F<emph.end type="italics"/> trianguli <emph type="italics"/>abc<emph.end type="italics"/> latera <lb/><emph type="italics"/>ab, ac, bc<emph.end type="italics"/> re&longs;pective, compleri pote&longs;t Figura <emph type="italics"/>ABCdef<emph.end type="italics"/> Figuræ <lb/><emph type="italics"/>abc DEF<emph.end type="italics"/> &longs;imilis & æqualis, atque eam complendo &longs;olvetur Pro­<lb/>blema. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s> </p> <pb pagenum="91"/> <figure id="fig61"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Hinc recta duci pote&longs;t cujus partes longitudine datæ rectis <lb/> <arrow.to.target n="note67"></arrow.to.target><lb/>tribus po&longs;itione datis interjacebunt. </s> <s>Concipe Triangulum <emph type="italics"/>DEF,<emph.end type="italics"/><lb/>puncto <emph type="italics"/>D<emph.end type="italics"/> ad latus <emph type="italics"/>EF<emph.end type="italics"/> accedente, & lateribus <emph type="italics"/>DE, DF<emph.end type="italics"/> in di­<lb/>rectum po&longs;itis, mutari in lineam rectam, cujus pars data <emph type="italics"/>DE<emph.end type="italics"/> rec­<lb/>tis po&longs;itione datis <emph type="italics"/>AB, AC,<emph.end type="italics"/> & pars data <emph type="italics"/>DF<emph.end type="italics"/> rectis po&longs;itione da­<lb/>tis <emph type="italics"/>AB, BC<emph.end type="italics"/> interponi debet; & applicando con&longs;tructionem præ­<lb/>cedentem ad hunc ca&longs;um &longs;olvetur Problema. </s> </p> <p type="margin"> <s><margin.target id="note67"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVIII. PROBLEMA XX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam &longs;pecie & magnitudine datam de&longs;cribere, cujus partes da­<lb/>tæ rectis tribus po&longs;itione datis interjacebunt.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>De&longs;cribenda &longs;it Trajectoria quæ &longs;it &longs;imilis & æqualis Lineæ cur­<lb/>væ <emph type="italics"/>DEF,<emph.end type="italics"/> quæque a rectis tribus <emph type="italics"/>AB, AC, BC<emph.end type="italics"/> po&longs;itione datis, in <lb/> <arrow.to.target n="fig62"></arrow.to.target><lb/>partes datis hujus partibus <emph type="italics"/>DE<emph.end type="italics"/> & <emph type="italics"/>EF<emph.end type="italics"/> &longs;imiles & æquales &longs;eca­<lb/>bitur. </s> </p> <figure id="fig62"></figure> <p type="main"> <s>Age rectas <emph type="italics"/>DE, EF, DF,<emph.end type="italics"/> & trianguli hujus <emph type="italics"/>DEF<emph.end type="italics"/> pone an­<lb/>los <emph type="italics"/>D, E, F<emph.end type="italics"/> ad rectas illas po&longs;itione datas (per Lem. </s> <s>XXVI) Dein <lb/>circa triangulum de&longs;cribe Trajectoriam Curvæ <emph type="italics"/>DEF<emph.end type="italics"/> &longs;imilem & <lb/>æqualem. <emph type="italics"/>q.E.F.<emph.end type="italics"/> <pb pagenum="92"/> <arrow.to.target n="note68"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note68"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>LEMMA XXVII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Trapezium &longs;pecie datum de&longs;cribere cujus anguli ad rectas quatuor po­<lb/>&longs;itione datas, quæ neque omnes parallelæ &longs;unt, neque ad commune <lb/>punctum convergunt, &longs;inguli ad &longs;ingulas con&longs;i&longs;tent.<emph.end type="italics"/></s> </p> <p type="main"> <s>Dentur po&longs;itione rectæ quatuor <emph type="italics"/>ABC, AD, BD, CE,<emph.end type="italics"/> qua­<lb/>rum prima &longs;ecet &longs;ecundam in <emph type="italics"/>A,<emph.end type="italics"/> tertiam in <emph type="italics"/>B,<emph.end type="italics"/> & quartam in <emph type="italics"/>C:<emph.end type="italics"/><lb/>& de&longs;cribendum &longs;it Trapezium <emph type="italics"/>fghi<emph.end type="italics"/> quod &longs;it Trapezio <emph type="italics"/>FGHI<emph.end type="italics"/><lb/> <arrow.to.target n="fig63"></arrow.to.target><lb/>&longs;imile, & cujus angulus <emph type="italics"/>f,<emph.end type="italics"/> angulo dato <emph type="italics"/>F<emph.end type="italics"/> æqualis, tangat rectam <lb/><emph type="italics"/>ABC,<emph.end type="italics"/> cæterique anguli <emph type="italics"/>g, h, i,<emph.end type="italics"/> cæteris angulis datis <emph type="italics"/>G, H, I<emph.end type="italics"/> æqua­<lb/>les, tangant cæteras lineas <emph type="italics"/>AD, BD, CE<emph.end type="italics"/> re&longs;pective. </s> <s>Jungatur <lb/><emph type="italics"/>FH<emph.end type="italics"/> & &longs;uper <emph type="italics"/>FG, FH, FI<emph.end type="italics"/> de&longs;cribantur totidem circulorum &longs;eg­<lb/>menta <emph type="italics"/>FSG, FTH, FVI<emph.end type="italics"/>; quorum primum <emph type="italics"/>FSG<emph.end type="italics"/> capiat angu- <pb pagenum="93"/>lum æqualem angulo <emph type="italics"/>BAD,<emph.end type="italics"/> &longs;ecundum <emph type="italics"/>FTH<emph.end type="italics"/> capiat angulum æ­<lb/> <arrow.to.target n="note69"></arrow.to.target><lb/>qualem angulo <emph type="italics"/>CBD,<emph.end type="italics"/> ac tertium <emph type="italics"/>FVI<emph.end type="italics"/> capiat angulum æqualem <lb/>angulo <emph type="italics"/>ACE.<emph.end type="italics"/> De&longs;cribi autem debent &longs;egmenta ad eas partes li­<lb/>nearum <emph type="italics"/>FG, FH, FI,<emph.end type="italics"/> ut literarum <emph type="italics"/>FSGF<emph.end type="italics"/> idem &longs;it ordo circula­<lb/>ris qui literarum <emph type="italics"/>BADB,<emph.end type="italics"/> utque literæ <emph type="italics"/>FTHF<emph.end type="italics"/> eodem ordine cum <lb/>literis <emph type="italics"/>CBDC,<emph.end type="italics"/> & literæ <emph type="italics"/>FVIF<emph.end type="italics"/> eodem cum literis <emph type="italics"/>ACEA<emph.end type="italics"/> in or­<lb/>bem redeant. </s> <s>Compleantur &longs;egmenta in circulos integros, &longs;itque <emph type="italics"/>P<emph.end type="italics"/><lb/>centrum circuli primi <emph type="italics"/>FSG,<emph.end type="italics"/> & <emph type="italics"/>Q<emph.end type="italics"/> centrum &longs;ecundi <emph type="italics"/>FTH.<emph.end type="italics"/> Jungatur <lb/>& utrinque producatur <emph type="italics"/>PQ,<emph.end type="italics"/> & in ea capiatur <emph type="italics"/>QR<emph.end type="italics"/> in ea ratione ad <lb/><emph type="italics"/>PQ<emph.end type="italics"/> quam habet <emph type="italics"/>BC<emph.end type="italics"/> ad <emph type="italics"/>AB.<emph.end type="italics"/> Capiatur autem <emph type="italics"/>QR<emph.end type="italics"/> ad eas partes <lb/>puncti <emph type="italics"/>Q<emph.end type="italics"/> ut literarum <emph type="italics"/>P, Q, R<emph.end type="italics"/> idem &longs;it ordo atque literarum <lb/><emph type="italics"/>A, B, C:<emph.end type="italics"/> centroque <emph type="italics"/>R<emph.end type="italics"/> & intervallo <emph type="italics"/>RF<emph.end type="italics"/> de&longs;cribatur circulus quartus <lb/><emph type="italics"/>FNc<emph.end type="italics"/> &longs;ecans circulum tertium <emph type="italics"/>FVI<emph.end type="italics"/> in <emph type="italics"/>c.<emph.end type="italics"/> Jungatur <emph type="italics"/>Fc<emph.end type="italics"/> &longs;ecans <lb/>circulum primum in <emph type="italics"/>a<emph.end type="italics"/> & &longs;ecundum in <emph type="italics"/>b.<emph.end type="italics"/> Agantur <emph type="italics"/>a G, b H, c I,<emph.end type="italics"/> & <lb/>Figuræ <emph type="italics"/>abc FGHI<emph.end type="italics"/> &longs;imilis con&longs;tituatur Figura <emph type="italics"/>ABCfghi:<emph.end type="italics"/> Eritque <lb/>Trapezium <emph type="italics"/>fghi<emph.end type="italics"/> illud ip&longs;um quod con&longs;tituere oportebat. </s> </p> <p type="margin"> <s><margin.target id="note69"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig63"></figure> <p type="main"> <s>Secent enim circuli duo primi <emph type="italics"/>FSG, FTH<emph.end type="italics"/> &longs;e mutuo in <emph type="italics"/>K.<emph.end type="italics"/><lb/>Jungantur <emph type="italics"/>PK, QK, RK, a K, b K, c K,<emph.end type="italics"/> & producatur <emph type="italics"/>QP<emph.end type="italics"/> ad <emph type="italics"/>L.<emph.end type="italics"/><lb/>Anguli ad circumferentias <emph type="italics"/>FaK, FbK, FcK<emph.end type="italics"/> &longs;unt &longs;emi&longs;&longs;es an­<lb/>gulorum <emph type="italics"/>FPK, FQK, FRK<emph.end type="italics"/> ad centra, adeoque angulorum <lb/>illorum dimidiis <emph type="italics"/>LPK, LQK, LRK<emph.end type="italics"/> æquales. </s> <s>E&longs;t ergo Figura <lb/><emph type="italics"/>PQRK<emph.end type="italics"/> Figuræ <emph type="italics"/>abcK<emph.end type="italics"/> æquiangula & &longs;imilis, & propterea <emph type="italics"/>ab<emph.end type="italics"/> e&longs;t <lb/>ad <emph type="italics"/>bc<emph.end type="italics"/> ut <emph type="italics"/>PQ<emph.end type="italics"/> ad <emph type="italics"/>QR,<emph.end type="italics"/> id e&longs;t, ut <emph type="italics"/>AB<emph.end type="italics"/> ad <emph type="italics"/>BC.<emph.end type="italics"/> Angulis in&longs;uper <emph type="italics"/>FaG, <lb/>FbH, FcI<emph.end type="italics"/> æquantur <emph type="italics"/>fAg, fBh, fCi<emph.end type="italics"/> per con&longs;tructionem. </s> <s>Er­<lb/>go Figuræ <emph type="italics"/>abcFGHI<emph.end type="italics"/> Figura &longs;imilis <emph type="italics"/>ABCfghi<emph.end type="italics"/> compleri pote&longs;t <lb/>Quo facto Trapezium <emph type="italics"/>fghi<emph.end type="italics"/> con&longs;tituetur &longs;imile Trapezio <emph type="italics"/>FGHI<emph.end type="italics"/><lb/>& angulis &longs;uis <emph type="italics"/>f, g, h, i<emph.end type="italics"/> tanget rectas <emph type="italics"/>ABC, AD, BD, CE <lb/>q.E.F.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Hinc recta duci pote&longs;t cujus partes, rectis quatuor po&longs;i­<lb/>tione datis dato ordine interjectæ, datam habebunt proportionem <lb/>ad invicem. </s> <s>Augeantur anguli <emph type="italics"/>FGH, GHI<emph.end type="italics"/> u&longs;que eo, ut rectæ <emph type="italics"/>FG, <lb/>GH, HI<emph.end type="italics"/> in directum jaceant, & in hoc ca&longs;u con&longs;truendo Proble­<lb/>ma, ducetur recta <emph type="italics"/>fghi<emph.end type="italics"/> cujus partes <emph type="italics"/>fg, gh, hi,<emph.end type="italics"/> rectis quatuor po­<lb/>&longs;itione datis <emph type="italics"/>AB<emph.end type="italics"/> & <emph type="italics"/>AD, AD<emph.end type="italics"/> & <emph type="italics"/>BD, BD<emph.end type="italics"/> & <emph type="italics"/>CE<emph.end type="italics"/> interjectæ, e­<lb/>runt ad invicem ut lineæ <emph type="italics"/>FG, GH, HI,<emph.end type="italics"/> eundemque &longs;ervabunt or­<lb/>dinem inter &longs;e. </s> <s>Idem vero &longs;ic fit expeditius. <pb pagenum="94"/> <arrow.to.target n="note70"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note70"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s>Producantur <emph type="italics"/>AB<emph.end type="italics"/> ad <emph type="italics"/>K,<emph.end type="italics"/> & <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>L,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>BK<emph.end type="italics"/> ad <emph type="italics"/>AB<emph.end type="italics"/> ut <lb/><emph type="italics"/>HI<emph.end type="italics"/> ad <emph type="italics"/>GH<emph.end type="italics"/>; & <emph type="italics"/>DL<emph.end type="italics"/> ad <emph type="italics"/>BD<emph.end type="italics"/> ut <emph type="italics"/>GI<emph.end type="italics"/> ad <emph type="italics"/>FG<emph.end type="italics"/>; & jungatur <emph type="italics"/>KL<emph.end type="italics"/><lb/>occurrens rectæ <emph type="italics"/>CE<emph.end type="italics"/> in <emph type="italics"/>i.<emph.end type="italics"/> Producatur <emph type="italics"/>iL<emph.end type="italics"/> ad <emph type="italics"/>M,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>LM<emph.end type="italics"/> ad <emph type="italics"/>iL<emph.end type="italics"/><lb/>ut <emph type="italics"/>GH<emph.end type="italics"/> ad <emph type="italics"/>HI,<emph.end type="italics"/> & agatur tum <emph type="italics"/>MQ<emph.end type="italics"/> ip&longs;i <emph type="italics"/>LB<emph.end type="italics"/> parallela rectæque <lb/><emph type="italics"/>AD<emph.end type="italics"/> occurrens in <emph type="italics"/>g,<emph.end type="italics"/> tum <emph type="italics"/>gi<emph.end type="italics"/> &longs;ecans <emph type="italics"/>AB, BD<emph.end type="italics"/> in <emph type="italics"/>f, h.<emph.end type="italics"/> Dico <lb/>factum. </s> </p> <p type="main"> <s>Secet enim <emph type="italics"/>Mg<emph.end type="italics"/> rectam <emph type="italics"/>AB<emph.end type="italics"/> in <emph type="italics"/>Q,<emph.end type="italics"/> & <emph type="italics"/>AD<emph.end type="italics"/> rectam <emph type="italics"/>KL<emph.end type="italics"/> in <emph type="italics"/>S,<emph.end type="italics"/> & <lb/>agatur <emph type="italics"/>AP<emph.end type="italics"/> quæ &longs;it ip&longs;i <emph type="italics"/>BD<emph.end type="italics"/> parallela & occurrat <emph type="italics"/>iL<emph.end type="italics"/> in <emph type="italics"/>P,<emph.end type="italics"/> & <lb/>erunt <emph type="italics"/>gM<emph.end type="italics"/> ad <emph type="italics"/>Lh (gi<emph.end type="italics"/> ad <emph type="italics"/>hi, Mi<emph.end type="italics"/> ad <emph type="italics"/>Li, GI<emph.end type="italics"/> ad <emph type="italics"/>HI, AK<emph.end type="italics"/> ad <lb/><emph type="italics"/>BK<emph.end type="italics"/>) & <emph type="italics"/>AP<emph.end type="italics"/> ad <emph type="italics"/>BL<emph.end type="italics"/> in eadem ratione. </s> <s>Secetur <emph type="italics"/>DL<emph.end type="italics"/> in <emph type="italics"/>R<emph.end type="italics"/> ut &longs;it <lb/> <arrow.to.target n="fig64"></arrow.to.target><lb/><emph type="italics"/>DL<emph.end type="italics"/> ad <emph type="italics"/>RL<emph.end type="italics"/> in eadem illa ratione, & ob proportionales <emph type="italics"/>gS<emph.end type="italics"/> ad <lb/><emph type="italics"/>gM, AS<emph.end type="italics"/> ad <emph type="italics"/>AP,<emph.end type="italics"/> & <emph type="italics"/>DS<emph.end type="italics"/> ad <emph type="italics"/>DL<emph.end type="italics"/>; erit, ex æquo, ut <emph type="italics"/>gS<emph.end type="italics"/> ad <emph type="italics"/>Lh<emph.end type="italics"/> ita <lb/><emph type="italics"/>AS<emph.end type="italics"/> ad <emph type="italics"/>BL<emph.end type="italics"/> & <emph type="italics"/>DS<emph.end type="italics"/> ad <emph type="italics"/>RL<emph.end type="italics"/>; & mixtim, <emph type="italics"/>BL-RL<emph.end type="italics"/> ad <emph type="italics"/>Lh-BL<emph.end type="italics"/><lb/>ut <emph type="italics"/>AS-DS<emph.end type="italics"/> ad <emph type="italics"/>gS-AS.<emph.end type="italics"/> Id e&longs;t <emph type="italics"/>BR<emph.end type="italics"/> ad <emph type="italics"/>Bh<emph.end type="italics"/> ut <emph type="italics"/>AD<emph.end type="italics"/> ad <emph type="italics"/>Ag<emph.end type="italics"/> ad­<lb/>eoque ut <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/><expan abbr="gq.">gque</expan><emph.end type="italics"/> Et vici&longs;&longs;im <emph type="italics"/>BR<emph.end type="italics"/> ad <emph type="italics"/>BD<emph.end type="italics"/> ut <emph type="italics"/>Bh<emph.end type="italics"/> ad <emph type="italics"/>gQ,<emph.end type="italics"/> &longs;eu <lb/><emph type="italics"/>fh<emph.end type="italics"/> ad <emph type="italics"/>fg.<emph.end type="italics"/> Sed ex con&longs;tructione linea <emph type="italics"/>RL<emph.end type="italics"/> eadem ratione &longs;ecta fuit <lb/>in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>R<emph.end type="italics"/> atque linea <emph type="italics"/>FI<emph.end type="italics"/> in <emph type="italics"/>G<emph.end type="italics"/> & <emph type="italics"/>H:<emph.end type="italics"/> ideoque e&longs;t <emph type="italics"/>BR<emph.end type="italics"/> ad <emph type="italics"/>BD<emph.end type="italics"/><lb/>ut <emph type="italics"/>FH<emph.end type="italics"/> ad <emph type="italics"/>FG.<emph.end type="italics"/> Ergo <emph type="italics"/>fh<emph.end type="italics"/> e&longs;t ad <emph type="italics"/>fg<emph.end type="italics"/> ut <emph type="italics"/>FH<emph.end type="italics"/> ad <emph type="italics"/>FG.<emph.end type="italics"/> Cum igitur <lb/>&longs;it etiam <emph type="italics"/>gi<emph.end type="italics"/> ad <emph type="italics"/>hi<emph.end type="italics"/> ut <emph type="italics"/>Mi<emph.end type="italics"/> ad <emph type="italics"/>Li,<emph.end type="italics"/> id e&longs;t, ut <emph type="italics"/>GI<emph.end type="italics"/> ad <emph type="italics"/>HI,<emph.end type="italics"/> patet li­<lb/>neas <emph type="italics"/>FI, fi<emph.end type="italics"/> in <emph type="italics"/>g<emph.end type="italics"/> & <emph type="italics"/>h, G<emph.end type="italics"/> & <emph type="italics"/>H<emph.end type="italics"/> &longs;imiliter &longs;ectas e&longs;&longs;e. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s> </p> <pb pagenum="95"/> <figure id="fig64"></figure> <p type="main"> <s>In con&longs;tructione Corollarii hujus po&longs;tquam ducitur <emph type="italics"/>LK<emph.end type="italics"/> &longs;ecans </s> </p> <p type="main"> <s> <arrow.to.target n="note71"></arrow.to.target><lb/><emph type="italics"/>CE<emph.end type="italics"/> in <emph type="italics"/>i,<emph.end type="italics"/> producere licet <emph type="italics"/>iE<emph.end type="italics"/> ad <emph type="italics"/>V,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>EV<emph.end type="italics"/> ad <emph type="italics"/>Ei<emph.end type="italics"/> ut <emph type="italics"/>FH<emph.end type="italics"/> ad <emph type="italics"/>HI,<emph.end type="italics"/><lb/>& agere <emph type="italics"/>Vf<emph.end type="italics"/> parallelam ip&longs;i <emph type="italics"/>BD.<emph.end type="italics"/> Eodem recidit &longs;i centro <emph type="italics"/>i,<emph.end type="italics"/> in­<lb/>tervallo <emph type="italics"/>IH,<emph.end type="italics"/> de&longs;cribatur circulus &longs;ecans <emph type="italics"/>BD<emph.end type="italics"/> in <emph type="italics"/>X,<emph.end type="italics"/> & producatur <lb/><emph type="italics"/>iX<emph.end type="italics"/> ad <emph type="italics"/>Y,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>iY<emph.end type="italics"/> æqualis <emph type="italics"/>IF,<emph.end type="italics"/> & agatur <emph type="italics"/>Yf<emph.end type="italics"/> ip&longs;i <emph type="italics"/>BD<emph.end type="italics"/> parallela. </s> </p> <p type="margin"> <s><margin.target id="note71"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s>Problematis hujus &longs;olutiones alias <emph type="italics"/>Wrennus<emph.end type="italics"/> & <emph type="italics"/>Walli&longs;ius<emph.end type="italics"/> olim ex­<lb/>cogitarunt. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIX. PROBLEMA XXI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam &longs;pecie datam de&longs;cribere, quæ a rectis quatuor po&longs;itione <lb/>datis in partes &longs;ecabitur, ordine, &longs;pecie & proportione datas.<emph.end type="italics"/><emph.end type="center"/></s> </p> <figure></figure> <p type="main"> <s>De&longs;cribenda &longs;it Trajectoria <lb/> <arrow.to.target n="fig65"></arrow.to.target><lb/><emph type="italics"/>fghi,<emph.end type="italics"/> quæ &longs;imilis &longs;it Lincæ curvæ <lb/><emph type="italics"/>FGHI,<emph.end type="italics"/> & cujus partes <emph type="italics"/>fg, gh, hi<emph.end type="italics"/><lb/>illius partibus <emph type="italics"/>FG, GH, HI<emph.end type="italics"/> &longs;i­<lb/>miles & proportionales, rectis <lb/><emph type="italics"/>AB<emph.end type="italics"/> & <emph type="italics"/>AD, AD<emph.end type="italics"/> & <emph type="italics"/>BD, BD<emph.end type="italics"/><lb/>& <emph type="italics"/>CE<emph.end type="italics"/> po&longs;itione datis, prima pri­<lb/>mis, &longs;ecunda &longs;ecundis, tertia ter­<lb/>tiis interjaceant. </s> <s>Actis rectis <emph type="italics"/>FG, <lb/>GH, HI, FI,<emph.end type="italics"/> de&longs;cribatur (per <lb/>Lem. </s> <s>XXVII.) Trapezium <emph type="italics"/>fghi<emph.end type="italics"/><lb/>quod &longs;it Trapezio <emph type="italics"/>FGHI<emph.end type="italics"/> &longs;imile & cujus anguli <emph type="italics"/>f, g, h, i<emph.end type="italics"/> tangant <lb/>rectas illas po&longs;itione datas <emph type="italics"/>AB, AD, BD, CE,<emph.end type="italics"/> &longs;inguli &longs;ingulas <lb/>dicto ordine. </s> <s>Dein circa hoc Trapezium de&longs;cribatur Trajectoria <lb/>curvæ Lineæ <emph type="italics"/>FGHI<emph.end type="italics"/> con&longs;imilis. <pb pagenum="96"/> <arrow.to.target n="note72"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note72"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig65"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Con&longs;trui etiam pote&longs;t hoc Problema ut &longs;equitur. </s> <s>Junctis <emph type="italics"/>FG, <lb/>GH, HI, FI<emph.end type="italics"/> produc <emph type="italics"/>GF<emph.end type="italics"/> ad <emph type="italics"/>V,<emph.end type="italics"/> jungeque <emph type="italics"/>FH, IG,<emph.end type="italics"/> & angulis <lb/><emph type="italics"/>FGH, VFH<emph.end type="italics"/> fac angulos <emph type="italics"/>CAK, DAL<emph.end type="italics"/> æquales. </s> <s>Concurrant <lb/><emph type="italics"/>AK, AL<emph.end type="italics"/> cum recta <emph type="italics"/>BD<emph.end type="italics"/> in <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>L,<emph.end type="italics"/> & inde agantur <emph type="italics"/>KM, LN,<emph.end type="italics"/><lb/>quarum <emph type="italics"/>KM<emph.end type="italics"/> con&longs;tituat angulum <emph type="italics"/>AKM<emph.end type="italics"/> æqualem angulo <emph type="italics"/>GHI,<emph.end type="italics"/><lb/>&longs;itque ad <emph type="italics"/>AK<emph.end type="italics"/> ut e&longs;t <emph type="italics"/>HI<emph.end type="italics"/> ad <emph type="italics"/>GH<emph.end type="italics"/>; & <emph type="italics"/>LN<emph.end type="italics"/> con&longs;tituat angulum <lb/><emph type="italics"/>ALN<emph.end type="italics"/> æqualem angulo <emph type="italics"/>FHI,<emph.end type="italics"/> &longs;itque ad <emph type="italics"/>AL<emph.end type="italics"/> ut <emph type="italics"/>HI<emph.end type="italics"/> ad <emph type="italics"/>FH.<emph.end type="italics"/> Du­<lb/>cantur autem <emph type="italics"/>AK, KM, AL, LN<emph.end type="italics"/> ad eas partes linearum <emph type="italics"/>AD, <lb/>AK, AL,<emph.end type="italics"/> ut literæ <emph type="italics"/>CAKMC, ALKA, DALND<emph.end type="italics"/> eodem <lb/>ordine cum literis <emph type="italics"/>FGHIF<emph.end type="italics"/> in orbem redeant; & act <emph type="italics"/>MN<emph.end type="italics"/> oc­<lb/>currat rectæ <emph type="italics"/>CE<emph.end type="italics"/> in <emph type="italics"/>i.<emph.end type="italics"/> Fac angulum <emph type="italics"/>iEP<emph.end type="italics"/> æqualem angulo <emph type="italics"/>IGF,<emph.end type="italics"/><lb/> <arrow.to.target n="fig66"></arrow.to.target><lb/>&longs;itque <emph type="italics"/>PE<emph.end type="italics"/> ad <emph type="italics"/>Ei<emph.end type="italics"/> ut <emph type="italics"/>FG<emph.end type="italics"/> ad <emph type="italics"/>GI;<emph.end type="italics"/> & per <emph type="italics"/>P<emph.end type="italics"/> agatur <emph type="italics"/>PQf,<emph.end type="italics"/> quæ <lb/>cum recta <emph type="italics"/>ADE<emph.end type="italics"/> contineat angulum <emph type="italics"/>PQE<emph.end type="italics"/> æqualem angulo <lb/><emph type="italics"/>FIG,<emph.end type="italics"/> rectæque <emph type="italics"/>AB<emph.end type="italics"/> occurrat in <emph type="italics"/>f,<emph.end type="italics"/> & jungatur <emph type="italics"/>fi.<emph.end type="italics"/> Agantur au­<lb/>rem <emph type="italics"/>PE<emph.end type="italics"/> & <emph type="italics"/>PQ<emph.end type="italics"/> ad eas partes linearum <emph type="italics"/>CE, PE,<emph.end type="italics"/> ut literarum <lb/><emph type="italics"/>PEiP<emph.end type="italics"/> & <emph type="italics"/>PEQP<emph.end type="italics"/> idem &longs;it ordo circularis qui literarum <emph type="italics"/>FGHIF,<emph.end type="italics"/><lb/>& &longs;i &longs;uper linea <emph type="italics"/>fi<emph.end type="italics"/> eodem quoque literarum ordine con&longs;tituatur <lb/>Trapezium <emph type="italics"/>fghi<emph.end type="italics"/> Trapezio <emph type="italics"/>FGHI<emph.end type="italics"/> &longs;imile, & circum&longs;cribatur Tra­<lb/>jectoria &longs;pecie data, &longs;olvetur Problema. </s> </p> <figure id="fig66"></figure> <p type="main"> <s>Hactenus de Orbibus inveniendis. </s> <s>Supere&longs;t ut Motus corpo­<lb/>rum in Orbibus inventis determinemus. <pb pagenum="97"/> <arrow.to.target n="note73"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note73"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>SECTIO VI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Inventione Motuum in Orbibus datis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXX. PROBLEMA XXII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corporis in data Trajectoria Parabolica moti invenire locum ad <lb/>tempus a&longs;&longs;ignatum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Sit <emph type="italics"/>S<emph.end type="italics"/> umbilicus & <emph type="italics"/>A<emph.end type="italics"/> vertex principa­<lb/> <arrow.to.target n="fig67"></arrow.to.target><lb/>lis Parabolæ, &longs;itque 4 <emph type="italics"/>ASXM<emph.end type="italics"/> æquale <lb/>areæ Parabolicæ ab&longs;cindendæ <emph type="italics"/>APS,<emph.end type="italics"/><lb/>quæ radio <emph type="italics"/>SP,<emph.end type="italics"/> vel po&longs;t exce&longs;&longs;um cor­<lb/>poris de vertice de&longs;cripta fuit, vel an­<lb/>te appul&longs;um ejus ad verticem de&longs;cri­<lb/>benda e&longs;t. </s> <s>Innote&longs;cit quantitas areæ il­<lb/>lius ab&longs;cindendæ ex tempore ip&longs;i pro­<lb/>portionali. </s> <s>Bi&longs;eca <emph type="italics"/>AS<emph.end type="italics"/> in <emph type="italics"/>G,<emph.end type="italics"/> erigeque <lb/>perpendiculum <emph type="italics"/>GH<emph.end type="italics"/> æquale 3 M, & <lb/>Circulus centro <emph type="italics"/>H,<emph.end type="italics"/> intervallo <emph type="italics"/>HS<emph.end type="italics"/><lb/>de&longs;criptus &longs;ecabit Parabolam in loco <lb/>quæ&longs;ito <emph type="italics"/>P.<emph.end type="italics"/> Nam, demi&longs;&longs;a ad axem <lb/>perpendiculari <emph type="italics"/>PO<emph.end type="italics"/> & ducta <emph type="italics"/>PH,<emph.end type="italics"/> e&longs;t <lb/><emph type="italics"/>AGq+GHq (=HP q=―AO-AG: quad.+―PO-GH: quad.)= <lb/>AOq+POq-2 <expan abbr="GAO-2GHXPO+AGq+GHq.">GAO-2GHXPO+AGq+GHque</expan><emph.end type="italics"/> Unde <lb/>2 <emph type="italics"/>GHXPO (=AOq+POq-2GAO)=AOq+1/4 <expan abbr="POq.">POque</expan><emph.end type="italics"/><lb/>Pro <emph type="italics"/>AOq<emph.end type="italics"/> &longs;cribe (<emph type="italics"/>AOXPOq/4AS<emph.end type="italics"/>); &, applicatis terminis omnibus ad <lb/>3<emph type="italics"/>PO<emph.end type="italics"/> ducti&longs;que in 2<emph type="italics"/>AS,<emph.end type="italics"/> fiet 4/3 <emph type="italics"/>GHXAS(=1/6AOXPO+1/2 ASXPO <lb/>=(AO+3AS/6)XPO=(4AO-3SO/6)XPO<emph.end type="italics"/>=areæ ―<emph type="italics"/>APO-SPO)<emph.end type="italics"/><lb/>=areæ <emph type="italics"/>APS.<emph.end type="italics"/> Sed <emph type="italics"/>GH<emph.end type="italics"/> erat 3 M, & inde 4/3 <emph type="italics"/>GHXAS<emph.end type="italics"/> e&longs;t 4 <emph type="italics"/>AS<emph.end type="italics"/>XM. </s> <s><lb/>Ergo area ab&longs;ci&longs;&longs;a <emph type="italics"/>APS<emph.end type="italics"/> æqualis e&longs;t ab&longs;cindendæ 4<emph type="italics"/>ASXM. q.E.D.<emph.end type="italics"/></s> </p> <figure id="fig67"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc <emph type="italics"/>GH<emph.end type="italics"/> e&longs;t ad <emph type="italics"/>AS,<emph.end type="italics"/> ut tempus quo corpùs de&longs;crip­<lb/>&longs;it arcum <emph type="italics"/>AP<emph.end type="italics"/> ad tempus quo corpus de&longs;crip&longs;it arcum inter verti­<lb/>cem <emph type="italics"/>A<emph.end type="italics"/> & perpendiculum ad axem ab umbilico <emph type="italics"/>S<emph.end type="italics"/> erectum. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et Circulo <emph type="italics"/>ASP<emph.end type="italics"/> per corpus motum <emph type="italics"/>P<emph.end type="italics"/> perpetuo tran&longs;­<lb/>eunte, velocitas puncti <emph type="italics"/>H<emph.end type="italics"/> e&longs;t ad velocitatem quam corpus habuit <pb pagenum="98"/> <arrow.to.target n="note74"></arrow.to.target><lb/>in vertice <emph type="italics"/>A,<emph.end type="italics"/> ut 3 ad 8; adeoque in ea etiam ratione e&longs;t linea <emph type="italics"/>GH<emph.end type="italics"/><lb/>ad lineam rectam quam corpus tempore motus &longs;ui ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>P,<emph.end type="italics"/> ea <lb/>cum velocitate quam habuit in vertice <emph type="italics"/>A,<emph.end type="italics"/> de&longs;cribere po&longs;&longs;et. </s> </p> <p type="margin"> <s><margin.target id="note74"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Hinc etiam vice ver&longs;a inveniri pote&longs;t tempus quo cor­<lb/>pus de&longs;crip&longs;it arcum quemvis a&longs;&longs;ignatum <emph type="italics"/>AP.<emph.end type="italics"/> Junge <emph type="italics"/>AP<emph.end type="italics"/> & ad <lb/>medium ejus punctum erige perpendiculum rectæ <emph type="italics"/>GH<emph.end type="italics"/> occur­<lb/>rens in <emph type="italics"/>H.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>LEMMA XXVIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Nulla extat Figura Ovalis cujus area, rectis pro lubitu ab&longs;ci&longs;&longs;a, po&longs;&longs;it <lb/>per æquationes numero terminorum ac dimen&longs;ionum finitas genera­<lb/>liter inveniri.<emph.end type="italics"/></s> </p> <p type="main"> <s>Intra Ovalem detur punctum quodvis, circa quod ceu polum re­<lb/>volvatur perpetuo linea recta, uniformi cum motu, & interea in rec­<lb/>ta illa exeat punctum mobile de polo, pergatque &longs;emper ea cum <lb/>velocitate, quæ &longs;it ut rectæ illius intra Ovalem quadratum. </s> <s>Hoc <lb/>motu punctum illud de&longs;cribet Spiralem gyris infinitis. </s> <s>Jam &longs;i areæ <lb/>Ovalis a recta illa ab&longs;ci&longs;&longs;æ incrementum per finitam æquationem <lb/>inveniri pote&longs;t, invenietur etiam per eandem æquationem di&longs;tantia <lb/>puncti a polo, quæ huic areæ proportionalis e&longs;t, adeoque om­<lb/>nia Spiralis puncta per æquationem finitam inveniri po&longs;&longs;unt: & <lb/>propterea rectæ cuju&longs;vis po&longs;itione datæ inter&longs;ectio cum Spirali in­<lb/>veniri etiam pote&longs;t per æquationem finitam. </s> <s>Atqui recta omnis <lb/>infinite producta Spiralem &longs;ecat in punctis numero infinitis, & æqua­<lb/>tio, qua inter&longs;ectio aliqua duarum linearum invenitur, exhibet ea­<lb/>rum inter&longs;ectiones omnes radicibus totidem, adeoque a&longs;cendit ad <lb/>rot dimen&longs;iones quot &longs;unt inter&longs;ectiones. </s> <s>Quoniam Circuli duo &longs;e <lb/>mutuo &longs;ecant in punctis duobus, inter&longs;ectio una non invenietur <lb/>ni&longs;i per æquationem duarum dimen&longs;ionum, qua inter&longs;ectio altera <lb/>etiam inveniatur. </s> <s>Quoniam duarum &longs;ectionum Conicarum quatuor <lb/>e&longs;&longs;e po&longs;&longs;unt inter&longs;ectiones, non pote&longs;t aliqua earum generaliter in­<lb/>veniri ni&longs;i per æquationem quatuor dimen&longs;ionum, qua omnes &longs;i­<lb/>mul inveniantur. </s> <s>Nam &longs;i inter&longs;ectiones illæ &longs;eor&longs;im quærantur, quo­<lb/>niam eadem e&longs;t omnium lex & conditio, idem erit calculus in ca&longs;u <lb/>unoquoque & propterea eadem &longs;emper conclu&longs;io, quæ igitur de­<lb/>bet omnes inter&longs;ectiones &longs;imul complecti & indifferenter exhibere. <pb pagenum="99"/>Unde etiam inter&longs;ectiones Sectionum Conicarum & Curvarum ter­<lb/> <arrow.to.target n="note75"></arrow.to.target><lb/>tiæ pote&longs;tatis, eo quod &longs;ex e&longs;&longs;e po&longs;&longs;unt, &longs;imul prodeunt per æqua­<lb/>tiones &longs;ex dimen&longs;ionum, & inter&longs;ectiones duarum Curvarum tertiæ <lb/>pote&longs;tatis, quia novem e&longs;&longs;e po&longs;&longs;unt, &longs;imul prodeunt per æqua­<lb/>tiones dimen&longs;ionum novem. </s> <s>Id ni&longs;i nece&longs;&longs;ario fieret, reducere licc­<lb/>ret Problemata omnia Solida ad Plana, & plu&longs;quam Solida ad Soli­<lb/>da. </s> <s>Loquor hic de Curvis pote&longs;tate irreducibilibus. </s> <s>Nam &longs;i æqua­<lb/>tio per quam Curva definitur, ad inferiorem pote&longs;tatem reduci <lb/>po&longs;&longs;it: Curva non erit unica, &longs;ed ex duabus vel pluribus compo&longs;i­<lb/>ta, quarum inter&longs;ectiones per calculos diver&longs;os &longs;eor&longs;im inveniri <lb/>po&longs;&longs;unt. </s> <s>Ad eundem modum inter&longs;ectiones binæ rectarum & &longs;ecti­<lb/>onum Conicarum prodeunt &longs;emper per æquationes duarum dimen­<lb/>&longs;ionum; ternæ rectarum & Curvarum irreducibilium tertiæ pote&longs;tatis <lb/>per æquationes trium, quaternæ rectarum & Curvarvm irreducibi­<lb/>lium quartæ pote&longs;tatis per æquationes dimen&longs;ionum quatuor, & &longs;ic <lb/>in infinitum. </s> <s>Ergo rectæ & Spiralis inter&longs;ectiones numero infinitæ, cum <lb/>Curva hæc &longs;it &longs;implex & in Curvas plures irreducibilis, requirunt æ­<lb/>quationes numero dimen&longs;ionum & radicum infinitas, quibus omnes <lb/>po&longs;&longs;unt &longs;imul exhiberi. </s> <s>E&longs;t enim eadem omnium lex & idem calculus. </s> <s><lb/>Nam &longs;i a polo in rectam illam &longs;ecantem demittatur perpendiculum, <lb/>& perpendiculum illud una cum &longs;ecante revolvatur circa polum, in­<lb/>ter&longs;ectiones Spiralis tran&longs;ibunt in &longs;e mutuo, quæque prima erat &longs;eu <lb/>proxima, po&longs;t unam revolutionem &longs;ecunda erit, po&longs;t duas tertia, <lb/>& &longs;ic deinceps: nec interea mutabitur æquatio ni&longs;i pro mutata mag­<lb/>nitudine quantitatum per quas po&longs;itio &longs;ecantis determinatur. </s> <s>Unde <lb/>cum quantitates illæ po&longs;t &longs;ingulas revolutiones redeunt ad magni­<lb/>tudines primas, æquatio redibit ad formam primam, adeoque una <lb/>eademque exhibebit inter&longs;ectiones omnes, & propterea radices ha­<lb/>bebit numero infinitas, quibus omnes exhiberi po&longs;&longs;unt. </s> <s>Nequit <lb/>ergo inter&longs;ectio rectæ & Spiralis per æquationem finitam generali­<lb/>ter inveniri, & idcirco nulla extat Ovalis cujus area, rectis impe­<lb/>ratis ab&longs;ci&longs;&longs;a, po&longs;&longs;it per talem æquationem generaliter exhiberi. </s> </p> <p type="margin"> <s><margin.target id="note75"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s>Eodem argumento, &longs;i intervallum poli & puncti, quo Spiralis de­<lb/>&longs;cribitur, capiatur Ovalis perimetro ab&longs;ci&longs;&longs;æ proportionale, pro­<lb/>bari pote&longs;t quod longitudo perimetri nequit per finitam æquatio­<lb/>nem generaliter exhiberi. </s> <s>De Ovalibus autem hic loquor quæ non <lb/>tanguntur a figuris conjugatis in infinitum pergentibus. <pb pagenum="100"/> <arrow.to.target n="note76"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note76"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Hinc area Ellip&longs;eos, quæ radio ab umbilico ad corpus mobile <lb/>ducto de&longs;cribitur, non prodit ex dato tempore, per æquationem <lb/>finitam; & propterea per de&longs;criptionem Curvarum Geometrice ra­<lb/>tionalium determinari nequit. </s> <s>Curvas Geometrice rationales ap­<lb/>pello quarum puncta omnia per longitudines æquationibus defini­<lb/>tas, id e&longs;t, per longitudinum rationes complicatas, determinari <lb/>po&longs;&longs;unt; cætera&longs;que (ut Spirales, Quadratrices, Trochoides) Geo­<lb/>metrice irrationales. </s> <s>Nam longitudines quæ &longs;unt vel non &longs;unt ut <lb/>numerus ad numerum (quemadmodum in decimo Elementorum) <lb/>&longs;unt Arithmetice rationales vel irrationales. </s> <s>Aream igitur Ellip&longs;eos <lb/>tempori proportionalem ab&longs;cindo per Curvam Geometrice irratio­<lb/>nalem ut &longs;equitur. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXI. PROBLEMA XXIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corporis in data Trajectoria Elliptica moti invenire locum ad <lb/>tempus a&longs;&longs;ignatum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Ellip&longs;eos <emph type="italics"/>APB<emph.end type="italics"/> &longs;it <emph type="italics"/>A<emph.end type="italics"/> vertex principalis, <emph type="italics"/>S<emph.end type="italics"/> umbilicus, & <emph type="italics"/>O<emph.end type="italics"/><lb/>centrum, &longs;itque <emph type="italics"/>P<emph.end type="italics"/> corporis locus inveniendus. </s> <s>Produc <emph type="italics"/>OA<emph.end type="italics"/> ad <emph type="italics"/>G,<emph.end type="italics"/><lb/>ut &longs;it <emph type="italics"/>OG<emph.end type="italics"/> ad <emph type="italics"/>OA<emph.end type="italics"/> ut <emph type="italics"/>OA<emph.end type="italics"/> ad <emph type="italics"/>OS.<emph.end type="italics"/> Erige perpendiculum <emph type="italics"/>GH,<emph.end type="italics"/> centroque <lb/> <arrow.to.target n="fig68"></arrow.to.target><lb/><emph type="italics"/>O<emph.end type="italics"/> & intervallo <emph type="italics"/>OG<emph.end type="italics"/> de&longs;cribe circulum <emph type="italics"/>EFG,<emph.end type="italics"/> & &longs;uper regula <emph type="italics"/>GH,<emph.end type="italics"/><lb/>ceu fundo, progrediatur Rota <emph type="italics"/>GEF<emph.end type="italics"/> revolvendo circa axem <lb/>&longs;uum, & interea puncto &longs;uo <emph type="italics"/>A<emph.end type="italics"/> de&longs;cribendo Trochoidem <emph type="italics"/>ALI.<emph.end type="italics"/> <pb pagenum="101"/>Quo facto, cape <emph type="italics"/>GK<emph.end type="italics"/> in ratione ad Rotæ perimetrum <emph type="italics"/>GEFG,<emph.end type="italics"/> ut <lb/> <arrow.to.target n="note77"></arrow.to.target><lb/>e&longs;t tempus quo corpus progrediendo ab <emph type="italics"/>A<emph.end type="italics"/> de&longs;crip&longs;it arcum <emph type="italics"/>AP,<emph.end type="italics"/> ad <lb/>tempus revolutionis unius in Ellip&longs;i. </s> <s>Erigatur perpendiculum <emph type="italics"/>KL<emph.end type="italics"/><lb/>occurrens Trochoidi in <emph type="italics"/>L,<emph.end type="italics"/> & acta <emph type="italics"/>LP<emph.end type="italics"/> ip&longs;i <emph type="italics"/>KG<emph.end type="italics"/> parallela occurret <lb/>Ellip&longs;i in corporis loco quæ&longs;ito <emph type="italics"/>P.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note77"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig68"></figure> <p type="main"> <s>Nam centro <emph type="italics"/>O,<emph.end type="italics"/> intervallo <emph type="italics"/>OA<emph.end type="italics"/> de&longs;cribatur &longs;emicirculus <emph type="italics"/>AQB,<emph.end type="italics"/><lb/>& arcui <emph type="italics"/>AQ<emph.end type="italics"/> occurrat <emph type="italics"/>LP<emph.end type="italics"/> producta in <emph type="italics"/>Q,<emph.end type="italics"/> junganturque <emph type="italics"/>SQ, <expan abbr="Oq.">Oque</expan><emph.end type="italics"/><lb/>Arcui <emph type="italics"/>EFG<emph.end type="italics"/> occurrat <emph type="italics"/>OQ<emph.end type="italics"/> in <emph type="italics"/>F,<emph.end type="italics"/> & in eandem <emph type="italics"/>OQ<emph.end type="italics"/> demittatur per­<lb/>pendiculum <emph type="italics"/>SR.<emph.end type="italics"/> Area <emph type="italics"/>APS<emph.end type="italics"/> e&longs;t ut area <emph type="italics"/>AQS,<emph.end type="italics"/> id e&longs;t, ut diffe­<lb/>rentia inter &longs;ectorem <emph type="italics"/>OQA<emph.end type="italics"/> & triangulum <emph type="italics"/>OQS,<emph.end type="italics"/> &longs;ive ut differen­<lb/>tia rectangulorum 1/2 <emph type="italics"/>OQXAQ<emph.end type="italics"/> & 1/2 <emph type="italics"/>OQXSR,<emph.end type="italics"/> hoc e&longs;t, ob datam <lb/>1/2 <emph type="italics"/>OQ,<emph.end type="italics"/> ut differentia inter arcum <emph type="italics"/>AQ<emph.end type="italics"/> & rectam <emph type="italics"/>SR,<emph.end type="italics"/> adeoque (ob <lb/>æqualitatem datarum rationum <emph type="italics"/>SR<emph.end type="italics"/> ad &longs;inum arcus <emph type="italics"/>AQ, OS<emph.end type="italics"/> ad <emph type="italics"/>OA, <lb/>OA<emph.end type="italics"/> ad <emph type="italics"/>OG, AQ<emph.end type="italics"/> ad <emph type="italics"/>GF,<emph.end type="italics"/> & divi&longs;im <emph type="italics"/>AQ-SR<emph.end type="italics"/> ad <emph type="italics"/>GF<emph.end type="italics"/>-&longs;in. </s> <s>arc. <emph type="italics"/>AQ<emph.end type="italics"/>) <lb/>ut <emph type="italics"/>GK<emph.end type="italics"/> differentia inter arcum <emph type="italics"/>GF<emph.end type="italics"/> & &longs;inum arcus <emph type="italics"/><expan abbr="Aq.">Aque</expan> <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Cæterum, cum difficilis &longs;it hujus Curvæ de&longs;criptio, præ&longs;tat &longs;olu­<lb/>tionem vero proximam adhibere. </s> <s>Inveniatur tum angulus quidam <lb/>B, qui &longs;it ad angulum graduum 57,29578, quem arcus radio æqualis <lb/>&longs;ubtendit, ut e&longs;t umbilicorum di&longs;tantia <emph type="italics"/>SH<emph.end type="italics"/> ad Ellip&longs;eos diame­<lb/>trum <emph type="italics"/>AB<emph.end type="italics"/>; tum etiam longitudo quædam L, quæ &longs;it ad radium in <lb/>eadem ratione inver&longs;e. </s> <s>Quibus &longs;emel inventis, Problema deinceps <lb/>confit per &longs;equentem Analy&longs;in. </s> <s>Per con&longs;tructionem quamvis (vel. </s> <s><lb/>utcunque conjec­<lb/> <arrow.to.target n="fig69"></arrow.to.target><lb/>turam faciendo) <lb/>cogno&longs;catur cor­<lb/>poris locus <emph type="italics"/>P<emph.end type="italics"/> pro­<lb/>ximus vero ejus lo­<lb/>co <emph type="italics"/>p.<emph.end type="italics"/> Demi&longs;&longs;aque ad <lb/>axem Ellip&longs;eos or­<lb/>dinatim applicata <lb/><emph type="italics"/>PR,<emph.end type="italics"/> ex propor­<lb/>tione diametrorum <lb/>Ellip&longs;eos, dabitur <lb/>Circuli circum&longs;cri­<lb/>pti <emph type="italics"/>AQB<emph.end type="italics"/> ordinatim applicata <emph type="italics"/>RQ,<emph.end type="italics"/> quæ &longs;inus e&longs;t anguli <emph type="italics"/>AOQ<emph.end type="italics"/> exi­<lb/>&longs;tente <emph type="italics"/>AO<emph.end type="italics"/> radio. </s> <s>Sufficit angulum illum rudi calculo in numeris <lb/>proximis invenire. </s> <s>Cogno&longs;catur etiam angulus tempori propor- <pb pagenum="102"/> <arrow.to.target n="note78"></arrow.to.target><lb/>tionalis, id e&longs;t, qui &longs;it ad quatuor rectos, ut e&longs;t tempus quo corpus <lb/>de&longs;crip&longs;it arcum <emph type="italics"/>Ap,<emph.end type="italics"/> ad tempus revolutionis unius in Ellip&longs;i. </s> <s>Sit <lb/>angulus i&longs;te N. </s> <s>Tum capiatur & angulus D ad angulum B, ut <lb/>e&longs;t &longs;inus i&longs;te anguli <emph type="italics"/>AOQ<emph.end type="italics"/> ad radium, & angulus E ad angulum <lb/>N-<emph type="italics"/>AOQ<emph.end type="italics"/>+D, ut e&longs;t longitudo L ad longitudinem eandem L <lb/>co&longs;inu anguli <emph type="italics"/>AOQ<emph.end type="italics"/> diminutam, ubi angulus i&longs;te recto minor e&longs;t, <lb/>auctam ubi major. </s> <s>Po&longs;tea capiatur tum angulus F ad angulum B, <lb/>ut e&longs;t &longs;inus anguli <emph type="italics"/>AOQ<emph.end type="italics"/>+E ad radium, tum angulus G ad angu­<lb/>lum N-<emph type="italics"/>AOQ<emph.end type="italics"/>-E+F ut e&longs;t longitudo L ad longitudinem ean­<lb/>dem co&longs;inu anguli <emph type="italics"/>AOQ<emph.end type="italics"/>+E diminutam ubi angulus i&longs;te recto mi­<lb/>nor e&longs;t, auctam ubi major. </s> <s>Tertia vice capiatur angulus H ad an­<lb/>gulum B, ut e&longs;t &longs;inus anguli <emph type="italics"/>AOQ<emph.end type="italics"/>+E+G ad radium; & angu­<lb/>lus I ad angulum N-<emph type="italics"/>AOQ<emph.end type="italics"/>-E-G+H, ut e&longs;t longitudo L ad <lb/>eandem longitudinem co&longs;inu anguli <emph type="italics"/>AOQ<emph.end type="italics"/>+E+G diminutam, <lb/>ubi angulus i&longs;te re­<lb/> <arrow.to.target n="fig70"></arrow.to.target><lb/>cto minor e&longs;t, auc­<lb/>tam ubi major. </s> <s>Et <lb/>&longs;ic pergere licet in <lb/>infinitum. </s> <s>Deni­<lb/>que capiatur angu­<lb/>lus <emph type="italics"/>AOq<emph.end type="italics"/> æqualis <lb/>angulo <emph type="italics"/>AOQ<emph.end type="italics"/>+E <lb/>+G+I+&c. </s> <s>e t <lb/>ex co&longs;inu ejus <emph type="italics"/>Or<emph.end type="italics"/><lb/>& ordinata <emph type="italics"/>pr,<emph.end type="italics"/> quæ <lb/>e&longs;t ad &longs;inum ejus <lb/><emph type="italics"/>qr<emph.end type="italics"/> ut Ellip&longs;eos axis minor ad axem majorem, habebitur corporis <lb/>locus correctus <emph type="italics"/>p.<emph.end type="italics"/> Si quando angulus N-<emph type="italics"/>AOQ<emph.end type="italics"/>+D negativus <lb/>e&longs;t, debet &longs;ignum+ip&longs;ius E ubique mutari in-, & &longs;ignum-in+. <lb/>Idem intelligendum e&longs;t de &longs;ignis ip&longs;orum G & I, ubi anguli <lb/>N-<emph type="italics"/>AOQ<emph.end type="italics"/>-E+F, & N-<emph type="italics"/>AOQ<emph.end type="italics"/>-E-G+H negativi prodeunt. </s> <s><lb/>Convergit autem &longs;eries infinita <emph type="italics"/>AOQ<emph.end type="italics"/>+E+G+I+&c. </s> <s>quam <lb/>celerrime, adeo ut vix unquam opus fuerit ultra progredi quam <lb/>ad terminum &longs;ecundum E. </s> <s>Et fundatur calculus in hoc Theore­<lb/>mate, quod area <emph type="italics"/>APS<emph.end type="italics"/> &longs;it ut differentia inter arcum <emph type="italics"/>AQ<emph.end type="italics"/> & <lb/>rectam ab umbilico <emph type="italics"/>S<emph.end type="italics"/> in Radium <emph type="italics"/>OQ<emph.end type="italics"/> perpendiculariter de­<lb/>mi&longs;&longs;am. </s> </p> <p type="margin"> <s><margin.target id="note78"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig69"></figure> <figure id="fig70"></figure> <p type="main"> <s>Non di&longs;&longs;imili calculo conficitur Problema in Hyperbola. </s> <s>Sit <lb/>ejus Centrum <emph type="italics"/>O,<emph.end type="italics"/> Vertex <emph type="italics"/>A,<emph.end type="italics"/> Umbilicus <emph type="italics"/>S<emph.end type="italics"/> & A&longs;ymptotos <emph type="italics"/>OK.<emph.end type="italics"/> Cog- <pb pagenum="103"/>no&longs;catur quantitas areæ ab&longs;cindendæ tempori proportionalis. </s> <s>Sit ea <lb/> <arrow.to.target n="note79"></arrow.to.target><lb/>A, & fiat conjectura de po&longs;itione rectæ <emph type="italics"/>SP,<emph.end type="italics"/> quæ aream <emph type="italics"/>APS<emph.end type="italics"/><lb/>ab&longs;cindat veræ proximam. </s> <s>Jun­<lb/> <arrow.to.target n="fig71"></arrow.to.target><lb/>gatur <emph type="italics"/>OP,<emph.end type="italics"/> & ab <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> ad <lb/>A&longs;ymptoton agantur <emph type="italics"/>AI, PK<emph.end type="italics"/><lb/>A&longs;ymptoto alteri parallelæ, & per <lb/>Tabulam Logarithmorum dabi­<lb/>tur Area <emph type="italics"/>AIKP,<emph.end type="italics"/> eique æqualis <lb/>area <emph type="italics"/>OPA,<emph.end type="italics"/> quæ &longs;ubducta de tri­<lb/>angulo <emph type="italics"/>OPS<emph.end type="italics"/> relinquet aream ab­<lb/>&longs;ci&longs;&longs;am <emph type="italics"/>APS.<emph.end type="italics"/> Applicando areæ <lb/>ab&longs;cindendæ A & ab&longs;ci&longs;&longs;æ <emph type="italics"/>APS<emph.end type="italics"/><lb/>differentiam duplam 2 <emph type="italics"/>APS<emph.end type="italics"/>-2 A <lb/>vel 2 A-2 <emph type="italics"/>APS<emph.end type="italics"/> ad lineam <emph type="italics"/>SN,<emph.end type="italics"/> quæ ab umbilico <emph type="italics"/>S<emph.end type="italics"/> in tangentem <lb/><emph type="italics"/>PT<emph.end type="italics"/> perpendicularis e&longs;t, orietur longitudo chordæ <emph type="italics"/><expan abbr="Pq.">Pque</expan><emph.end type="italics"/> In&longs;cri­<lb/>batur autem chorda illa <emph type="italics"/>PQ<emph.end type="italics"/> inter <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P,<emph.end type="italics"/> &longs;i area ab&longs;ci&longs;&longs;a <emph type="italics"/>APS<emph.end type="italics"/><lb/>major &longs;it area ab&longs;cindenda A, &longs;ecus ad puncti <emph type="italics"/>P<emph.end type="italics"/> contrarias partes: <lb/>& punctum <emph type="italics"/>Q<emph.end type="italics"/> erit locus corporis accuratior. </s> <s>Et computatione <lb/>repetita invenietur idem accuratior in perpetuum. </s> </p> <p type="margin"> <s><margin.target id="note79"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig71"></figure> <p type="main"> <s>Atque his calculis Problema generaliter confit Analytice. </s> <s>Ve­<lb/>rum u&longs;ibus A&longs;tronomicis accommodatior e&longs;t calculus particularis <lb/>qui &longs;equitur. </s> <s>Exi&longs;tentibus <emph type="italics"/>AO, OB, OD<emph.end type="italics"/> &longs;emiaxibus Ellip&longs;eos, & <lb/>L ip&longs;ius latere recto, ac D differentia inter &longs;emiaxem minorem <emph type="italics"/>OD<emph.end type="italics"/><lb/>& lateris recti &longs;emi&longs;&longs;em 1/2 L; quære tum angulum Y, cujus &longs;inus <lb/>&longs;it ad Radium ut e&longs;t rectangu­<lb/> <arrow.to.target n="fig72"></arrow.to.target><lb/>lum &longs;ub differentia illa D, & <lb/>&longs;emi&longs;umma axium <emph type="italics"/>AO+OD<emph.end type="italics"/><lb/>ad quadratum axis majoris <emph type="italics"/>AB<emph.end type="italics"/>; <lb/>tum angulum Z, cujus &longs;inus <lb/>&longs;it ad Radium ut e&longs;t duplum <lb/>rectangulum &longs;ub umbilicorum <lb/>di&longs;tantia <emph type="italics"/>SH<emph.end type="italics"/> & differentia <lb/>illa D ad triplum quadratum <lb/>&longs;emiaxis majoris <emph type="italics"/>AO.<emph.end type="italics"/> His <lb/>angulis &longs;emel inventis; locus corporis &longs;ic deinceps determinabitur. </s> <s><lb/>Sume angulum T proportionalem tempori quo arcus <emph type="italics"/>BP<emph.end type="italics"/> de&longs;crip­<lb/>tus e&longs;t, &longs;cu motui medio (ut loquuntur) æqualem; & angulum <lb/>V (primam medii motus æquationem) ad angulum Y (æquatio­<lb/>nem maximam primam) ut e&longs;t &longs;inus dupli anguli T ad Radium; <pb pagenum="104"/> <arrow.to.target n="note80"></arrow.to.target><lb/>atque angulum X (æquationem &longs;ecundam) ad angulum Z (æqua­<lb/>tionem maximam &longs;ecundam) ut e&longs;t cubus &longs;inus anguli T ad cubum <lb/>Radii. </s> <s>Angulorum T, V, X vel &longs;ummæ T+X+V, &longs;i angulus <lb/>T recto minor e&longs;t, vel differentiæ T+X-V, &longs;i is recto major e&longs;t <lb/>recti&longs;que duobus minor, æqualem cape angulum <emph type="italics"/>BHP<emph.end type="italics"/> (motum <lb/>medium æquatum;) &, &longs;i <emph type="italics"/>HP<emph.end type="italics"/> occurrat Ellip&longs;i in <emph type="italics"/>P,<emph.end type="italics"/> acta <emph type="italics"/>SP<emph.end type="italics"/> ab­<lb/>&longs;cindet aream <emph type="italics"/>BSP<emph.end type="italics"/> tempori proportionalem quamproxime. </s> <s>Hæc <lb/>Praxis &longs;atis expedita videtur, <lb/> <arrow.to.target n="fig73"></arrow.to.target><lb/>propterea quod angulorum per­<lb/>exiguorum V & X (in minutis <lb/>&longs;ecundis, &longs;i placet, po&longs;itorum) <lb/>figuras duas ter&longs;ve primas in­<lb/>venire &longs;ufficit. </s> <s>Sed & &longs;atis ac­<lb/>curata e&longs;t ad Theoriam Planeta­<lb/>rum. </s> <s>Nam in Orbe vel Martis <lb/>ip&longs;ius, cujus Æquatio centri ma­<lb/>xima e&longs;t graduum decem, error <lb/>vix &longs;uperabit minutum unum <lb/>&longs;ecundum. </s> <s>Invento autem angulo motus medii æquati <emph type="italics"/>BHP,<emph.end type="italics"/> an­<lb/>gulus veri motus <emph type="italics"/>BSP<emph.end type="italics"/> & di&longs;tantia <emph type="italics"/>SP<emph.end type="italics"/> in promptu &longs;unt per <lb/><emph type="italics"/>Wardi<emph.end type="italics"/> methodum noti&longs;&longs;imam. </s> </p> <p type="margin"> <s><margin.target id="note80"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig72"></figure> <figure id="fig73"></figure> <p type="main"> <s>Hactenus de Motu corporum in lineis Curvis. </s> <s>Fieri autem po­<lb/>te&longs;t ut mobile recta de&longs;cendat vel recta a&longs;cendat, & quæ ad i&longs;tiu&longs;­<lb/>modi Motus &longs;pectant, pergo jam exponere. <pb pagenum="105"/> <arrow.to.target n="note81"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note81"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>SECTIO VII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Corporum A&longs;cen&longs;u & De&longs;cen&longs;u Rectilineo.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXII. PROBLEMA XXIV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod Vis centripeta &longs;it reciproce proportionalis quadrato di­<lb/>&longs;tantiæ locorum a centro, Spatia definire quæ corpus recta cadendo <lb/>datis temporibus de&longs;cribit.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Si Corpus non cadit perpendicu­<lb/> <arrow.to.target n="fig74"></arrow.to.target><lb/>lariter de&longs;cribet id, per Corol. </s> <s>1. Prop. </s> <s>XIII, <lb/>Sectionem aliquam Conicam cujus umbili­<lb/>cus congruit cum centro virium. </s> <s>Sit Sec­<lb/>tio illa Conica <emph type="italics"/>ARPB<emph.end type="italics"/> & umbilicus ejus <emph type="italics"/>S.<emph.end type="italics"/><lb/>Et primo &longs;i Figura Ellip&longs;is e&longs;t, &longs;uper hu­<lb/>jus axe majore <emph type="italics"/>AB<emph.end type="italics"/> de&longs;cribatur Semicirculus <lb/><emph type="italics"/>ADB,<emph.end type="italics"/> & per corpus decidens tran&longs;eat rec­<lb/>ta <emph type="italics"/>DPC<emph.end type="italics"/> perpendicularis ad axem; acti&longs;que <lb/><emph type="italics"/>DS, PS<emph.end type="italics"/> erit area <emph type="italics"/>ASD<emph.end type="italics"/> areæ <emph type="italics"/>ASP<emph.end type="italics"/> at­<lb/>que adeo etiam tempori proportionalis. </s> <s>Ma­<lb/>nente axe <emph type="italics"/>AB<emph.end type="italics"/> minuatur perpetuo latitudo <lb/>Ellip&longs;eos, & &longs;emper manebit area <emph type="italics"/>ASD<emph.end type="italics"/><lb/>tempori proportionalis. </s> <s>Minuatur latitudo <lb/>illa in infinitum: &, Orbe <emph type="italics"/>APB<emph.end type="italics"/> jam coin­<lb/>cidente cum axe <emph type="italics"/>AB<emph.end type="italics"/> & umbilico <emph type="italics"/>S<emph.end type="italics"/> cum <lb/>axis termino <emph type="italics"/>B,<emph.end type="italics"/> de&longs;cendet corpus in recta <lb/><emph type="italics"/>AC,<emph.end type="italics"/> & area <emph type="italics"/>ABD<emph.end type="italics"/> evadet tempori pro­<lb/>portionalis. </s> <s>Dabitur itaque Spatium <emph type="italics"/>AC,<emph.end type="italics"/><lb/>quod corpus de loco <emph type="italics"/>A<emph.end type="italics"/> perpendiculariter <lb/>cadendo tempore dato de&longs;cribit, &longs;i modo tempori proportiona­<lb/>lis capiatur area <emph type="italics"/>ABD,<emph.end type="italics"/> & a puncto <emph type="italics"/>D<emph.end type="italics"/> ad rectam <emph type="italics"/>AB<emph.end type="italics"/> demit­<lb/>tatur perpendicularis <emph type="italics"/>DC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/> <pb pagenum="106"/> <arrow.to.target n="note82"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note82"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig74"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Si Figura illa <emph type="italics"/>RPB<emph.end type="italics"/> Hyperbola e&longs;t, de&longs;cribatur ad ean­<lb/>dem diametrum principalem <emph type="italics"/>AB<emph.end type="italics"/> Hyperbola rectangula <emph type="italics"/>BED:<emph.end type="italics"/><lb/>& quoniam areæ <emph type="italics"/>CSP, CBfP, SPfB<emph.end type="italics"/> &longs;unt ad areas <emph type="italics"/>CSD, <lb/>CBED, SDEB,<emph.end type="italics"/> &longs;ingulæ ad &longs;ingulas, in data ratione altitudi­<lb/>num <emph type="italics"/>CP, CD<emph.end type="italics"/>; & area <emph type="italics"/>SPfB<emph.end type="italics"/><lb/> <arrow.to.target n="fig75"></arrow.to.target><lb/>proportionalis e&longs;t tempori quo <lb/>corpus <emph type="italics"/>P<emph.end type="italics"/> movebitur per arcum <lb/><emph type="italics"/>PfB<emph.end type="italics"/>; erit etiam area <emph type="italics"/>SDEB<emph.end type="italics"/> ei­<lb/>dem tempori proportionalis. </s> <s><lb/>Minuatur latus rectum Hyper­<lb/>bolæ <emph type="italics"/>RPB<emph.end type="italics"/> in infinitum ma­<lb/>nente latere tran&longs;ver&longs;o, & coibit <lb/>arcus <emph type="italics"/>PB<emph.end type="italics"/> cum recta <emph type="italics"/>CB<emph.end type="italics"/> & um­<lb/>bilicus <emph type="italics"/>S<emph.end type="italics"/> cum vertice <emph type="italics"/>B<emph.end type="italics"/> & recta <lb/><emph type="italics"/>SD<emph.end type="italics"/> cum recta <emph type="italics"/>BD.<emph.end type="italics"/> Proinde a­<lb/>rea <emph type="italics"/>BDEB<emph.end type="italics"/> proportionalis erit <lb/>tempori quo corpus <emph type="italics"/>C<emph.end type="italics"/> recto <lb/>de&longs;cen&longs;u de&longs;cribit lineam <emph type="italics"/>CB. <lb/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <figure id="fig75"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 3. Et &longs;imili argumento &longs;i <lb/>Figura <emph type="italics"/>RPB<emph.end type="italics"/> Parabola e&longs;t, & <lb/>eodem vertice principali <emph type="italics"/>B<emph.end type="italics"/> de­<lb/>&longs;cribatur alia Parabola <emph type="italics"/>BED,<emph.end type="italics"/><lb/>quæ &longs;emper maneat data interea <lb/>dum Parabola prior in cujus perimetro corpus <emph type="italics"/>P<emph.end type="italics"/> movetur, dimi­<lb/>nuto & in nihilum redacto ejus latere recto, conveniat cum linea <lb/><emph type="italics"/>CB<emph.end type="italics"/>; fiet &longs;egmentum Parabolicum <emph type="italics"/>BDEB<emph.end type="italics"/> proportionale tempori <lb/>quo corpus illud <emph type="italics"/>P<emph.end type="italics"/> vel <emph type="italics"/>C<emph.end type="italics"/> de&longs;cendet ad centrum <emph type="italics"/>S<emph.end type="italics"/> vel <emph type="italics"/>B. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIII. THEOREMA IX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;itis jam inventis, dico quod corporis cadentis Velocitas in loco quo­<lb/>vis<emph.end type="italics"/> C <emph type="italics"/>est ad velocitatem corporis centro<emph.end type="italics"/> B <emph type="italics"/>intervallo<emph.end type="italics"/> BC <emph type="italics"/>Circu­<lb/>lum de&longs;cribentis, in &longs;ubduplicata ratione quam<emph.end type="italics"/> AC, <emph type="italics"/>di&longs;tantia cor­<lb/>poris a Circuli vel Hyperbolæ rect angulæ vertice ulteriore<emph.end type="italics"/> A, <emph type="italics"/>habet <lb/>ad Figuræ &longs;emidiametrum principalem<emph.end type="italics"/> 1/2 AB. </s> </p> <p type="main"> <s>Bi&longs;ecetur <emph type="italics"/>AB,<emph.end type="italics"/> communis utriu&longs;que Figuræ <emph type="italics"/>RPB, DEB<emph.end type="italics"/> dia­<lb/>meter, in <emph type="italics"/>O<emph.end type="italics"/>; & agatur recta <emph type="italics"/>PT<emph.end type="italics"/> quæ tangat Figuram <emph type="italics"/>RPB<emph.end type="italics"/> in <emph type="italics"/>P,<emph.end type="italics"/> atque <pb pagenum="107"/>etiam &longs;ecet communem illam diametrum <emph type="italics"/>AB<emph.end type="italics"/> (&longs;i opus e&longs;t productam) </s> </p> <p type="main"> <s> <arrow.to.target n="note83"></arrow.to.target><lb/>in <emph type="italics"/>T<emph.end type="italics"/>; &longs;itque <emph type="italics"/>SY<emph.end type="italics"/> ad hanc rectam, & <emph type="italics"/>BQ<emph.end type="italics"/> ad <lb/> <arrow.to.target n="fig76"></arrow.to.target><lb/>hanc diametrum perpendicularis, atque Figu­<lb/>ræ <emph type="italics"/>RPB<emph.end type="italics"/> latus rectum ponatur L. </s> <s>Con&longs;tat <lb/>per Cor. </s> <s>9. Prop. </s> <s>XVI, quod corporis in <lb/>linea <emph type="italics"/>RPB<emph.end type="italics"/> circa centrum <emph type="italics"/>S<emph.end type="italics"/> moventis velo­<lb/>citas in loco quovis <emph type="italics"/>P<emph.end type="italics"/> &longs;it ad velocitatem cor­<lb/>poris intervallo <emph type="italics"/>SP<emph.end type="italics"/> circa idem centrum Cir­<lb/>culum de&longs;cribentis in &longs;ubduplicata ratione rec­<lb/>tanguli 1/2 LX<emph type="italics"/>SP<emph.end type="italics"/> ad <emph type="italics"/>SY<emph.end type="italics"/> quadratum. </s> <s>E&longs;t au­<lb/>tem ex Conicis <emph type="italics"/>ACB<emph.end type="italics"/> ad <emph type="italics"/>CPq<emph.end type="italics"/> ut 2 <emph type="italics"/>AO<emph.end type="italics"/> ad L, <lb/>adeoque (2<emph type="italics"/>CPqXAO/ACB<emph.end type="italics"/>) æquale L. </s> <s>Ergo ve­<lb/>locitates illæ &longs;unt ad invicem in &longs;ubduplicata <lb/>ratione (<emph type="italics"/>CPqXAOXSP/ACB<emph.end type="italics"/>) ad <emph type="italics"/>SY quad.<emph.end type="italics"/> Por­<lb/>ro ex Conicis e&longs;t <emph type="italics"/>CO<emph.end type="italics"/> ad <emph type="italics"/>BO<emph.end type="italics"/> ut <emph type="italics"/>BO<emph.end type="italics"/> ad <emph type="italics"/>TO,<emph.end type="italics"/><lb/>& compo&longs;ite vel divi&longs;im ut <emph type="italics"/>CB<emph.end type="italics"/> ad <emph type="italics"/>BT.<emph.end type="italics"/><lb/>Unde vel dividendo vel componendo fit <lb/><emph type="italics"/>BO<emph.end type="italics"/>-vel+<emph type="italics"/>CO<emph.end type="italics"/> ad <emph type="italics"/>BO<emph.end type="italics"/> ut <emph type="italics"/>CT<emph.end type="italics"/> ad <emph type="italics"/>BT,<emph.end type="italics"/> id e&longs;t <lb/><emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>AO<emph.end type="italics"/> ut <emph type="italics"/>CP<emph.end type="italics"/> ad <emph type="italics"/>BQ<emph.end type="italics"/>; indeque (<emph type="italics"/>CPqXAOXSP/ACB<emph.end type="italics"/>) æquale e&longs;t <lb/>(<emph type="italics"/>BQqXACXSP/AOXBC.<emph.end type="italics"/>) Minuatur jam in infinitum Figuræ <emph type="italics"/>RPB<emph.end type="italics"/> latitu­<lb/>do <emph type="italics"/>CP,<emph.end type="italics"/> &longs;ic ut punctum <emph type="italics"/>P<emph.end type="italics"/> coeat cum puncto <emph type="italics"/>C,<emph.end type="italics"/> punctumque <emph type="italics"/>S<emph.end type="italics"/> cum <lb/>puncto <emph type="italics"/>B,<emph.end type="italics"/> & linea <emph type="italics"/>SP<emph.end type="italics"/> cum linea <emph type="italics"/>BC,<emph.end type="italics"/> lineaque <emph type="italics"/>SY<emph.end type="italics"/> cum linea <emph type="italics"/>BQ<emph.end type="italics"/>; <lb/>& corporis jam recta de&longs;cendentis in linea <emph type="italics"/>CB<emph.end type="italics"/> velocitas fiet ad <lb/>velocitatem corporis centro <emph type="italics"/>B<emph.end type="italics"/> intervallo <emph type="italics"/>BC<emph.end type="italics"/> Circulum de&longs;cribentis, <lb/>in &longs;ubduplicata ratione ip&longs;ius (<emph type="italics"/>BQqXACXSP/AOXBC<emph.end type="italics"/>) ad <emph type="italics"/>SYq,<emph.end type="italics"/> hoc e&longs;t (neg­<lb/>lectis æqualitatis rationibus <emph type="italics"/>SP<emph.end type="italics"/> ad <emph type="italics"/>BC<emph.end type="italics"/> & <emph type="italics"/>BQq<emph.end type="italics"/> ad <emph type="italics"/>SYq<emph.end type="italics"/>) in &longs;ub­<lb/>duplicata ratione <emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>AO<emph.end type="italics"/> &longs;ive 1/2 <emph type="italics"/>AB. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note83"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig76"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Punctis <emph type="italics"/>B<emph.end type="italics"/> & <emph type="italics"/>S<emph.end type="italics"/> coeuntibus, fit <emph type="italics"/>TC<emph.end type="italics"/> ad <emph type="italics"/>TS<emph.end type="italics"/> ut <emph type="italics"/>AC<emph.end type="italics"/><lb/>ad <emph type="italics"/>AO.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Corpus ad datam a centro di&longs;tantiam in Circulo quo­<lb/>vis revolvens, motu &longs;uo &longs;ur&longs;um ver&longs;o a&longs;cendet ad duplam &longs;uam a <lb/>centro di&longs;tantiam. <pb pagenum="108"/> <arrow.to.target n="note84"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note84"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIV. THEOREMA X.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si Figura<emph.end type="italics"/> BED <emph type="italics"/>Parabola e&longs;t, dico<emph.end type="italics"/><lb/> <arrow.to.target n="fig77"></arrow.to.target><lb/><emph type="italics"/>quod corporis cadentis Veloci­<lb/>tas in loco quovis<emph.end type="italics"/> C <emph type="italics"/>æqualis e&longs;t <lb/>velocitati qua corpus centro<emph.end type="italics"/> B <lb/><emph type="italics"/>dimidio intervalli &longs;ui<emph.end type="italics"/> BC <emph type="italics"/>Cir­<lb/>culum uniformiter de&longs;cribere <lb/>potest.<emph.end type="italics"/></s> </p> <figure id="fig77"></figure> <p type="main"> <s>Nam corporis Parabolam <lb/><emph type="italics"/>RPB<emph.end type="italics"/> circa centrum <emph type="italics"/>S<emph.end type="italics"/> de&longs;cri­<lb/>bentis velocitas in loco quovis <lb/><emph type="italics"/>P<emph.end type="italics"/> (per Corol. </s> <s>7. Prop. </s> <s>XVI) æ­<lb/>qualis e&longs;t velocitati corporis di­<lb/>midio intervalli <emph type="italics"/>SP<emph.end type="italics"/> Circulum cir­<lb/>ca idem centrum <emph type="italics"/>S<emph.end type="italics"/> uniformiter <lb/>de&longs;cribentis. </s> <s>Minuatur Parabolæ <lb/>latitudo <emph type="italics"/>CP<emph.end type="italics"/> in infinitum eo, ut <lb/>arcus Parabolicus <emph type="italics"/>PfB<emph.end type="italics"/> cum rec­<lb/>ta <emph type="italics"/>CB,<emph.end type="italics"/> centrum <emph type="italics"/>S<emph.end type="italics"/> cum vertice <emph type="italics"/>B,<emph.end type="italics"/><lb/>& intervallum <emph type="italics"/>SP<emph.end type="italics"/> cum intervallo <emph type="italics"/>BC<emph.end type="italics"/> coincidat, & con&longs;tabit Pro­<lb/>po&longs;itio. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXV. THEOREMA XI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod area Figuræ<emph.end type="italics"/> DES, <emph type="italics"/>radio indefinito<emph.end type="italics"/> SD <emph type="italics"/>de­<lb/>&longs;cripta, æqualis &longs;it areæ quam corpus, radio dimidium lateris recti <lb/>Figuræ<emph.end type="italics"/> DES <emph type="italics"/>æquante, circa centrum<emph.end type="italics"/> S <emph type="italics"/>uniformiter gyrando, eo­<lb/>dem tempore de&longs;cribere potest.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam concipe corpus <emph type="italics"/>C<emph.end type="italics"/> quam minima temporis particula lineo­<lb/>lam <emph type="italics"/>Cc<emph.end type="italics"/> cadendo de&longs;cribere, & interea corpus aliud <emph type="italics"/>K,<emph.end type="italics"/> uniformi­<lb/>ter in Circulo <emph type="italics"/>OKk<emph.end type="italics"/> circa centrum <emph type="italics"/>S<emph.end type="italics"/> gyrando, arcum <emph type="italics"/>Kk<emph.end type="italics"/> de&longs;cri­<lb/>bere. </s> <s>Erigantur perpendicula <emph type="italics"/>CD, cd<emph.end type="italics"/> occurrentia Figuræ <emph type="italics"/>DES<emph.end type="italics"/><lb/>in <emph type="italics"/>D, d.<emph.end type="italics"/> Jungantur <emph type="italics"/>SD, Sd, SK, Sk<emph.end type="italics"/> & ducatur <emph type="italics"/>Dd<emph.end type="italics"/> axi <emph type="italics"/>AS<emph.end type="italics"/> oc­<lb/>rens in <emph type="italics"/>T,<emph.end type="italics"/> & ad eam demittatur perpendiculum <emph type="italics"/>SY.<emph.end type="italics"/></s> </p> <pb pagenum="109"/> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/> 1. Jam &longs;i Figura <emph type="italics"/>DES<emph.end type="italics"/> Circulus e&longs;t vel Hyperbola, bi&longs;ece­<lb/> <arrow.to.target n="note85"></arrow.to.target><lb/>tur ejus tran&longs;ver&longs;a diameter <emph type="italics"/>AS<emph.end type="italics"/> in <emph type="italics"/>O,<emph.end type="italics"/> & erit <lb/> <arrow.to.target n="fig78"></arrow.to.target><lb/><emph type="italics"/>SO<emph.end type="italics"/> dimidium lateris recti. </s> <s>Et quoniam e&longs;t <lb/><emph type="italics"/>TC<emph.end type="italics"/> ad <emph type="italics"/>TD<emph.end type="italics"/> ut <emph type="italics"/>Cc<emph.end type="italics"/> ad <emph type="italics"/>Dd,<emph.end type="italics"/> & <emph type="italics"/>TD<emph.end type="italics"/> ad <emph type="italics"/>TS<emph.end type="italics"/> ut <lb/><emph type="italics"/>CD<emph.end type="italics"/> ad <emph type="italics"/>SY,<emph.end type="italics"/> erit ex æquo <emph type="italics"/>TC<emph.end type="italics"/> ad <emph type="italics"/>TS<emph.end type="italics"/> ut <lb/><emph type="italics"/>CDXCc<emph.end type="italics"/> ad <emph type="italics"/>SYXDd.<emph.end type="italics"/> Sed per Corol. </s> <s>1. Prop. </s> <s><lb/>XXXIII, e&longs;t <emph type="italics"/>TC<emph.end type="italics"/> ad <emph type="italics"/>TS<emph.end type="italics"/> ut <emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>AO,<emph.end type="italics"/> puta &longs;i <lb/>in coitu punctorum <emph type="italics"/>D, d<emph.end type="italics"/> capiantur linearum <lb/>rationes ultimæ. </s> <s>Ergo <emph type="italics"/>AC<emph.end type="italics"/> e&longs;t ad (<emph type="italics"/>AO<emph.end type="italics"/> &longs;eu) <emph type="italics"/>SK<emph.end type="italics"/><lb/>ut <emph type="italics"/>CDXCc<emph.end type="italics"/> ad <emph type="italics"/>SYXDd.<emph.end type="italics"/> Porro corporis <lb/>de&longs;cendentis velocitas in <emph type="italics"/>C<emph.end type="italics"/> e&longs;t ad velocitatem <lb/>corporis Circulum intervallo <emph type="italics"/>SC<emph.end type="italics"/> circa cen­<lb/>trum <emph type="italics"/>S<emph.end type="italics"/> de&longs;cribentis in &longs;ubduplicata ratione <lb/><emph type="italics"/>AC<emph.end type="italics"/> ad (<emph type="italics"/>AO<emph.end type="italics"/> vel) <emph type="italics"/>SK<emph.end type="italics"/> (per Prop. </s> <s>XXXIII.) Et <lb/>hæc velocitas ad velocitatem corporis de&longs;cri­<lb/>bentis Circulum <emph type="italics"/>OKk<emph.end type="italics"/> in &longs;ubduplicata ratione <lb/><emph type="italics"/>SK<emph.end type="italics"/> ad <emph type="italics"/>SC<emph.end type="italics"/> per Cor. </s> <s>6. Prop. </s> <s>IV, & ex æquo velo­<lb/>citas prima ad ultimam, hoc e&longs;t lineola <emph type="italics"/>Cc<emph.end type="italics"/> ad <lb/>arcum <emph type="italics"/>Kk<emph.end type="italics"/> in &longs;ubduplicata ratione <emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>SC,<emph.end type="italics"/><lb/>id e&longs;t in ratione <emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>CD.<emph.end type="italics"/> Quare e&longs;t <emph type="italics"/>CDXCc<emph.end type="italics"/><lb/>æquale <emph type="italics"/>ACXKk,<emph.end type="italics"/> & propterea <emph type="italics"/>AC<emph.end type="italics"/> ad <emph type="italics"/>SK<emph.end type="italics"/> ut <lb/><emph type="italics"/>ACXKk<emph.end type="italics"/> ad <emph type="italics"/>SYXDd,<emph.end type="italics"/> <expan abbr="indeq;">indeque</expan> <emph type="italics"/>SKXKk<emph.end type="italics"/> æqua­<lb/>le <emph type="italics"/>SYXDd,<emph.end type="italics"/> & 1/2 <emph type="italics"/>SKXKk<emph.end type="italics"/> æquale 1/2 <emph type="italics"/>SYXDd,<emph.end type="italics"/><lb/>id e&longs;t area <emph type="italics"/>KSk<emph.end type="italics"/> æqualis areæ <emph type="italics"/>SDd.<emph.end type="italics"/> Singulis <lb/>igitur temporis particulis generantur arearum <lb/>duarum particulæ <emph type="italics"/>KSk,<emph.end type="italics"/> & <emph type="italics"/>SDd,<emph.end type="italics"/> quæ, &longs;i mag­<lb/>nitudo earum minuatur & numerus augeatur in infinitum, ratio­<lb/>nem obtinent æqualitatis, & propterea (per Corollarium Lem­<lb/>matis IV) areæ totæ &longs;imul genitæ &longs;unt &longs;emper æquales, <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note85"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig78"></figure> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/> 2. Quod &longs;i Figura <emph type="italics"/>DES<emph.end type="italics"/> Parabola &longs;it, invenietur e&longs;&longs;e ut &longs;u­<lb/>pra <emph type="italics"/>CDXCc<emph.end type="italics"/> ad <emph type="italics"/>SYXDd<emph.end type="italics"/> ut <emph type="italics"/>TC<emph.end type="italics"/> ad <emph type="italics"/>TS,<emph.end type="italics"/> hoc e&longs;t ut 2 ad 1, ad­<lb/>eoque 1/4 <emph type="italics"/>CDXCc<emph.end type="italics"/> æquale e&longs;&longs;e 1/2 <emph type="italics"/>SYXDd.<emph.end type="italics"/> Sed corporis caden­<lb/>tis velocitas in <emph type="italics"/>C<emph.end type="italics"/> æqualis e&longs;t velocitati qua Circulus intervallo 1/2 <emph type="italics"/>SC<emph.end type="italics"/><lb/>uniformiter de&longs;cribi po&longs;&longs;it (per Prop. </s> <s>XXXIV) Et hæc velocitas ad ve­<lb/>locitatem qua Circulus radio <emph type="italics"/>SK<emph.end type="italics"/> de&longs;cribi po&longs;&longs;it, hoc e&longs;t, lineola <lb/><emph type="italics"/>Cc<emph.end type="italics"/> ad arcum <emph type="italics"/>Kk<emph.end type="italics"/> (per Corol. </s> <s>6. Prop. </s> <s>IV) e&longs;t in &longs;ubduplicata ratione <lb/><emph type="italics"/>SK<emph.end type="italics"/> ad 1/2 <emph type="italics"/>SC,<emph.end type="italics"/> id e&longs;t, in ratione <emph type="italics"/>SK<emph.end type="italics"/> ad 1/2 <emph type="italics"/>CD.<emph.end type="italics"/> Quare e&longs;t 1/2 <emph type="italics"/>SKXKk<emph.end type="italics"/><lb/>æquale 1/4 <emph type="italics"/>CDXCc,<emph.end type="italics"/> adeoque æquale 1/2 <emph type="italics"/>SYXDd,<emph.end type="italics"/> hoc e&longs;t, area <emph type="italics"/>KSk<emph.end type="italics"/><lb/>æqualis areæ <emph type="italics"/>SDd,<emph.end type="italics"/> ut &longs;upra. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/> <pb pagenum="110"/> <arrow.to.target n="note86"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note86"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVI. PROBLEMA XXV.<emph.end type="center"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Corporis de loco dato<emph.end type="italics"/> A <emph type="italics"/>cadentis determinare Tem­<lb/>pora de&longs;cen&longs;us.<emph.end type="italics"/></s> </p> <p type="main"> <s>Super diametro <emph type="italics"/>AS<emph.end type="italics"/> (di&longs;tantia corporis a cen­<lb/>tro &longs;ub initio) de&longs;cribe Semicirculum <emph type="italics"/>ADS,<emph.end type="italics"/> ut & <lb/>huic æqualem Semicirculum <emph type="italics"/>OKH<emph.end type="italics"/> circa centrum <lb/><emph type="italics"/>S.<emph.end type="italics"/> De corporis loco quovis <emph type="italics"/>C<emph.end type="italics"/> erige ordinatim ap­<lb/>plicatam <emph type="italics"/>CD.<emph.end type="italics"/> Junge <emph type="italics"/>SD,<emph.end type="italics"/> & areæ <emph type="italics"/>ASD<emph.end type="italics"/> æqua­<lb/>lem con&longs;titue &longs;ectorem <emph type="italics"/>OSK.<emph.end type="italics"/> Patet per Prop­<lb/>XXXV, quod corpus cadendo de&longs;cribet &longs;patium <emph type="italics"/>AC<emph.end type="italics"/><lb/>eodem Tempore quo corpus aliud uniformiter cir­<lb/>ca centrum <emph type="italics"/>S<emph.end type="italics"/> gyrando, de&longs;cribere pote&longs;t arcum <lb/><emph type="italics"/>OK. <expan abbr="q.">que</expan> E. F.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVII. PROBLEMA XXVI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corporis de loco dato &longs;ur&longs;um vel deor&longs;um projecti definire Tempora <lb/>a&longs;cen&longs;us vel de&longs;cen&longs;us.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Exeat corpus de loco dato <emph type="italics"/>G<emph.end type="italics"/> &longs;ecundum <lb/> <arrow.to.target n="fig79"></arrow.to.target><lb/>lineam <emph type="italics"/>ASG<emph.end type="italics"/> cum velocitate quacunque. </s> <s><lb/>In duplicata ratione hujus velocitatis ad <lb/>uniformem in Circulo velocitatem, qua cor­<lb/>pus ad intervallum datum <emph type="italics"/>SG<emph.end type="italics"/> circa centrum <lb/><emph type="italics"/>S<emph.end type="italics"/> revolvi po&longs;&longs;et, cape <emph type="italics"/>GA<emph.end type="italics"/> ad 1/2 <emph type="italics"/>AS.<emph.end type="italics"/><lb/>Si ratio illa e&longs;t numeri binarii ad unita­<lb/>tem, punctum <emph type="italics"/>A<emph.end type="italics"/> infinite di&longs;tat, quo ca­<lb/>&longs;u Parabola vertice <emph type="italics"/>S,<emph.end type="italics"/> axe <emph type="italics"/>SC,<emph.end type="italics"/> latere quo­<lb/>vis recto de&longs;cribenda e&longs;t. </s> <s>Patet hoc per <lb/>Prop. </s> <s>XXXIV. </s> <s>Sin ratio illa minor vel ma­<lb/>jor e&longs;t quam 2 ad 1, priore ca&longs;u Circulus, <lb/>po&longs;teriore Hyperbola rectangula &longs;uper di­<lb/>ametro <emph type="italics"/>SA<emph.end type="italics"/> de&longs;cribi debet. </s> <s>Patet per <lb/>Prop. </s> <s>XXXIII. </s> <s>Tum centro <emph type="italics"/>S,<emph.end type="italics"/> intervallo <lb/>æquante dimidium lateris recti, de&longs;cribatur <lb/>Circulus <emph type="italics"/>HKk,<emph.end type="italics"/> & ad corporis a&longs;cenden­<lb/>tis vel de&longs;cendentis loca duo quævis <emph type="italics"/>G, C,<emph.end type="italics"/><lb/>erigantur perpendicula <emph type="italics"/>GI, CD<emph.end type="italics"/> occurren­<lb/>tia Conicæ Sectioni vel Circulo in <emph type="italics"/>I<emph.end type="italics"/> ac <emph type="italics"/>D.<emph.end type="italics"/> <pb pagenum="111"/>Dein junctis <emph type="italics"/>SI, SD,<emph.end type="italics"/> fiant &longs;egmentis <emph type="italics"/>SEIS, SEDS,<emph.end type="italics"/> &longs;ec­<lb/> <arrow.to.target n="note87"></arrow.to.target><lb/>tores <emph type="italics"/>HSK, HSk<emph.end type="italics"/> æquales, & per Prop. </s> <s>XXXV, corpus <emph type="italics"/>G<emph.end type="italics"/> de&longs;cri­<lb/>bet &longs;patium <emph type="italics"/>GC<emph.end type="italics"/> eodem Tempore quo corpus <emph type="italics"/>K<emph.end type="italics"/> de&longs;cribere po­<lb/>te&longs;t arcum <emph type="italics"/>Kk. </s> <s><expan abbr="q.">que</expan> E. F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note87"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig79"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVIII. THEOREMA XII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod Vis centripeta proportionalis &longs;it altitudini &longs;eu di&longs;tantiæ lo­<lb/>corum a centro, dico quod cadentium Tempora, Velocitates & Spa­<lb/>tia de&longs;cripta &longs;unt arcubus, arcuumque finibus rectis & &longs;inibus <lb/>ver&longs;is re&longs;pective proportionalia.<emph.end type="italics"/></s> </p> <p type="main"> <s>Cadat corpus de loco quovis <emph type="italics"/>A<emph.end type="italics"/> &longs;ecun­<lb/> <arrow.to.target n="fig80"></arrow.to.target><lb/>dum rectam <emph type="italics"/>AS<emph.end type="italics"/>; & centro virium <emph type="italics"/>S,<emph.end type="italics"/> in­<lb/>tervallo <emph type="italics"/>AS,<emph.end type="italics"/> de&longs;cribatur Circuli quadrans <lb/><emph type="italics"/>AE,<emph.end type="italics"/> &longs;itque <emph type="italics"/>CD<emph.end type="italics"/> &longs;inus rectus arcus cuju&longs;­<lb/>vis <emph type="italics"/>AD<emph.end type="italics"/>; & corpus <emph type="italics"/>A,<emph.end type="italics"/> Tempore <emph type="italics"/>AD,<emph.end type="italics"/> ca­<lb/>dendo de&longs;cribet Spatium <emph type="italics"/>AC,<emph.end type="italics"/> inque loco <lb/><emph type="italics"/>C<emph.end type="italics"/> acquiret Velocitatem <emph type="italics"/>CD.<emph.end type="italics"/></s> </p> <figure id="fig80"></figure> <p type="main"> <s>Demon&longs;tratur eodem modo ex Propo&longs;i­<lb/>tione X, quo Propo&longs;itio XXXII, ex Propo­<lb/>&longs;itione XI demon&longs;trata fuit. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc æqualia &longs;unt Tempora quibus corpus unum de loco <lb/><emph type="italics"/>A<emph.end type="italics"/> cadendo pervenit ad centrum <emph type="italics"/>S,<emph.end type="italics"/> & corpus aliud revolvendo de­<lb/>&longs;cribit arcum quadrantalem <emph type="italics"/>ADE.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Proinde æqualia &longs;unt Tempora omnia quibus corpora de <lb/>locis quibu&longs;vis ad u&longs;que centrum cadunt. </s> <s>Nam revolventium tem­<lb/>pora omnia periodica (per Corol. </s> <s>3. Prop. </s> <s>IV.) æquantur. <pb pagenum="112"/> <arrow.to.target n="note88"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note88"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIX. PROBLEMA XXVII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;ita cuju&longs;cunque generis Vi centripeta, & conce&longs;&longs;is figurarum <lb/>curvilinearum quadraturis, requiritu, corporis recta a&longs;cenden­<lb/>tis vel de&longs;cendentis tum Velocitas in locis &longs;ingulis, tum Tempus <lb/>quo corpus ad locum quemvis perveniet: Et contra.<emph.end type="italics"/></s> </p> <p type="main"> <s>De loco quovis <emph type="italics"/>A<emph.end type="italics"/> in recta <emph type="italics"/>ADEC<emph.end type="italics"/> cadat corpus <emph type="italics"/>E,<emph.end type="italics"/> deque loco <lb/>ejus <emph type="italics"/>E<emph.end type="italics"/> erigatur &longs;emper perpendicularis <emph type="italics"/>EG,<emph.end type="italics"/> vi centripetæ in loco <lb/>illo ad centrum <emph type="italics"/>C<emph.end type="italics"/> tendenti proportio­<lb/> <arrow.to.target n="fig81"></arrow.to.target><lb/>nalis: Sitque <emph type="italics"/>BFG<emph.end type="italics"/> linea curva quam <lb/>punctum <emph type="italics"/>G<emph.end type="italics"/> perpetuo tangit. </s> <s>Coinci­<lb/>dat autem <emph type="italics"/>EG<emph.end type="italics"/> ip&longs;o motus initio cum <lb/>perpendiculari <emph type="italics"/>AB,<emph.end type="italics"/> & erit corporis Ve­<lb/>locitas in loco quovis <emph type="italics"/>E<emph.end type="italics"/> ut areæ cur­<lb/>vilineæ <emph type="italics"/>ABGE<emph.end type="italics"/> latus quadratum. <lb/><emph type="italics"/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <figure id="fig81"></figure> <p type="main"> <s>In <emph type="italics"/>EG<emph.end type="italics"/> capiatur <emph type="italics"/>EM<emph.end type="italics"/> lateri quadra­<lb/>to areæ <emph type="italics"/>ABGE<emph.end type="italics"/> reciproce proportio­<lb/>nalis, & &longs;it <emph type="italics"/>ALM<emph.end type="italics"/> linea curva quam <lb/>punctum <emph type="italics"/>M<emph.end type="italics"/>perpetuotangit, & erit Tem­<lb/>pus quo corpus cadendo de&longs;cribit li­<lb/>neam <emph type="italics"/>AE<emph.end type="italics"/> ut area curvilinea <emph type="italics"/>ALME. <lb/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <p type="main"> <s>Etenim in recta <emph type="italics"/>AE<emph.end type="italics"/> capiatur linea <lb/>quam minima <emph type="italics"/>DE<emph.end type="italics"/> datæ longitudinis, <lb/>&longs;itque <emph type="italics"/>DLF<emph.end type="italics"/> locus lineæ <emph type="italics"/>EMG<emph.end type="italics"/> ubi <lb/>corpus ver&longs;abatur in <emph type="italics"/>D<emph.end type="italics"/>; & &longs;i ea &longs;it vis centripeta, ut areæ <emph type="italics"/>ABGE<emph.end type="italics"/><lb/>latus quadratum &longs;it ut de&longs;cendentis velocitas, erit area ip&longs;a in du­<lb/>plicata ratione velocitatis, id e&longs;t, &longs;i pro velocitatibus in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>E<emph.end type="italics"/><lb/>&longs;cribantur V & V+I, erit area <emph type="italics"/>ABFD<emph.end type="italics"/> ut VV, & area <emph type="italics"/>ABGE<emph.end type="italics"/> ut <lb/>VV+2 VI+II, & divi&longs;im area <emph type="italics"/>DFGE<emph.end type="italics"/> ut 2 VI+II, adeoque <lb/>(<emph type="italics"/>DFGE/DE<emph.end type="italics"/>) ut (2VI+II/<emph type="italics"/>DE<emph.end type="italics"/>), id e&longs;t, &longs;i primæ quantitatum na&longs;centium <lb/>rationes &longs;umantur, longitudo <emph type="italics"/>DF<emph.end type="italics"/> ut quantitas (2VI/<emph type="italics"/>DE<emph.end type="italics"/>), adeoque e­<lb/>tiam ut quantitatis hujus dimidium (IXV/<emph type="italics"/>DE<emph.end type="italics"/>). E&longs;t autem tempus quo <pb pagenum="113"/>corpus cadendo de&longs;cribit lineolam <emph type="italics"/>DE,<emph.end type="italics"/> ut lineola illa directe & <lb/> <arrow.to.target n="note89"></arrow.to.target><lb/>velocitas V inver&longs;e, e&longs;tque vis ut velocitatis incrementum I directe <lb/>& tempus inver&longs;e, adeoque &longs;i primæ na&longs;centium rationes &longs;uman­<lb/>tur, ut (IXV/<emph type="italics"/>DE<emph.end type="italics"/>), hoc e&longs;t, ut longitudo <emph type="italics"/>DF.<emph.end type="italics"/> Ergo Vis ip&longs;i <emph type="italics"/>DF<emph.end type="italics"/> vel <emph type="italics"/>EG<emph.end type="italics"/><lb/>proportionalis facit ut corpus ea cum Velocitate de&longs;cendat quæ &longs;it <lb/>ut areæ <emph type="italics"/>ABGE<emph.end type="italics"/> latus quadratum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note89"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s>Porro cum tempus, quo quælibet longitudinis datæ lineola <emph type="italics"/>DE<emph.end type="italics"/><lb/>de&longs;cribatur, &longs;it ut velocitas inver&longs;e adeoque ut areæ <emph type="italics"/>ABFD<emph.end type="italics"/> latus <lb/>quadratum inver&longs;e; &longs;itque <emph type="italics"/>DL,<emph.end type="italics"/> atque adeo area na&longs;cens <emph type="italics"/>DLME,<emph.end type="italics"/><lb/>ut idem latus quadratum inver&longs;e: erit tempus ut area <emph type="italics"/>DLME,<emph.end type="italics"/> & <lb/>&longs;umma omnium temporum ut &longs;umma omnium arearum, hoc e&longs;t <lb/>(per Corol. </s> <s>Lem. </s> <s>IV) Tempus totum quo linea <emph type="italics"/>AE<emph.end type="italics"/> de&longs;cribitur ut <lb/>area tota <emph type="italics"/>AME. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Si <emph type="italics"/>P<emph.end type="italics"/> &longs;it locus de quo corpus cadere debet, ut, urgen­<lb/>te aliqua uniformi vi centripeta nota (qualis vulgo &longs;upponitur <lb/>Gravitas) velocitatem acquirat in loco <emph type="italics"/>D<emph.end type="italics"/> æqualem velocitati <lb/>quam corpus aliud vi quacunque cadens acqui&longs;ivit eodem loco <emph type="italics"/>D,<emph.end type="italics"/><lb/>& in perpendiculari <emph type="italics"/>DF<emph.end type="italics"/> capiatur <emph type="italics"/>DR,<emph.end type="italics"/> quæ &longs;it ad <emph type="italics"/>DF<emph.end type="italics"/> ut vis illa <lb/>uniformis ad vim alteram in loco <emph type="italics"/>D,<emph.end type="italics"/> & compleatur rectangulum <lb/><emph type="italics"/>PDRQ,<emph.end type="italics"/> eique æqualis ab&longs;cindatur area <emph type="italics"/>ABFD;<emph.end type="italics"/> erit <emph type="italics"/>A<emph.end type="italics"/> locus <lb/>de quo corpus alterum cecidit. </s> <s>Namque completo rectangulo <lb/><emph type="italics"/>DRSE,<emph.end type="italics"/> cum &longs;it area <emph type="italics"/>ABFD<emph.end type="italics"/> ad aream <emph type="italics"/>DFGE<emph.end type="italics"/> ut VV ad <lb/>2VI, adeoque ut 1/2 V ad I, id e&longs;t, ut &longs;emi&longs;&longs;is velocitatis totius <lb/>ad incrementum velocitatis corporis vi inæquabili cadentis; & &longs;i­<lb/>militer area <emph type="italics"/>PQRD<emph.end type="italics"/> ad aream <emph type="italics"/>DRSE<emph.end type="italics"/> ut &longs;emi&longs;&longs;is velocitatis to­<lb/>tius ad incrementum velocitatis corporis uniformi vi cadentis; <lb/>&longs;intque incrementa illa (ob æqualitatem temporum na&longs;centium) <lb/>ut vires generatrices, id e&longs;t, ut ordinatim applicatæ <emph type="italics"/>DF, DR,<emph.end type="italics"/><lb/>adeoque ut areæ na&longs;centes <emph type="italics"/>DFGE, DRSE<emph.end type="italics"/>; erunt (ex æquo) <lb/>areæ totæ <emph type="italics"/>ABFD, PQRD<emph.end type="italics"/> ad invicem ut &longs;emi&longs;&longs;es totarum ve­<lb/>locitatum, & propterea (ob æqualitatem velocitatum) æquantur. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Unde &longs;i corpus quodlibet de loco quocunque <emph type="italics"/>D<emph.end type="italics"/> data <lb/>cum velocitate vel &longs;ur&longs;um vel deor&longs;um projiciatur, & detur lex vis <lb/>centripetæ, invenietur velocitas ejus in alio quovis loco <emph type="italics"/>e,<emph.end type="italics"/> erigen­<lb/>do ordinatam <emph type="italics"/>eg,<emph.end type="italics"/> & capiendo velocitatem illam ad velocitatem in <lb/>loco <emph type="italics"/>D<emph.end type="italics"/> ut e&longs;t latus quadratum rectanguli <emph type="italics"/>PQRD<emph.end type="italics"/> area curvili­<lb/>nea <emph type="italics"/>DFge<emph.end type="italics"/> vel aucti, &longs;i locus <emph type="italics"/>e<emph.end type="italics"/> e&longs;t loco <emph type="italics"/>D<emph.end type="italics"/> inferior, vel diminuti, <lb/>&longs;i is &longs;uperior e&longs;t, ad latus quadratum rectanguli &longs;olius <emph type="italics"/>PQRD,<emph.end type="italics"/> id <lb/>e&longs;t, ut √<emph type="italics"/>PQRD<emph.end type="italics"/>+vel-<emph type="italics"/>DFge<emph.end type="italics"/> ad √<emph type="italics"/>PQRD.<emph.end type="italics"/> <pb pagenum="114"/> <arrow.to.target n="note90"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note90"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Tempus quoque innote&longs;cet erigendo ordinatam <emph type="italics"/>em<emph.end type="italics"/> re­<lb/>ciproce proportionalem lateri quadrato ex <emph type="italics"/>PQRD<emph.end type="italics"/>+vel-<emph type="italics"/>DFge,<emph.end type="italics"/><lb/>& capiendo tempus quo corpus de&longs;crip&longs;it lineam <emph type="italics"/>De<emph.end type="italics"/> ad tempus <lb/>quo corpus alterum vi uniformi cecidit a <emph type="italics"/>P<emph.end type="italics"/> & cadendo pervenit ad <lb/><emph type="italics"/>D,<emph.end type="italics"/> ut area curvilinea <emph type="italics"/>DLme<emph.end type="italics"/> ad rectangulum 2<emph type="italics"/>PDXDL.<emph.end type="italics"/> Nam­<lb/>que tempus quo corpus vi uniformi de&longs;cendens de&longs;crip&longs;it lineam <lb/><emph type="italics"/>PD<emph.end type="italics"/> e&longs;t ad tempus quo corpus idem de&longs;crip&longs;it lineam <emph type="italics"/>PE<emph.end type="italics"/> in &longs;ub­<lb/>duplicata ratione <emph type="italics"/>PD<emph.end type="italics"/> ad <emph type="italics"/>PE,<emph.end type="italics"/> id e&longs;t (lineola <emph type="italics"/>DE<emph.end type="italics"/> jamjam na&longs;cen­<lb/>te) in ratione <emph type="italics"/>PD<emph.end type="italics"/> ad <emph type="italics"/>PD<emph.end type="italics"/>+1/2 <emph type="italics"/>DE<emph.end type="italics"/> &longs;eu 2<emph type="italics"/>PD<emph.end type="italics"/> ad 2<emph type="italics"/>PD+DE,<emph.end type="italics"/><lb/>& divi&longs;im, ad tempus quo corpus idem de&longs;crip&longs;it lineolam <emph type="italics"/>DE<emph.end type="italics"/><lb/>ut 2<emph type="italics"/>PD<emph.end type="italics"/> ad <emph type="italics"/>DE,<emph.end type="italics"/> adeoque ut rectangulum 2<emph type="italics"/>PDXDL<emph.end type="italics"/> ad aream <lb/><emph type="italics"/>DLME<emph.end type="italics"/>; e&longs;tque tempus quo corpus utrumque de&longs;crip&longs;it lineo­<lb/>lam <emph type="italics"/>DE<emph.end type="italics"/> ad tempus quo corpus alterum inæquabili motu de&longs;crip­<lb/>&longs;it lineam <emph type="italics"/>De<emph.end type="italics"/> ut area <emph type="italics"/>DLME<emph.end type="italics"/> ad aream <emph type="italics"/>DLme,<emph.end type="italics"/> & ex æquo <lb/>tempus primum ad tempus ultimum ut rectangulum 2<emph type="italics"/>PDXDL<emph.end type="italics"/><lb/>ad aream <emph type="italics"/>DLme.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>SECTIO VIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Inventione Orbium in quibus corpora Viribus quibu&longs;cunque cen­<lb/>tripetis agitata revolvuntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XL. THEOREMA XIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si corpus, cogente Vi quacunque centripeta, moveatur utcunque, & <lb/>corpus aliud recta a&longs;cendat vel de&longs;cendat, &longs;intque eorum Velocita­<lb/>tes in aliquo æqualium altitudinum ca&longs;u æquales, Velocitates eorum <lb/>in omnibus æqualibus altitudinibus erunt æquales.<emph.end type="italics"/></s> </p> <p type="main"> <s>De&longs;cendat corpus aliquod ab <emph type="italics"/>A<emph.end type="italics"/> per <emph type="italics"/>D, E,<emph.end type="italics"/> ad centrum <emph type="italics"/>C,<emph.end type="italics"/> & <lb/>moveatur corpus aliud a <emph type="italics"/>V<emph.end type="italics"/> in linea curva <emph type="italics"/>VIKk,<emph.end type="italics"/> Centro <emph type="italics"/>C<emph.end type="italics"/> in­<lb/>tervallis quibu&longs;vis de&longs;cribantur circuli concentrici <emph type="italics"/>DI, EK<emph.end type="italics"/> rectæ <lb/><emph type="italics"/>AC<emph.end type="italics"/> in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>E,<emph.end type="italics"/> curvæque <emph type="italics"/>VIK<emph.end type="italics"/> in <emph type="italics"/>I<emph.end type="italics"/> & <emph type="italics"/>K<emph.end type="italics"/> occurrentes. </s> <s>Junga­<lb/>tur <emph type="italics"/>IC<emph.end type="italics"/> occurrens ip&longs;i <emph type="italics"/>KE<emph.end type="italics"/> in <emph type="italics"/>N;<emph.end type="italics"/> & in <emph type="italics"/>IK<emph.end type="italics"/> demittatur perpendi­<lb/>culum <emph type="italics"/>NT<emph.end type="italics"/>; &longs;itque circumferentiarum circulorum intervallum <emph type="italics"/>DE<emph.end type="italics"/><lb/>vel <emph type="italics"/>IN<emph.end type="italics"/> quam minimum, & habeant corpora in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>I<emph.end type="italics"/> velocita- <pb pagenum="115"/>tes æquales. </s> <s>Quoniam di&longs;tantiæ <emph type="italics"/>CD, CI<emph.end type="italics"/> æquantur, erunt vi­</s> </p> <p type="main"> <s> <arrow.to.target n="note91"></arrow.to.target><lb/>res centripetæ in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>I<emph.end type="italics"/> æquales. </s> <s>Exponantur hæ vires per æ­<lb/>quales lineolas <emph type="italics"/>DE, IN<emph.end type="italics"/>; & &longs;i vis una <emph type="italics"/>IN<emph.end type="italics"/> (per Legum Corol. </s> <s>2.) <lb/>re&longs;olvatur in duas <emph type="italics"/>NT<emph.end type="italics"/> & <emph type="italics"/>IT,<emph.end type="italics"/> vis <emph type="italics"/>NT,<emph.end type="italics"/> agendo &longs;ecundum lineam <lb/><emph type="italics"/>NT<emph.end type="italics"/> corporis cur&longs;ui <emph type="italics"/>ITK<emph.end type="italics"/> perpendicularem, nil mutabit velocita­<lb/>tem corporis in cur&longs;u illo, &longs;ed retrahet &longs;olummodo corpus a cur­<lb/>&longs;u rectilineo, facietque ip&longs;um de Orbis tangente perpetuo deflecte­<lb/>re, inque via curvilinea <emph type="italics"/>ITKk<emph.end type="italics"/> progredi. </s> <s>In hoc effectu produ­<lb/>cendo vis illa tota con&longs;umetur: vis autem altera <emph type="italics"/>IT,<emph.end type="italics"/> &longs;ecundum <lb/>corporis cur&longs;um agendo, tota accelerabit illud, ac dato tem­<lb/>pore quam minimo accelerationem generabit &longs;ibi ip&longs;i proportiona­<lb/>lem. </s> <s>Proinde corporum in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>I<emph.end type="italics"/> accelerationes æqualibus tem­<lb/>poribus factæ (&longs;i &longs;umantur linearum na&longs;centium <emph type="italics"/>DE, IN, IK, <lb/>IT, NT<emph.end type="italics"/> rationes primæ) &longs;unt ut lineæ <emph type="italics"/>DE, IT:<emph.end type="italics"/> temporibus au­<lb/>tem inæqualibus ut lineæ illæ & tempora conjunctim. </s> <s>Tempora <lb/>autem quibus <emph type="italics"/>DE<emph.end type="italics"/> & <emph type="italics"/>IK<emph.end type="italics"/> de&longs;cribuntur, ob æqualitatem velocita­<lb/> <arrow.to.target n="fig82"></arrow.to.target><lb/>tum &longs;unt ut viæ de&longs;criptæ <emph type="italics"/>DE<emph.end type="italics"/> & <emph type="italics"/>IK,<emph.end type="italics"/> adeoque accelerationes, in <lb/>cur&longs;u corporum per lineas <emph type="italics"/>DE<emph.end type="italics"/> & <emph type="italics"/>IK,<emph.end type="italics"/> funt ut <emph type="italics"/>DE<emph.end type="italics"/> & <emph type="italics"/>IT, DE<emph.end type="italics"/> & <lb/><emph type="italics"/>IK<emph.end type="italics"/> conjunctim, id e&longs;t ut <emph type="italics"/>DE quad<emph.end type="italics"/> & <emph type="italics"/>ITXIK rectangulum.<emph.end type="italics"/> Sed <lb/><emph type="italics"/>rectangulum ITXIK<emph.end type="italics"/> æquale e&longs;t <emph type="italics"/>IN quadrato,<emph.end type="italics"/> hoc e&longs;t, æquale <lb/><emph type="italics"/>DE quadrato;<emph.end type="italics"/> & propterea accelerationes in tran&longs;itu corporum a <lb/><emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>I<emph.end type="italics"/> ad <emph type="italics"/>E<emph.end type="italics"/> & <emph type="italics"/>K<emph.end type="italics"/> æquales gcnerantur. </s> <s>Æquales igitur &longs;unt cor- <pb pagenum="116"/> <arrow.to.target n="note92"></arrow.to.target><lb/>porum velocitates in <emph type="italics"/>E<emph.end type="italics"/> & <emph type="italics"/>K<emph.end type="italics"/> & eodem argumento &longs;emper reperi­<lb/>entur æquales in &longs;ub&longs;equentibus æqualibus di&longs;tantiis. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note91"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="margin"> <s><margin.target id="note92"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig82"></figure> <p type="main"> <s>Sed & eodem argumento corpora æquivelocia & æqualiter a cen­<lb/>tro di&longs;tantia, in a&longs;cen&longs;u ad æquales di&longs;tantias æqualiter retarda­<lb/>buntur. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i corpus vel funipendulum o&longs;cilletur, vel im­<lb/>pedimento quovis politi&longs;&longs;imo & perfecte lubrico cogatur in li­<lb/>nea curva moveri, & corpus aliud recta a&longs;cendat vel de&longs;cendat, <lb/>&longs;intque velocitates eorum in eadem quacunque altitudine æquales: <lb/>erunt velocitates eorum in aliis quibu&longs;cunque æqualibus altitudi­<lb/>nibus æquales. </s> <s>Namque impedimento va&longs;is ab&longs;olute lubrici idem <lb/>præ&longs;tatur quod vi tran&longs;ver&longs;a <emph type="italics"/>NT.<emph.end type="italics"/> Corpus eo non retardatur, <lb/>non acceleratur, &longs;ed tantum cogitur de cur&longs;u rectilineo di&longs;cedere. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Hinc etiam &longs;i quantitas P &longs;it mazima a centro di&longs;tan­<lb/>tia, ad quam corpus vel o&longs;cillans vel in Trajectoria quacunque re­<lb/>volvens, deque quovis Trajectoriæ puncto, ea quam ibi habet <lb/>velocitate &longs;ur&longs;um projectum a&longs;cendere po&longs;&longs;it; &longs;itque quantitas A <lb/>di&longs;tantia corporis a centro in alio quovis Orbitæ puncto, & vis <lb/>centripeta &longs;emper &longs;it ut ip&longs;ius A dignitas quælibet A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, cujus <lb/>Index <emph type="italics"/>n<emph.end type="italics"/>-1 e&longs;t numerus quilibet <emph type="italics"/>n<emph.end type="italics"/> unitate diminutus; velocitas <lb/>corporis in omni altitudine A erit ut √P<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>-A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, atque adeo da­<lb/>tur. </s> <s>Namque velocitas recta a&longs;cendentis ac de&longs;cendentis (per Prop. </s> <s><lb/>XXXIX) e&longs;t in hac ip&longs;a ratione. <pb pagenum="117"/> <arrow.to.target n="note93"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note93"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLI. PROBLEMA XXVIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;ita cuju&longs;cunque generis Vi centripeta & conce&longs;&longs;is Figurarum <lb/>curvilinearum quadraturis, requiruntur tum Trajectoriæ in qui­<lb/>bus corpora movebuntur, tum Tempora motuum in Trajectoriis <lb/>inventis.<emph.end type="italics"/></s> </p> <p type="main"> <s>Tendat vis quælibet ad centrum <emph type="italics"/>C<emph.end type="italics"/> & invenienda &longs;it Trajectoria <lb/><emph type="italics"/>VITKk.<emph.end type="italics"/> Detur Circulus <emph type="italics"/>VXY<emph.end type="italics"/> centro <emph type="italics"/>C<emph.end type="italics"/> intervallo quovis <emph type="italics"/>CV<emph.end type="italics"/><lb/>de&longs;criptus, centroque eodem de&longs;cribantur alii quivis circuli <emph type="italics"/>ID, <lb/>KE<emph.end type="italics"/> Trajectoriam &longs;ecantes in <emph type="italics"/>I<emph.end type="italics"/> & <emph type="italics"/>K<emph.end type="italics"/> rectamque <emph type="italics"/>CV<emph.end type="italics"/> in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>E.<emph.end type="italics"/><lb/>Age tum rectam <emph type="italics"/>CNIX<emph.end type="italics"/> &longs;ecantem circulos <emph type="italics"/>KE, VY<emph.end type="italics"/> in <emph type="italics"/>N<emph.end type="italics"/> & <emph type="italics"/>X,<emph.end type="italics"/><lb/>tum rectam <emph type="italics"/>CKY<emph.end type="italics"/> occurrentem circulo <emph type="italics"/>VXY<emph.end type="italics"/> in <emph type="italics"/>Y.<emph.end type="italics"/> Sint autem <lb/>puncta <emph type="italics"/>I<emph.end type="italics"/> & <emph type="italics"/>K<emph.end type="italics"/> &longs;ibi invicem vicini&longs;&longs;ima, & pergat corpus ab <emph type="italics"/>V<emph.end type="italics"/> per <lb/><emph type="italics"/>I, T<emph.end type="italics"/> & <emph type="italics"/>K<emph.end type="italics"/> ad <emph type="italics"/>k;<emph.end type="italics"/> &longs;itque punctum <emph type="italics"/>A<emph.end type="italics"/> locus ille de quo corpus aliud <lb/>cadere debet ut in loco <emph type="italics"/>D<emph.end type="italics"/> velocitatem acquirat æqualem veloci­<lb/>tati corporis prioris in <emph type="italics"/>I<emph.end type="italics"/>; & &longs;tantibus quæ in Propo&longs;itione XXXIX, <lb/>lineola <emph type="italics"/>IK,<emph.end type="italics"/> dato tempore quam minimo de&longs;cripta, erit ut ve­<lb/>locitas atque adeo ut latus quadratum areæ <emph type="italics"/>ABFD,<emph.end type="italics"/> & triangu­<lb/>lum <emph type="italics"/>ICK<emph.end type="italics"/> tempori proportionale dabitur, adeoque <emph type="italics"/>KN<emph.end type="italics"/> erit reci­<lb/>proce ut altitudo <emph type="italics"/>IC,<emph.end type="italics"/> id e&longs;t, &longs;i detur quantitas aliqua Q, & alti­<lb/>tudo <emph type="italics"/>IC<emph.end type="italics"/> nominetur A, ut Q/A. </s> <s>Hanc quantitatem Q/A nominemus Z, <lb/>& ponamus eam e&longs;&longs;e magnitudinem ip&longs;ius Q ut &longs;it in aliquo <lb/>ca&longs;u √ <emph type="italics"/>ABFD<emph.end type="italics"/> ad Z ut e&longs;t <emph type="italics"/>IK<emph.end type="italics"/> ad <emph type="italics"/>KN,<emph.end type="italics"/> & erit in omni ca&longs;u <lb/>√<emph type="italics"/>ABFD<emph.end type="italics"/> ad Z ut <emph type="italics"/>IK<emph.end type="italics"/> ad <emph type="italics"/>KN,<emph.end type="italics"/> & <emph type="italics"/>ABFD<emph.end type="italics"/> ad ZZ ut <emph type="italics"/><expan abbr="IKq.">IKque</expan><emph.end type="italics"/> ad <emph type="italics"/><expan abbr="KNq.">KNque</expan><emph.end type="italics"/><lb/>& divi&longs;im <emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ ad ZZ ut <emph type="italics"/>IN quad<emph.end type="italics"/> ad <emph type="italics"/>KN quad,<emph.end type="italics"/> ad­<lb/>eoque √<emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ ad (Z &longs;eu)Q/A ut <emph type="italics"/>IN<emph.end type="italics"/> ad <emph type="italics"/>KN,<emph.end type="italics"/> & propterea <lb/>AX<emph type="italics"/>KN<emph.end type="italics"/> æquale (QX<emph type="italics"/>IN/√ABFD<emph.end type="italics"/>-ZZ). Unde cum <emph type="italics"/>YXXXC<emph.end type="italics"/> &longs;it ad <lb/>AX<emph type="italics"/>KN<emph.end type="italics"/> ut <emph type="italics"/>CXq<emph.end type="italics"/> ad AA, erit rectangulum <emph type="italics"/>YXXXC<emph.end type="italics"/> æquale <lb/>(QX<emph type="italics"/>INXCX quad.<emph.end type="italics"/>/AA√<emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ). Igitur &longs;i in perpendiculo <emph type="italics"/>DF<emph.end type="italics"/> capiantur <lb/>&longs;emper <emph type="italics"/>Db, Dc<emph.end type="italics"/> ip&longs;is (Q/2√<emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ) & (QX<emph type="italics"/>CX quad.<emph.end type="italics"/>/2AA√<emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ) <lb/>æquales re&longs;pective, & de&longs;cribantur curvæ lineæ <emph type="italics"/>ab, cd<emph.end type="italics"/> quas <pb pagenum="118"/> <arrow.to.target n="note94"></arrow.to.target><lb/>puncta <emph type="italics"/>b, c<emph.end type="italics"/> perpetuo tangunt; deque puncto <emph type="italics"/>V<emph.end type="italics"/> ad lineam <emph type="italics"/>AC<emph.end type="italics"/> eri­<lb/>gatur perpendiculum <emph type="italics"/>Vad<emph.end type="italics"/> ab&longs;cindens areas curvilineas <emph type="italics"/>VDba, <lb/>VDcd,<emph.end type="italics"/> & erigantur etiam ordinatæ <emph type="italics"/>Ez, Ex:<emph.end type="italics"/> quoniam rectan­<lb/>gulum <emph type="italics"/>DbXIN<emph.end type="italics"/> &longs;eu <emph type="italics"/>DbzE<emph.end type="italics"/> æquale e&longs;t dimidio rectanguli <lb/>AX<emph type="italics"/>KN,<emph.end type="italics"/> &longs;eu triangulo <emph type="italics"/>ICK<emph.end type="italics"/>; & rectangulum <emph type="italics"/>DcXIN<emph.end type="italics"/> &longs;eu <lb/><emph type="italics"/>DcxE<emph.end type="italics"/> æquale e&longs;t dimidio rectanguli <emph type="italics"/>YXXXC,<emph.end type="italics"/> &longs;eu triangulo <lb/><emph type="italics"/>XCY;<emph.end type="italics"/> hoc e&longs;t, quoniam arearum <emph type="italics"/>VDba, VIC<emph.end type="italics"/> æquales &longs;emper <lb/>&longs;unt na&longs;centes particulæ <emph type="italics"/>DbzE, ICK,<emph.end type="italics"/> & arearum <emph type="italics"/>VDcd, <lb/>VCX<emph.end type="italics"/> æquales &longs;emper &longs;unt na&longs;centes particulæ <emph type="italics"/>DcxE, XCY,<emph.end type="italics"/><lb/>erit area genita <emph type="italics"/>VDba<emph.end type="italics"/> æqualis areæ genitæ <emph type="italics"/>VIC,<emph.end type="italics"/> adeoque tem­<lb/>pori proportionalis, & area genita <emph type="italics"/>VDcd<emph.end type="italics"/> æqualis Sectori ge­<lb/>nito <emph type="italics"/>VCX.<emph.end type="italics"/> Dato igitur tempore quovis ex quo corpus di&longs;ce&longs;­<lb/>&longs;it de loco <emph type="italics"/>V,<emph.end type="italics"/> dabitur area ip&longs;i proportionalis <emph type="italics"/>VDba,<emph.end type="italics"/> & inde <lb/>dabitur corporis altitudo <emph type="italics"/>CD<emph.end type="italics"/> vel <emph type="italics"/>CI<emph.end type="italics"/>; & area <emph type="italics"/>VDcd,<emph.end type="italics"/> eique <lb/>æqualis Sector <emph type="italics"/>VCX<emph.end type="italics"/> una cum ejus angulo <emph type="italics"/>VCI.<emph.end type="italics"/> Datis autem <lb/>angulo <emph type="italics"/>VCI<emph.end type="italics"/> & altitudine <emph type="italics"/>CI<emph.end type="italics"/> datur locus <emph type="italics"/>I,<emph.end type="italics"/> in quo corpus com­<lb/>pleto illo tempore reperietur. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note94"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc maximæ minimæque corporum altitudines, id e&longs;t <lb/>Ap&longs;ides Trajectoriarum expedite inveniri po&longs;&longs;unt. </s> <s>Sunt enim <lb/>Ap&longs;ides puncta illa in quibus recta <emph type="italics"/>IC<emph.end type="italics"/> per centrum ducta incidit <lb/>perpendiculariter in Trajectoriam <emph type="italics"/>VIK:<emph.end type="italics"/> id quod &longs;it ubi rectæ <emph type="italics"/>IK<emph.end type="italics"/><lb/>& <emph type="italics"/>NK<emph.end type="italics"/> æquantur, adeoque ubi area <emph type="italics"/>ABFD<emph.end type="italics"/> æqualis e&longs;t ZZ. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Sed & angulus <emph type="italics"/>KIN,<emph.end type="italics"/> in quo Trajectoria alibi &longs;ecat <lb/>lineam illam <emph type="italics"/>IC,<emph.end type="italics"/> ex data corporis altitudine <emph type="italics"/>IC<emph.end type="italics"/> expedite inveni­<lb/>tur; nimirum capiendo &longs;inum ejus ad radium ut <emph type="italics"/>KN<emph.end type="italics"/> ad <emph type="italics"/>IK,<emph.end type="italics"/> id <lb/>e&longs;t, ut Z ad latus quadratum areæ <emph type="italics"/>ABFD.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Si centro <emph type="italics"/>C<emph.end type="italics"/> & vertice principali <emph type="italics"/>V<emph.end type="italics"/> de&longs;cribatur Sectio quæ­<lb/>libet Conica <emph type="italics"/>VRS,<emph.end type="italics"/> & a quovis ejus puncto <emph type="italics"/>R<emph.end type="italics"/> agatur Tangens <emph type="italics"/>RT<emph.end type="italics"/><lb/>occurrens axi infinite producto <emph type="italics"/>CV<emph.end type="italics"/> in puncto <emph type="italics"/>T;<emph.end type="italics"/> dein juncta <emph type="italics"/>CR<emph.end type="italics"/><lb/>ducatur recta <emph type="italics"/>CP,<emph.end type="italics"/> quæ æqualis &longs;it ab&longs;ci&longs;&longs;æ <emph type="italics"/>CT,<emph.end type="italics"/> angulumque <emph type="italics"/>VCP<emph.end type="italics"/><lb/>Sectori <emph type="italics"/>VCR<emph.end type="italics"/> proportionalem con&longs;tituat; tendat autem ad centrum <emph type="italics"/>C<emph.end type="italics"/><lb/>Vis centripeta Cubo di&longs;tantiæ locorum a centro reciproce propor­<lb/>tionalis, & exeat corpus de loco <emph type="italics"/>V<emph.end type="italics"/> ju&longs;ta cum Velocitate &longs;ecundum <lb/>lineam rectæ <emph type="italics"/>CV<emph.end type="italics"/> perpendicularem: progredietur corpus illud in <lb/>Trajectoria quam punctum <emph type="italics"/>P<emph.end type="italics"/> perpetuo tangit; adeoque &longs;i Conica <lb/>&longs;ectio <emph type="italics"/>CVRS<emph.end type="italics"/> Hyperbola &longs;it, de&longs;cendet idem ad centrum: Sin <lb/>ea Ellip&longs;is &longs;it, a&longs;cendet illud perpetuo & abibit in infinitum. </s> <s>Et con­<lb/>tra, &longs;i corpus quacunque cum Velocitate exeat de loco <emph type="italics"/>V,<emph.end type="italics"/> & perin­<lb/>de ut incæperit vel oblique de&longs;cendere ad centrum, vel ab eo ob- <pb pagenum="119"/>lique a&longs;cendere, Figura <emph type="italics"/>CVRS<emph.end type="italics"/> vel Hyperbola &longs;it vel Ellip&longs;is, in­<lb/> <arrow.to.target n="note95"></arrow.to.target><lb/>veniri pote&longs;t Trajectoria augendo vel minuendo angulum <emph type="italics"/>VCP<emph.end type="italics"/><lb/>in data aliqua ratione. </s> <s>Sed &, Vi centripeta in centrifugam ver&longs;a, <lb/> <arrow.to.target n="fig83"></arrow.to.target><lb/>a&longs;cendet corpus oblique in Trajectoria <emph type="italics"/>VPQ<emph.end type="italics"/> quæ invenitur capi­<lb/>endo angulum <emph type="italics"/>VCP<emph.end type="italics"/> Sectori Elliptico <emph type="italics"/>CVRC<emph.end type="italics"/> proportionalem, & <lb/>longitudinem <emph type="italics"/>CP<emph.end type="italics"/> longitudini <emph type="italics"/>CT<emph.end type="italics"/> æqualem ut &longs;upra. </s> <s>Con&longs;equun­<lb/>tur hæc omnia ex Propo&longs;itione præcedente, per Curvæ cuju&longs;dam <lb/>quadraturam, cujus inventionem, ut &longs;atis facilem, brevitatis gratia <lb/>mi&longs;&longs;am facio. </s> </p> <p type="margin"> <s><margin.target id="note95"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig83"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO XLII. PROBLEMA XXIX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Data lege Vis centripetæ, requiritur motus corporis de loco dato <lb/>data cum Velocitate &longs;ecundum datam rectam egre&longs;&longs;i.<emph.end type="italics"/></s> </p> <p type="main"> <s>Stantibus quæ in tribus Propo&longs;itionibus præcedentibus: exeat <lb/>corpus de loco <emph type="italics"/>I<emph.end type="italics"/> &longs;ecundum lineolam <emph type="italics"/>IT,<emph.end type="italics"/> ea cum Velocitate quam <lb/>corpus aliud, vi aliqua uniformi centripeta, de loco <emph type="italics"/>P<emph.end type="italics"/> cadendo ac­<lb/>quirere po&longs;&longs;et in <emph type="italics"/>D:<emph.end type="italics"/> &longs;itque hæc vis uniformis ad vim qua corpus <pb pagenum="120"/> <arrow.to.target n="note96"></arrow.to.target><lb/>primum urgetur in <emph type="italics"/>I,<emph.end type="italics"/> ut <emph type="italics"/>DR<emph.end type="italics"/> ad <emph type="italics"/>DF.<emph.end type="italics"/> Pergat autem corpus ver&longs;us <lb/><emph type="italics"/>k;<emph.end type="italics"/> centroque <emph type="italics"/>C<emph.end type="italics"/> & intervallo <emph type="italics"/>Ck<emph.end type="italics"/> de&longs;cribatur circulus <emph type="italics"/>ke<emph.end type="italics"/> occurrens <lb/>rectæ <emph type="italics"/>PD<emph.end type="italics"/> in <emph type="italics"/>e,<emph.end type="italics"/> & erigantur curvarum <emph type="italics"/>ALMm, BFGg, abzv, dcxw<emph.end type="italics"/><lb/> <arrow.to.target n="fig84"></arrow.to.target><lb/>ordinatim applicatæ <emph type="italics"/>em, eg, ev, ew.<emph.end type="italics"/> Ex dato rectangulo <emph type="italics"/>PDRQ,<emph.end type="italics"/><lb/>dataque lege vis centripetæ qua corpus primum agitatur, dantur cur­<lb/>væ lineæ <emph type="italics"/>BFGg, ALMm,<emph.end type="italics"/> per con&longs;tructionem Problematis XXVII, <lb/>& ejus Corol. </s> <s>1. Deinde ex dato angulo <emph type="italics"/>CIT<emph.end type="italics"/> datur proportio na&longs;cen­<lb/>tium <emph type="italics"/>IK, KN,<emph.end type="italics"/> & inde, per con&longs;tructionem Prob. </s> <s>XXVIII, datur <lb/>quantitas Q, una cum curvis lineis <emph type="italics"/>abzv, dcxw:<emph.end type="italics"/> adeoque com­<lb/>pleto tempore quovis <emph type="italics"/>Dbve,<emph.end type="italics"/> datur tum corporis altitudo <emph type="italics"/>Ce<emph.end type="italics"/> vel <emph type="italics"/>Ck,<emph.end type="italics"/><lb/>tum area <emph type="italics"/>Dcwe,<emph.end type="italics"/> eique æqualis Sector <emph type="italics"/>XCy,<emph.end type="italics"/> angulu&longs;que <emph type="italics"/>ICk<emph.end type="italics"/> & <lb/>locus <emph type="italics"/>k<emph.end type="italics"/> in quo corpus tunc ver&longs;abitur. <emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note96"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig84"></figure> <p type="main"> <s>Supponimus autem in his Propo&longs;itionibus Vim centripetam in <lb/>rece&longs;&longs;u quidem a centro variari &longs;ecundum legem quamcunque quam <lb/>quis imaginari pote&longs;t, in æqualibus autem a centro di&longs;tantiis e&longs;&longs;e <lb/>undeque eandem. </s> <s>Atque hactenus Motum corporum in Orbibus <lb/>immobilibus con&longs;ideravimus. </s> <s>Supere&longs;t ut de Motu eorum in Orbi­<lb/>bus qui circa centrum virium revolvuntur adjiciamus pauca. <pb pagenum="121"/> <arrow.to.target n="note97"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note97"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>SECTIO IX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu Corporum in Orbibus mobilibus, deque motu Apfidum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIII. PROBLEMA XXX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Efficiendum est ut corpus in Trajectoria quacunque circa centrum <lb/>Virium revolvente perinde moveri po&longs;&longs;it, atque corpus aliud in <lb/>eadem Trajectoria quie&longs;cente.<emph.end type="italics"/></s> </p> <p type="main"> <s>In Orbe <emph type="italics"/>VPK<emph.end type="italics"/> po­<lb/> <arrow.to.target n="fig85"></arrow.to.target><lb/>&longs;itione dato revolvatur <lb/>corpus <emph type="italics"/>P<emph.end type="italics"/> pergendo a <lb/><emph type="italics"/>V<emph.end type="italics"/> ver&longs;us <emph type="italics"/>K.<emph.end type="italics"/> A centro <lb/><emph type="italics"/>C<emph.end type="italics"/> agatur &longs;emper <emph type="italics"/>Cp,<emph.end type="italics"/><lb/>quæ &longs;it ip&longs;i <emph type="italics"/>CP<emph.end type="italics"/> æqualis, <lb/>angulumque <emph type="italics"/>VCp<emph.end type="italics"/> an­<lb/>gulo <emph type="italics"/>VCP<emph.end type="italics"/> proportio­<lb/>nalem con&longs;tituat; & a­<lb/>rea quam linea <emph type="italics"/>Cp<emph.end type="italics"/> de­<lb/>&longs;cribit erit ad aream <lb/><emph type="italics"/>VCP<emph.end type="italics"/> quam linea <emph type="italics"/>CP<emph.end type="italics"/><lb/>&longs;imul de&longs;cribit, ut velo­<lb/>citas lineæ de&longs;cribentis <lb/><emph type="italics"/>Cp<emph.end type="italics"/> ad velocitatem li­<lb/>neæ de&longs;cribentis <emph type="italics"/>CP<emph.end type="italics"/>; <lb/>hoc e&longs;t, ut angulus <emph type="italics"/>VCp<emph.end type="italics"/> ad angulum <emph type="italics"/>VCP,<emph.end type="italics"/> adeoque in data ra­<lb/>tione, & propterea tempori proportionalis. </s> <s>Cum area tempori <lb/>proportionalis &longs;it quam linea <emph type="italics"/>Cp<emph.end type="italics"/> in plano immobili de&longs;cribit, ma­<lb/>nife&longs;tum e&longs;t quod corpus, cogente ju&longs;tæ quantitatis Vi centripeta, <lb/>revolvi po&longs;&longs;it una cum puncto <emph type="italics"/>p<emph.end type="italics"/> in Curva illa linea quam punctum <lb/>idem <emph type="italics"/>p<emph.end type="italics"/> ratione jam expo&longs;ita de&longs;cribit in plano immobili. </s> <s>Fiat angu­<lb/>lus <emph type="italics"/>VCu<emph.end type="italics"/> angulo <emph type="italics"/>PCp,<emph.end type="italics"/> & linea <emph type="italics"/>Cu<emph.end type="italics"/> lineæ <emph type="italics"/>CV,<emph.end type="italics"/> atque Figura <emph type="italics"/>uCp<emph.end type="italics"/> Fi­<lb/>guræ <emph type="italics"/>VCP<emph.end type="italics"/> æqualis, & corpus in <emph type="italics"/>p<emph.end type="italics"/> &longs;emper exi&longs;tens movebitur in <pb pagenum="122"/> <arrow.to.target n="note98"></arrow.to.target><lb/>perimetro Figuræ revolventis <emph type="italics"/>uCp,<emph.end type="italics"/> eodemque tempore de&longs;cribet <lb/>arcum ejus <emph type="italics"/>up<emph.end type="italics"/> quo corpus aliud <emph type="italics"/>P<emph.end type="italics"/> arcum ip&longs;i &longs;imilem & æqualem <lb/><emph type="italics"/>VP<emph.end type="italics"/> in Figura quie&longs;cente <emph type="italics"/>VPK<emph.end type="italics"/> de&longs;cribere pote&longs;t. </s> <s>Quæratur igi­<lb/>tur, per Corollarium quintum propo&longs;itionis VI, Vis centripeta qua <lb/>corpus revolvi po&longs;&longs;it in Curva illa linea quam punctum <emph type="italics"/>p<emph.end type="italics"/> de&longs;cribit <lb/>in plano immobili, & &longs;olvetur Problema. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note98"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig85"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIV. THEOREMA XIV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Differentia Virium, quibus corpus in Orbe quie&longs;cente, & corpus a­<lb/>liud in eodem Orbe revolvente æqualiter moveri po&longs;&longs;unt, est <lb/>in triplicata ratione communis altitudinis inver&longs;e.<emph.end type="italics"/></s> </p> <p type="main"> <s>Partibus Orbis quie­<lb/> <arrow.to.target n="fig86"></arrow.to.target><lb/>&longs;centis <emph type="italics"/>VP, PK<emph.end type="italics"/> &longs;unto <lb/>&longs;imiles & æquales Or­<lb/>bis revolventis partes <lb/><emph type="italics"/>up, pk<emph.end type="italics"/>; & punctorum <lb/><emph type="italics"/>P, K<emph.end type="italics"/> di&longs;tantia intelli­<lb/>gatur e&longs;&longs;e quam mini­<lb/>ma. </s> <s>A puncto <emph type="italics"/>k<emph.end type="italics"/> in re­<lb/>ctam <emph type="italics"/>pC<emph.end type="italics"/> demitte per­<lb/>pendiculum <emph type="italics"/>kr,<emph.end type="italics"/> idem­<lb/>que produc ad <emph type="italics"/>m,<emph.end type="italics"/> ut &longs;it <lb/><emph type="italics"/>mr<emph.end type="italics"/> ad <emph type="italics"/>kr<emph.end type="italics"/> ut angulus <lb/><emph type="italics"/>VCp<emph.end type="italics"/> ad angulum <emph type="italics"/>VCP.<emph.end type="italics"/><lb/>Quoniam corporum al­<lb/>titudines <emph type="italics"/>PC<emph.end type="italics"/> & <emph type="italics"/>pC, KC<emph.end type="italics"/><lb/>& <emph type="italics"/>kC<emph.end type="italics"/> &longs;emper æquan­<lb/>tur, manife&longs;tum e&longs;t quod linearum <emph type="italics"/>PC<emph.end type="italics"/> & <emph type="italics"/>pC<emph.end type="italics"/> incrementa vel <lb/>decrementa &longs;emper &longs;int æqualia, ideoque &longs;i corporum in locis <lb/><emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>p<emph.end type="italics"/> exi&longs;tentium diftinguantur motus &longs;inguli (per Legum <lb/>Corol. </s> <s>2.) in binos, quorum hi ver&longs;us centrum, &longs;ive &longs;ecundum <lb/>lineas <emph type="italics"/>PC, pC<emph.end type="italics"/> determinentur, & alteri prioribus tran&longs;ver&longs;i &longs;int, <lb/>& &longs;ecundum lineas ip&longs;is <emph type="italics"/>PC, pC<emph.end type="italics"/> perpendiculares directionem <lb/>habeant; motus ver&longs;us centrum erunt æquales, & motus tran&longs;­<lb/>ver&longs;us corporis <emph type="italics"/>p<emph.end type="italics"/> erit ad motum tran&longs;ver&longs;um corporis <emph type="italics"/>P,<emph.end type="italics"/> ut mo­<lb/>tus angularis lineæ <emph type="italics"/>pC,<emph.end type="italics"/> ad motum angularem lineæ <emph type="italics"/>PC,<emph.end type="italics"/> id e&longs;t, <pb pagenum="123"/>ut angulus <emph type="italics"/>VCp<emph.end type="italics"/> ad angulum <emph type="italics"/>VCP.<emph.end type="italics"/> Igitur eodem tempore quo <lb/> <arrow.to.target n="note99"></arrow.to.target><lb/>corpus <emph type="italics"/>P<emph.end type="italics"/> motu &longs;uo utroque pervenit ad punctum <emph type="italics"/>K,<emph.end type="italics"/> corpus <emph type="italics"/>p<emph.end type="italics"/> æ­<lb/>quali in centrum motu æqualiter movebitur a <emph type="italics"/>p<emph.end type="italics"/> ver&longs;us <emph type="italics"/>C,<emph.end type="italics"/> adeoque <lb/>completo illo tempore reperietur alicubi in linea <emph type="italics"/>mkr,<emph.end type="italics"/> quæ per <lb/>punctum <emph type="italics"/>k<emph.end type="italics"/> in lineam <emph type="italics"/>pC<emph.end type="italics"/> perpendicularis e&longs;t; & motu tran&longs;ver&longs;o <lb/>acquiret di&longs;tantiam a linea <emph type="italics"/>pC,<emph.end type="italics"/> quæ &longs;it ad di&longs;tantiam quam cor­<lb/>pus alterum <emph type="italics"/>P<emph.end type="italics"/> acquirit a linea <emph type="italics"/>PC,<emph.end type="italics"/> ut e&longs;t motus tran&longs;ver&longs;us cor­<lb/>poris <emph type="italics"/>p<emph.end type="italics"/> ad motum tran&longs;ver&longs;um corporis alterius <emph type="italics"/>P.<emph.end type="italics"/> Quare cum <lb/><emph type="italics"/>kr<emph.end type="italics"/> æqualis &longs;it di&longs;tantiæ quam corpus <emph type="italics"/>P<emph.end type="italics"/> acquirit a linea <emph type="italics"/>PC,<emph.end type="italics"/> &longs;itque <lb/><emph type="italics"/>mr<emph.end type="italics"/> ad <emph type="italics"/>kr<emph.end type="italics"/> ut angulus <emph type="italics"/>VCp<emph.end type="italics"/> ad angulum <emph type="italics"/>VCP,<emph.end type="italics"/> hoc e&longs;t, ut motus <lb/>tran&longs;ver&longs;us corporis <emph type="italics"/>p<emph.end type="italics"/> ad motum tran&longs;ver&longs;um corporis <emph type="italics"/>P,<emph.end type="italics"/> manife­<lb/>&longs;tum e&longs;t quod corpus <emph type="italics"/>p<emph.end type="italics"/> completo illo tempore reperietur in loco <lb/><emph type="italics"/>m.<emph.end type="italics"/> Hæc ita &longs;e habebunt ubi corpora <emph type="italics"/>p<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> æqualiter &longs;ecundum <lb/>lineas <emph type="italics"/>pC<emph.end type="italics"/> & <emph type="italics"/>PC<emph.end type="italics"/> moventur, adeoque æqualibus Viribus &longs;ecundum <lb/>lineas illas urgentur. </s> <s>Capiatur autem angulum <emph type="italics"/>pCn<emph.end type="italics"/> ad angulum <lb/><emph type="italics"/>pCk<emph.end type="italics"/> ut e&longs;t angulus <emph type="italics"/>VCp<emph.end type="italics"/> ad angulus <emph type="italics"/>VCP,<emph.end type="italics"/> &longs;itque <emph type="italics"/>nC<emph.end type="italics"/> æqualis <lb/><emph type="italics"/>kC,<emph.end type="italics"/> & corpus <emph type="italics"/>p<emph.end type="italics"/> completo illo tempore revera reperietur in <emph type="italics"/>n<emph.end type="italics"/>; ad­<lb/>eoque Vi majore urgetur quam corpus <emph type="italics"/>P,<emph.end type="italics"/> &longs;i modo angulus <emph type="italics"/>mCp<emph.end type="italics"/><lb/>angulo <emph type="italics"/>kCp<emph.end type="italics"/> major e&longs;t, id e&longs;t &longs;i Orbis <emph type="italics"/>upk<emph.end type="italics"/> vel movetur in con­<lb/>&longs;equentia, vel movetur in antecedentia majore celeritate quam <lb/>&longs;it dupla ejus qua linea <emph type="italics"/>CP<emph.end type="italics"/> in con&longs;equentia fertur; & Vi mino­<lb/>re &longs;i Orbis tardius movetur in antecedentia. </s> <s>E&longs;tque Virium dif­<lb/>ferentia ut locorum intervallum <emph type="italics"/>mn,<emph.end type="italics"/> per quod corpus illud <emph type="italics"/>p<emph.end type="italics"/><lb/>ip&longs;ius actione, dato illo temporis &longs;patio, transferri debet. </s> <s>Centro <lb/><emph type="italics"/>C<emph.end type="italics"/> intervallo <emph type="italics"/>Cn<emph.end type="italics"/> vel <emph type="italics"/>Ck<emph.end type="italics"/> de&longs;cribi intelligatur Circulus &longs;ecans <lb/>lineas <emph type="italics"/>mr, mn<emph.end type="italics"/> productas in <emph type="italics"/>s<emph.end type="italics"/> & <emph type="italics"/>t,<emph.end type="italics"/> & erit rectangulum <emph type="italics"/>mnXmt<emph.end type="italics"/> æ­<lb/>quale rectangulo <emph type="italics"/>mkXms,<emph.end type="italics"/> adeoque <emph type="italics"/>mn<emph.end type="italics"/> æquale (<emph type="italics"/>mkXms/mt<emph.end type="italics"/>). Cum <lb/>autem triangula <emph type="italics"/>pCk, pCn<emph.end type="italics"/> dentur magnitudine, &longs;unt <emph type="italics"/>kr<emph.end type="italics"/> & <emph type="italics"/>mr,<emph.end type="italics"/><lb/>earumque differentia <emph type="italics"/>mk<emph.end type="italics"/> & &longs;umma <emph type="italics"/>ms<emph.end type="italics"/> reciproce ut altitudo <emph type="italics"/>pC,<emph.end type="italics"/><lb/>adeoque rectangulum <emph type="italics"/>mkXms<emph.end type="italics"/> e&longs;t reciproce ut quadratum altitudi­<lb/>nis <emph type="italics"/>pC.<emph.end type="italics"/> E&longs;t & <emph type="italics"/>mt<emph.end type="italics"/> directe ut 1/2 <emph type="italics"/>mt,<emph.end type="italics"/> id e&longs;t, ut altitudo <emph type="italics"/>pC.<emph.end type="italics"/> Hæ <lb/>&longs;unt primæ rationes linearum na&longs;centium; & hinc fit (<emph type="italics"/>mkXms/mt<emph.end type="italics"/>), id <lb/>e&longs;t lineola na&longs;cens <emph type="italics"/>mn,<emph.end type="italics"/> eique proportionalis Virium differentia reci­<lb/>proce ut cubus altitudinis <emph type="italics"/>pC. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note99"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig86"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc differentia virium in locis <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>p<emph.end type="italics"/> vel <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>k,<emph.end type="italics"/> e&longs;t <lb/>ad vim qua corpus motu Circulari revolvi po&longs;&longs;it ab <emph type="italics"/>R<emph.end type="italics"/> ad <emph type="italics"/>K<emph.end type="italics"/> eodem <lb/>tempore quo corpus <emph type="italics"/>P<emph.end type="italics"/> in Orbe immobili de&longs;cribit arcum <emph type="italics"/>PK,<emph.end type="italics"/> ut <lb/>lineola na&longs;cens <emph type="italics"/>mn<emph.end type="italics"/> ad &longs;inum ver&longs;um arcus na&longs;centis <emph type="italics"/>RK,<emph.end type="italics"/> id e&longs;t <pb pagenum="124"/> <arrow.to.target n="note100"></arrow.to.target><lb/>ut (<emph type="italics"/>mkXms/mt<emph.end type="italics"/>) ad (<emph type="italics"/>rkq/2kC<emph.end type="italics"/>), vel ut <emph type="italics"/>mkXms<emph.end type="italics"/> ad <emph type="italics"/>rk<emph.end type="italics"/> quadratum; hoc e&longs;t, &longs;i <lb/>capiantur datæ quantitates F, G in ea ratione ad invicem quam <lb/>habet angulus <emph type="italics"/>VCP<emph.end type="italics"/> ad angulum <emph type="italics"/>VCp,<emph.end type="italics"/> ut GG-FF ad FF. </s> <s>Et <lb/>propterea, &longs;i centro <emph type="italics"/>C<emph.end type="italics"/> intervallo quovis <emph type="italics"/>CP<emph.end type="italics"/> vel <emph type="italics"/>Cp<emph.end type="italics"/> de&longs;cribatur <lb/>Sector circularis æqualis areæ toti <emph type="italics"/>VPC,<emph.end type="italics"/> quam corpus <emph type="italics"/>P<emph.end type="italics"/> tempore <lb/>quovis in Orbe immobili revolvens radio ad centrum ducto de­<lb/>&longs;crip &longs;it: differentia virium, quibus corpus <emph type="italics"/>P<emph.end type="italics"/> in Orbe immobili & <lb/>corpus <emph type="italics"/>p<emph.end type="italics"/> in Orbe mobili revolvuntur, erit ad vim centripetam qua <lb/>corpus aliquod radio ad centrum ducto Sectorem illum, eodem tem­<lb/>pore quo de&longs;cripta &longs;it area <emph type="italics"/>VPC<emph.end type="italics"/> uniformiter de&longs;eribere potui&longs;&longs;et, <lb/>ut GG-FF ad FF. </s> <s>Namque Sector ille & area <emph type="italics"/>pCk<emph.end type="italics"/> &longs;unt ad in­<lb/>vicem ut tempora quibus de&longs;cribuntur. </s> </p> <p type="margin"> <s><margin.target id="note100"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Si Orbis <emph type="italics"/>VPK<emph.end type="italics"/> Ellip&longs;is &longs;it umbilicum habens <emph type="italics"/>C<emph.end type="italics"/> & Ap­<lb/>&longs;idem &longs;ummam <emph type="italics"/>V;<emph.end type="italics"/> eique &longs;imilis & æqualis ponatur Ellip&longs;is <emph type="italics"/>upk,<emph.end type="italics"/><lb/>ita ut &longs;it &longs;emper <emph type="italics"/>pC<emph.end type="italics"/> æqualis <emph type="italics"/>PC,<emph.end type="italics"/> & angulus <emph type="italics"/>VCp<emph.end type="italics"/> &longs;it ad angulum <lb/><emph type="italics"/>VCP<emph.end type="italics"/> in data ratione G ad F; pro altitudine autem <emph type="italics"/>PC<emph.end type="italics"/> vel <emph type="italics"/>pC<emph.end type="italics"/><lb/>&longs;cribatur A, & pro Ellip&longs;eos latere recto ponatur 2 R: erit vis qua <lb/>corpus in Ellip&longs;i mobili revolvi pote&longs;t, ut (FF/AA)+(RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>) <lb/>& contra. </s> <s>Exponatur enim vis qua corpus revolvatur in immota <lb/>Ellip&longs;i per quantitatem (FF/AA), & vis in <emph type="italics"/>V<emph.end type="italics"/> erit (FF/<emph type="italics"/>CV quad.<emph.end type="italics"/>). Vis au­<lb/>tem qua corpus in Circulo ad di&longs;tantiam <emph type="italics"/>CV<emph.end type="italics"/> ea cum velocitate <lb/>revolvi po&longs;&longs;et quam corpus in Ellip&longs;i revolvens habet in <emph type="italics"/>V,<emph.end type="italics"/><lb/>e&longs;t ad vim qua corpus in Ellip&longs;i revolvens urgetur in Ap&longs;ide <emph type="italics"/>V,<emph.end type="italics"/><lb/>ut dimidium lateris recti Ellip&longs;eos. </s> <s>ad Circuli &longs;emidiametrum <emph type="italics"/>CV,<emph.end type="italics"/><lb/>adeoque valet (RFF/<emph type="italics"/>CV cub.<emph.end type="italics"/>): & vis quæ &longs;it ad hanc ut GG-FF ad <lb/>FF, valet (RGG-RFF/<emph type="italics"/>CV cub.<emph.end type="italics"/>): e&longs;tque hæc vis (per hujus Corol. </s> <s>1.) <lb/>differentia virium in <emph type="italics"/>V<emph.end type="italics"/> quibus corpus <emph type="italics"/>P<emph.end type="italics"/> in Ellipfi immota <emph type="italics"/>VPK,<emph.end type="italics"/><lb/>& corpus <emph type="italics"/>p<emph.end type="italics"/> in Ellip&longs;i mobili <emph type="italics"/>upk<emph.end type="italics"/> revolvuntur. </s> <s>Unde cum (per <lb/>hanc Prop.) differentia illa in alia quavis altitudine A &longs;it ad &longs;e­<lb/>ip&longs;am in altitudine <emph type="italics"/>CV<emph.end type="italics"/> ut (1/A <emph type="italics"/>cub.<emph.end type="italics"/>) ad (1/<emph type="italics"/>CV cub.<emph.end type="italics"/>), eadem differentia <lb/>in omni altitudine. </s> <s>A valebit (RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>). Igitur ad vim (FF/AA) <lb/>qua corpus revolvi pote&longs;t in Ellip&longs;i immobili <emph type="italics"/>VPK,<emph.end type="italics"/> addatur ex­<lb/>ce&longs;&longs;us (RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>) & componetur vis tota (FF/AA)+(RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>) <pb pagenum="125"/>qua corpus in Ellip&longs;i mobili <emph type="italics"/>upk<emph.end type="italics"/> ii&longs;dem temporibus revolvi <lb/> <arrow.to.target n="note101"></arrow.to.target><lb/>po&longs;&longs;it. </s> </p> <p type="margin"> <s><margin.target id="note101"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Ad eundem modum colligetur quod, &longs;i Orbis immo­<lb/>bilis <emph type="italics"/>VPK<emph.end type="italics"/> Ellip&longs;is &longs;it centrum habens in virium centro <emph type="italics"/>C<emph.end type="italics"/>; ei­<lb/>que &longs;imilis, æqualis & concentrica ponatur Ellip&longs;is mobilis <emph type="italics"/>upk;<emph.end type="italics"/><lb/>&longs;itque 2 R Ellip&longs;eos hujus latus rectum principale, & 2T latus <lb/>tran&longs;ver&longs;um &longs;ive axis major, atque angulus <emph type="italics"/>VCp<emph.end type="italics"/> &longs;emper &longs;it ad <lb/>angulum <emph type="italics"/>VCP<emph.end type="italics"/> ut G ad F; vires quibus corpora in Ellip&longs;i im­<lb/>mobili & mobili temporibus æqualibus revolvi po&longs;&longs;unt, erunt ut <lb/>(FFA/T <emph type="italics"/>cub.<emph.end type="italics"/>) & (FFA/T <emph type="italics"/>cub.<emph.end type="italics"/>)+(RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>) re&longs;pective. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Et univer&longs;aliter, &longs;i corporis altitudo maxima <emph type="italics"/>CV<emph.end type="italics"/> no­<lb/>minetur T, & radius curvaturæ quam Orbis <emph type="italics"/>VPK<emph.end type="italics"/> habet in <emph type="italics"/>V,<emph.end type="italics"/> id <lb/>e&longs;t radius Circuli æqualiter curvi, nominetur R, & vis centripeta <lb/>qua corpus in Trajectoria quacunque immobili <emph type="italics"/>VPK<emph.end type="italics"/> revolvi po­<lb/>te&longs;t, in loco <emph type="italics"/>V<emph.end type="italics"/> dicatur (VFF/TT), atque aliis in locis <emph type="italics"/>P<emph.end type="italics"/> indefinite dica­<lb/>tur X, altitudine <emph type="italics"/>CP<emph.end type="italics"/> nominata A, & capiatur G ad F in data <lb/>ratione anguli <emph type="italics"/>VCp<emph.end type="italics"/> ad angulum <emph type="italics"/>VCP:<emph.end type="italics"/> erit vis centripeta qua <lb/>corpus idem eo&longs;dem motus in eadem Trajectoria <emph type="italics"/>upk<emph.end type="italics"/> circula­<lb/>riter mota temporibus ii&longs;dem peragere pote&longs;t, ut &longs;umma virium <lb/>X+(VRGG-VRFF/A <emph type="italics"/>cub.<emph.end type="italics"/>). </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Dato igitur motu corporis in Orbe quocunque immo­<lb/>bili, augeri vel minui pote&longs;t ejus motus angularis circa centrum <lb/>virium in ratione data, & inde inveniri novi Orbes immobiles in <lb/>quibus corpora novis viribus centripetis gyrentur. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 6. Igitur &longs;i ad rectam <emph type="italics"/>CV<emph.end type="italics"/> po­<lb/> <arrow.to.target n="fig87"></arrow.to.target><lb/>&longs;itione datam erigatur perpendiculum <lb/><emph type="italics"/>VP<emph.end type="italics"/> longitudinis indeterminatæ, jun­<lb/>gaturque <emph type="italics"/>CP,<emph.end type="italics"/> & ip&longs;i æqualis agatur <lb/><emph type="italics"/>Cp,<emph.end type="italics"/> con&longs;tituens angulum <emph type="italics"/>VCp,<emph.end type="italics"/> qui &longs;it <lb/>ad angulum <emph type="italics"/>VCP<emph.end type="italics"/> in data ratione; <lb/>vis qua corpus gyrari pote&longs;t in Curva <lb/>illa <emph type="italics"/>Vpk<emph.end type="italics"/> quam punctum <emph type="italics"/>p<emph.end type="italics"/> perpetuo <lb/>tangit, erit reciproce ut cubus altitu­<lb/>dinis <emph type="italics"/>Cp.<emph.end type="italics"/> Nam corpus <emph type="italics"/>P,<emph.end type="italics"/> per vim inertiæ, nulla alia vi urgente, <lb/>uniformiter progredi pote&longs;t in recta <emph type="italics"/>VP.<emph.end type="italics"/> Addatur vis in centrum <lb/><emph type="italics"/>C,<emph.end type="italics"/> cubo altitudinis <emph type="italics"/>CP<emph.end type="italics"/> vel <emph type="italics"/>Cp<emph.end type="italics"/> reciproce proportionalis, & (per <lb/>jam demon&longs;trata) detorquebitur motus ille rectilineus in lineam <pb pagenum="126"/> <arrow.to.target n="note102"></arrow.to.target><lb/>curvam <emph type="italics"/>Vpk.<emph.end type="italics"/> E&longs;t autem hæc Curva <emph type="italics"/>Vpk<emph.end type="italics"/> eadem cum Curva illa <lb/><emph type="italics"/>VPQ<emph.end type="italics"/> in Corol. </s> <s>3. Prop. </s> <s>XLI inventa, in qua ibi diximus corpora <lb/>huju&longs;modi viribus attracta oblique a&longs;cendere. </s> </p> <p type="margin"> <s><margin.target id="note102"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig87"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO XLV. PROBLEMA XXXI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Orbium qui &longs;unt Circulis maxime &longs;initimi requiruntur motus Ap­<lb/>&longs;idum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Problema &longs;olvitur Arithmetice faciendo ut Orbis, quem corpus <lb/>in Ellip&longs;i mobili (ut in Propo&longs;itionis &longs;uperioris Corol. </s> <s>2, vel 3) <lb/>revolvens de&longs;cribit in plano immobili, accedat ad formam Orbis <lb/>cujus Ap&longs;ides requiruatur, & quærendo Ap&longs;ides Orbis quem cor­<lb/>pus illud in plano immobili de&longs;cribit. </s> <s>Orbes autem eandem ac­<lb/>quirent formam, &longs;i vires centripetæ quibus de&longs;cribuntur, inter &longs;e <lb/>collatæ, in æqualibus altitudinibus reddantur proportionales. </s> <s>Sit <lb/>punctum <emph type="italics"/>V<emph.end type="italics"/> Ap&longs;is &longs;umma, & &longs;cribantur T pro altitudine maxima <lb/><emph type="italics"/>CV,<emph.end type="italics"/> A pro altitudine quavis alia <emph type="italics"/>CP<emph.end type="italics"/> vel <emph type="italics"/>Cp,<emph.end type="italics"/> & X pro alti­<lb/>titudinum differentia <emph type="italics"/>CV-CP<emph.end type="italics"/>; & vis qua corpus in Ellip&longs;i <lb/>circa umbilicum &longs;uum <emph type="italics"/>C<emph.end type="italics"/> (ut in Corollario 2.) revolvente move­<lb/>tur, quæque in Corollario 2. erat ut (FF/AA)+(RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>), id e&longs;t <lb/>ut (FFA+RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>), &longs;ub&longs;tituendo T-X pro A, erit ut <lb/>(RGG-RFF+TFF-FFX/A <emph type="italics"/>cub.<emph.end type="italics"/>). Reducenda &longs;imiliter e&longs;t vis alia <lb/>quævis centripeta ad fractionem cujus denominator &longs;it A <emph type="italics"/>cub.,<emph.end type="italics"/> & <lb/>numeratores, facta homologorum terminorum collatione, &longs;tatuendi <lb/>&longs;unt analogi. </s> <s>Res Exemplis patebit. </s> </p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/> 1. Ponamus vim centripetam uniformem e&longs;&longs;e, adeoque <lb/>ut (A <emph type="italics"/>cub.<emph.end type="italics"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>), &longs;ive (&longs;cribendo T-X pro A in Numeratore) ut <lb/>(T <emph type="italics"/>cub.<emph.end type="italics"/>-3TTX+3TXX-X <emph type="italics"/>cub.<emph.end type="italics"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>); & collatis Numeratorum ter­<lb/>minis corre&longs;pondentibus, nimirum datis cum datis & non datis <lb/>cum non datis, fiet RGG-RFF+TFF ad T <emph type="italics"/>cub.<emph.end type="italics"/> ut-FFX ad <lb/>-3TTX+3TXX-X<emph type="italics"/>cub.<emph.end type="italics"/> &longs;ive ut-FF ad-3TT+3TX <lb/>-XX. </s> <s>Jam cum Orbis ponatur Circulo quam maxime finitimus, <lb/>coeat Orbis cum Circulo; & ob factas R, T æquales, atque X in infi- <pb pagenum="127"/>nitum diminutam, rationes ultimæ erunt RGG ad T <emph type="italics"/>cub.<emph.end type="italics"/> ut-FF <lb/> <arrow.to.target n="note103"></arrow.to.target><lb/>ad-3TT &longs;eu GG ad TT ut FF ad 3TT & vici&longs;&longs;im GG ad <lb/>FF ut TT ad 3 TT id e&longs;t, ut 1 ad 3; adeoque G ad F, <lb/>hoc e&longs;t angulus <emph type="italics"/>VCp<emph.end type="italics"/> ad angulum <emph type="italics"/>VCP,<emph.end type="italics"/> ut 1 ad √3. Er­<lb/>go cum corpus in Ellip&longs;i immobili, ab Ap&longs;ide &longs;umma ad Ap­<lb/>&longs;idem imam de&longs;cendendo conficiat angulum <emph type="italics"/>VCP<emph.end type="italics"/> (ut ita di­<lb/>cam) gradum 180; corpus aliud in Ellip&longs;i mobili, atque adeo in <lb/>Orbe immobili de quo agimus, ab Ap&longs;ide &longs;umma ad Ap&longs;idem <lb/>imam de&longs;cendendo conficiet angulum <emph type="italics"/>VCp<emph.end type="italics"/> gradum (180/√3): id <lb/>adeo ob &longs;imilitudinem Orbis hujus, quem corpus agente uniformi <lb/>vi centripeta de&longs;cribit, & Orbis illius quem corpus in Ellip&longs;i re­<lb/>volvente gyros peragens de&longs;cribit in plano quie&longs;cente. </s> <s>Per &longs;u­<lb/>periorem terminorum collationem &longs;imiles redduntur hi Orbes, non <lb/>univer&longs;aliter, &longs;ed tunc cum ad formam circularem quam maxime <lb/>appropinquant. </s> <s>Corpus igitur uniformi cum vi centripeta in <lb/>Orbe propemodum circulari revolvens, inter Ap&longs;idem &longs;ummam <lb/>& Ap&longs;idem imam conficiet &longs;emper angulum (180/√3) graduum, &longs;eu <lb/>103 <emph type="italics"/>gr.<emph.end type="italics"/> 55 <emph type="italics"/>m.<emph.end type="italics"/> 23 <emph type="italics"/>&longs;ec.<emph.end type="italics"/> ad centrum; perveniens ab Ap&longs;ide &longs;umma ad <lb/>Ap&longs;idem imam ubi &longs;emel confecit hunc angulum, & inde ad Ap&longs;i­<lb/>dem &longs;ummam rediens ubi iterum confecit eundem angulum; & <lb/>&longs;ic deinceps in infinitum. </s> </p> <p type="margin"> <s><margin.target id="note103"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/> 2. Ponamus vim centripetam e&longs;&longs;e ut altitudinis A dig­<lb/>nitas quælibet A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-3<emph.end type="sup"/> &longs;eu (A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A<emph type="sup"/>3<emph.end type="sup"/>): ubi <emph type="italics"/>n<emph.end type="italics"/>-3 & <emph type="italics"/>n<emph.end type="italics"/> &longs;ignificant digni­<lb/>tatum indices quo&longs;cunque integros vel fractos, rationales vel irratio­<lb/>nales, affirmativos vel negativos. </s> <s>Numerator ille A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> &longs;eu ―T-X<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/><lb/>in &longs;eriem indeterminatam per Methodum no&longs;tram Serierum conver­<lb/>gentium reducta, evadit T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>-<emph type="italics"/>n<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>+(<emph type="italics"/>nn-n<emph.end type="italics"/>/2)XXT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-2<emph.end type="sup"/> &c. </s> <s><lb/>Et collatis hujus terminis cum terminis Numeratoris alterius <lb/>RGG-RFF+TFF-FFX, fit RGG-RFF+TFF ad T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/><lb/>ut-FF ad-<emph type="italics"/>n<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>+(<emph type="italics"/>nn-n<emph.end type="italics"/>/2)XT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-2<emph.end type="sup"/> &c. </s> <s>Et &longs;umendo ratio­<lb/>nes ultimas ubi Orbes ad formam circularem accedunt, fit RGG <lb/>ad T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> ut-FF ad-<emph type="italics"/>n<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, &longs;eu GG ad T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> ut FF ad <emph type="italics"/>n<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, <lb/>& vici&longs;&longs;im GG ad FF ut T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> ad <emph type="italics"/>n<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> id e&longs;t ut 1 ad <emph type="italics"/>n<emph.end type="italics"/>; <lb/>adeoque G ad F, id e&longs;t angulus <emph type="italics"/>VCp<emph.end type="italics"/> ad angulum <emph type="italics"/>VCP,<emph.end type="italics"/> <pb pagenum="128"/> <arrow.to.target n="note104"></arrow.to.target><lb/>ut 1 ad √<emph type="italics"/>n.<emph.end type="italics"/> Quare cum angulus <emph type="italics"/>VCP,<emph.end type="italics"/> in de&longs;cen&longs;u corporis <lb/>ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam in Ellip&longs;i confectus, &longs;it <lb/>graduum 180; conficietur angulus <emph type="italics"/>VCp,<emph.end type="italics"/> in de&longs;cen&longs;u corporis <lb/>ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam, in Orbe propemodum Cir­<lb/>culari quem corpus quodvis vi centripeta dignitati A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-3<emph.end type="sup"/> pro­<lb/>portionali de&longs;cribit, æqualis angulo graduum (180/√<emph type="italics"/>n<emph.end type="italics"/>); & hoc angulo <lb/>repetito corpus redibit ab Ap&longs;ide ima ad Ap&longs;idem &longs;ummam, & <lb/>&longs;ic deinceps in infinitum. </s> <s>Ut &longs;i vis centripeta &longs;it ut di&longs;tantia cor­<lb/>poris a centro, id e&longs;t, ut A &longs;eu (A<emph type="sup"/>4<emph.end type="sup"/>/A<emph type="sup"/>3<emph.end type="sup"/>), erit <emph type="italics"/>n<emph.end type="italics"/> æqualis 4 & √<emph type="italics"/>n<emph.end type="italics"/> æqualis 2; <lb/>adeoque angulus inter Ap&longs;idem &longs;ummam & Ap&longs;idem imam æ­<lb/>qualis (180/2) <emph type="italics"/>gr.<emph.end type="italics"/> &longs;eu 90 <emph type="italics"/>gr.<emph.end type="italics"/> Completa igitur quarta parte revolutio­<lb/>nis unius corpus perveniet ad Ap&longs;idem imam, & completa alia <lb/>quarta parte ad Ap&longs;idem &longs;ummam, & &longs;ic deinceps per vices in <lb/>infinitum. </s> <s>Id quod etiam ex Propo&longs;itione x. </s> <s>manife&longs;tum e&longs;t. </s> <s>Nam <lb/>corpus urgente hac vi centripeta revolvetur in Ellip&longs;i immobili, <lb/>cujus centrum e&longs;t in centro virium. </s> <s>Quod &longs;i vis centripeta &longs;it reci­<lb/>proce ut di&longs;tantia, id e&longs;t directe ut 1/A &longs;eu (A<emph type="sup"/>2<emph.end type="sup"/>/A<emph type="sup"/>3<emph.end type="sup"/>), erit <emph type="italics"/>n<emph.end type="italics"/> æqualis 2, ad­<lb/>eoque inter Ap&longs;idem &longs;ummam & imam angulus erit graduum (180/√2) <lb/>&longs;eu 127 <emph type="italics"/>gr.<emph.end type="italics"/> 16 <emph type="italics"/>m.<emph.end type="italics"/> 45 <emph type="italics"/>&longs;ec.<emph.end type="italics"/> & propterea corpus tali vi revolvens, perpe­<lb/>tua anguli hujus repetitione, vicibus alternis ab Ap&longs;ide &longs;umma ad <lb/>imam & ab ima ad &longs;ummam perveniet in æternum. </s> <s>Porro &longs;i vis <lb/>centripeta &longs;it reciproce ut latus quadrato-quadratum undecimæ <lb/>dignitatis altitudinis, id e&longs;t reciproce ut A (11/4), adeoque directe ut <lb/>(1/A<emph type="sup"/>11/4<emph.end type="sup"/>) &longs;eu ut (A<emph type="sup"/>1/4<emph.end type="sup"/>/A<emph type="sup"/>3<emph.end type="sup"/>) erit <emph type="italics"/>n<emph.end type="italics"/> æqualis 1/4, & (180/√<emph type="italics"/>n<emph.end type="italics"/>) <emph type="italics"/>gr.<emph.end type="italics"/> æqualis 360 <emph type="italics"/>gr.<emph.end type="italics"/> & prop­<lb/>terea corpus de Ap&longs;ide &longs;umma di&longs;cedens & &longs;ubinde perpetuo de­<lb/>&longs;cendens, perveniet ad Ap&longs;idem imam ubi complevit revolutionem <lb/>integram, dein perpetuo a&longs;cen&longs;u complendo aliam revolutionem in­<lb/>regram, redibit ad Ap&longs;idem &longs;ummam: & &longs;ic per vices in æternum. </s> </p> <p type="margin"> <s><margin.target id="note104"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/> 3. A&longs;&longs;umentes <emph type="italics"/>m<emph.end type="italics"/> & <emph type="italics"/>n<emph.end type="italics"/> pro quibu&longs;vis indicibus dignitatum <lb/>Altitudinis, & <emph type="italics"/>b, c<emph.end type="italics"/> pro numeris quibu&longs;vis datis, ponamus vim cen­<lb/>tripetam e&longs;&longs;e ut (<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>), id e&longs;t, ut (<emph type="italics"/>b<emph.end type="italics"/> in ―T-X<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/> in ―T-X<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>) <lb/>&longs;eu (per eandem Methodum no&longs;tram Serierum convergentium) ut <lb/>(<emph type="italics"/>b<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>-<emph type="italics"/>mb<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>-<emph type="italics"/>nc<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>+(<emph type="italics"/>mm-mb<emph.end type="italics"/>/2)XXT<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-2<emph.end type="sup"/>+(<emph type="italics"/>nn-nc<emph.end type="italics"/>/2)XXT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-2<emph.end type="sup"/> <emph type="italics"/>&c.<emph.end type="italics"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>) <pb pagenum="129"/>& collatis numeratorum terminis, fiet RGG-RFF+TFF <lb/> <arrow.to.target n="note105"></arrow.to.target><lb/>ad <emph type="italics"/>b<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, ut -FF ad -<emph type="italics"/>mb<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>-<emph type="italics"/>nc<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/><lb/>+(<emph type="italics"/>mm-m<emph.end type="italics"/>/2)<emph type="italics"/>b<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-2<emph.end type="sup"/>+(<emph type="italics"/>nn-n<emph.end type="italics"/>/2)<emph type="italics"/>c<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-2<emph.end type="sup"/> &c. </s> <s>Et &longs;umendo rationes ulti­<lb/>mas quæ prodeunt ubi Orbes ad formam circularem accedunt, fit <lb/>GG ad <emph type="italics"/>b<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, ut FF ad <emph type="italics"/>mb<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>+<emph type="italics"/>nc<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, & <lb/>vici&longs;&longs;im GG ad FF ut <emph type="italics"/>b<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> ad <emph type="italics"/>mb<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>+<emph type="italics"/>nc<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>. </s> <s><lb/>Quæ proportio, exponendo altitudinem maximam <emph type="italics"/>CV<emph.end type="italics"/> &longs;eu T Arith­<lb/>metice per Unitatem, fit GG ad FF ut <emph type="italics"/>b+c<emph.end type="italics"/> ad <emph type="italics"/>mb+nc,<emph.end type="italics"/> adeoque ut <lb/>1 ad (<emph type="italics"/>mb+nc/b+c<emph.end type="italics"/>). Unde e&longs;t G ad F, id e&longs;t angulus <emph type="italics"/>VCp<emph.end type="italics"/> ad angulum <lb/><emph type="italics"/>VCP,<emph.end type="italics"/> ut 1 ad √(<emph type="italics"/>mb+nc/b+c<emph.end type="italics"/>). Et propterea cum angulus <emph type="italics"/>VCP<emph.end type="italics"/> inter <lb/>Ap&longs;idem &longs;ummam & Ap&longs;idem imam in Ellip&longs;i immobili &longs;it 180 <emph type="italics"/>gr.<emph.end type="italics"/><lb/>erit angulus <emph type="italics"/>VCp<emph.end type="italics"/> inter ea&longs;dem Ap&longs;ides, in Orbe quem corpus vi <lb/>centripeta quantitati (<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>) proportionali de&longs;cribit, æqua­<lb/>lis angulo graduum 180 √(<emph type="italics"/>b+c/mb+nc<emph.end type="italics"/>). Et eodem argumento &longs;i vis cen­<lb/>tripeta &longs;it ut (<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>-<emph type="italics"/>c<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>), angulus inter Ap&longs;ides invenietur graduum <lb/>180 √(<emph type="italics"/>b-c/mb-nc<emph.end type="italics"/>). Nec &longs;ecus re&longs;olvetur Problema in ca&longs;ibus diffi­<lb/>cilioribus. </s> <s>Quantitas cui vis centripeta proportionalis e&longs;t, re­<lb/>&longs;olvi &longs;emper debet in Series convergentes denominatorem ha­<lb/>bentes A <emph type="italics"/>cub.<emph.end type="italics"/> Dein pars data numeratoris qui ex illa operatione <lb/>provenit ad ip&longs;ius partem alteram non datam, & pars data nu­<lb/>meratoris hujus RGG-RFF+TFF-FFX ad ip&longs;ius partem <lb/>alteram non datam in eadem ratione ponendæ &longs;unt: Et quantitates <lb/>&longs;uperfluas delendo, &longs;cribendoque Unitatem pro T, obtinebitur <lb/>proportio G ad F. </s> </p> <p type="margin"> <s><margin.target id="note105"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i vis centripeta &longs;it ut aliqua altitudinis digni­<lb/>tas, inveniri pote&longs;t dignitas illa ex motu Ap&longs;idum; & contra. </s> <s><lb/>Nimirum &longs;i motus totus angularis, quo corpus redit ad Ap&longs;idem <lb/>eandem, &longs;it ad motum angularem revolutionis unius, &longs;eu graduum <lb/>360, ut numerus aliquis <emph type="italics"/>m<emph.end type="italics"/> ad numerum alium <emph type="italics"/>n,<emph.end type="italics"/> & altitudo no­<lb/>minetur A: erit vis ut altitudinis dignitas illa A<emph type="sup"/>(<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3<emph.end type="sup"/>, cujus In- <pb pagenum="130"/> <arrow.to.target n="note106"></arrow.to.target><lb/>dex e&longs;t (<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3. Id quod per Exempla &longs;ecunda manife&longs;tum e&longs;t. </s> <s><lb/>Unde liquet vim illam in majore quam triplicata altitudinis ratione, <lb/>in rece&longs;&longs;u a centro, decre&longs;cere non po&longs;&longs;e: Corpus tali vi revolvens <lb/>deque Ap&longs;ide di&longs;cedens, &longs;i cæperit de&longs;cendere nunquam perveniet <lb/>ad Ap&longs;idem imam &longs;eu altitudinem minimam, &longs;ed de&longs;cendet u&longs;que ad <lb/>centrum, de&longs;cribens Curvam illam lineam de qua egimus in Cor. </s> <s>3. <lb/>Prop. </s> <s>XLI. </s> <s>Sin cæperit illud, de Ap&longs;ide di&longs;cedens, vel minimum <lb/>a&longs;cendere; a&longs;cendet in infinitum, neque unquam perveniet ad Ap­<lb/>&longs;idem &longs;ummam. </s> <s>De&longs;cribet enim Curvam illam lineam de qua ac­<lb/>tum e&longs;t in eodem Corol. </s> <s>& in Corol. </s> <s>6, Prop. </s> <s>XLIV. </s> <s>Sic & ubi <lb/>vis, in rece&longs;&longs;u a centro, decre&longs;cit in majore quam triplicata ratione <lb/>altitudinis, corpus de Ap&longs;ide di&longs;cedens, perinde ut cæperit de&longs;cen­<lb/>dere vel a&longs;cendere, vel de&longs;cendet ad centrum u&longs;que vel a&longs;cendet <lb/>in infinitum. </s> <s>At &longs;i vis, in rece&longs;&longs;u a centro, vel decre&longs;cat in minore <lb/>quam triplicata ratione altitudinis, vel crefcat in altitudinis ratione <lb/>quacunque; corpus nunquam de&longs;cendet ad centrum u&longs;que, &longs;ed ad <lb/>Ap&longs;idem imam aliquando perveniet: & contra, &longs;i corpus de Ap&longs;i­<lb/>de ad Ap&longs;idem alternis vicibus de&longs;cendens & a&longs;cendens nunquam <lb/>appellat ad centrum; vis in rece&longs;&longs;u a centro aut augebitur, aut in <lb/>minore quam triplicata altitudinis ratione decre&longs;cet: & quo ci­<lb/>tius corpus de Ap&longs;ide ad Ap&longs;idem redierit, eo longius ratio virium <lb/>recedet a ratione illa triplicata. </s> <s>Ut &longs;i corpus revolutionibus 8 vel <lb/>4 vel 2 vel 1 1/2 de Ap&longs;ide &longs;umma ad Ap&longs;idem &longs;ummam alterno de­<lb/>&longs;cen&longs;u & a&longs;cen&longs;u redierit; hoc e&longs;t, &longs;i fuerit <emph type="italics"/>m<emph.end type="italics"/> ad <emph type="italics"/>n<emph.end type="italics"/> ut 8 vel 4 vel <lb/>2 vel 1 1/2 ad 1, adeoque (<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3 valeat (1/64)-3 vel (1/16) -3 vel 1/4-3 <lb/>vel 4/9-3: erit vis ut A<emph type="sup"/>(1/64)-3<emph.end type="sup"/> vel A<emph type="sup"/>(1/16)-3<emph.end type="sup"/> vel A<emph type="sup"/>1/4-3<emph.end type="sup"/> vel A<emph type="sup"/>4/9-3<emph.end type="sup"/>, <lb/>id e&longs;t, reciproce ut A<emph type="sup"/>3-(1/64)<emph.end type="sup"/> vel A<emph type="sup"/>3-(1/16)<emph.end type="sup"/> vel A<emph type="sup"/>3-1/4<emph.end type="sup"/> vel A<emph type="sup"/>3-4/9<emph.end type="sup"/>. </s> <s><lb/>Si corpus &longs;ingulis revolutionibus redierit ad Ap&longs;idem eandem immo­<lb/>tam; erit <emph type="italics"/>m<emph.end type="italics"/> ad <emph type="italics"/>n<emph.end type="italics"/> ut 1 ad 1, adeoque A (<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3 æqualis A<emph type="sup"/>-2<emph.end type="sup"/> &longs;eu (1/AA<gap/>) <lb/>& propterea decrementum virium in ratione duplicata altitudinis, <lb/>ut in præcedentibus demon&longs;tratum e&longs;t. </s> <s>Si corpus partibus revo­<lb/>lutionis unius vel tribus quartis, vel duabus tertiis, vel una ter­<lb/>tia, vel una quarta, ad Ap&longs;idem eandem redierit; erit <emph type="italics"/>m<emph.end type="italics"/> ad <emph type="italics"/>n<emph.end type="italics"/> ut <lb/>1/4 vel 2/3 vel 1/3 vel 1/4 ad 1, adeoque A(<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3 æqualis A<emph type="sup"/>(16/9)-3<emph.end type="sup"/> vel <lb/>A<emph type="sup"/>9/4-3<emph.end type="sup"/> vel A<emph type="sup"/>9-3<emph.end type="sup"/> vel A<emph type="sup"/>16-3<emph.end type="sup"/>; & propterea vis aut reciproce ut <pb pagenum="131"/>A<emph type="sup"/>(11/9)<emph.end type="sup"/> vel A<emph type="sup"/>1/4<emph.end type="sup"/>, aut directe ut A<emph type="sup"/>6<emph.end type="sup"/> vel A <emph type="sup"/>13<emph.end type="sup"/>. </s> <s>Denique &longs;i corpus pergendo <lb/> <arrow.to.target n="note107"></arrow.to.target><lb/>ab Ap&longs;ide &longs;umma ad Ap&longs;idem &longs;ummam confecerit revolutionem in­<lb/>tegram, & præterea gradus tres, adeoque Ap&longs;is illa &longs;ingulis corporis <lb/>revolutionibus confecerit in con&longs;equentia gradus tres; erit <emph type="italics"/>m<emph.end type="italics"/> ad <emph type="italics"/>n<emph.end type="italics"/> ut <lb/>363 <emph type="italics"/>gr.<emph.end type="italics"/> ad 360<emph type="italics"/>gr.<emph.end type="italics"/> &longs;ive ut 121 ad 120, adeoque A<emph type="sup"/>(<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3<emph.end type="sup"/> erit æquale <lb/>A<emph type="sup"/>-(29523/14641)<emph.end type="sup"/>; & propterea vis centripeta reciproce ut A <emph type="sup"/>(29523/14641)<emph.end type="sup"/> &longs;eu re­<lb/>ciproce ut A<emph type="sup"/>2 (4/2+3)<emph.end type="sup"/> proxime. </s> <s>Decre&longs;cit igitur vis centripeta in ratio­<lb/>ne paulo majore quam duplicata, &longs;ed quæ vicibus 59 3/4 propius ad <lb/>duplicatam quam ad triplicatam accedit. </s> </p> <p type="margin"> <s><margin.target id="note106"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="margin"> <s><margin.target id="note107"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Hinc etiam &longs;i corpus, vi centripeta quæ &longs;it reciproce <lb/>ut quadratum altitudinis, revolvatur in Ellip&longs;i umbilicum haben­<lb/>te in centro virium, & huic vi centripetæ addatur vel auferatur <lb/>vis alia quævis extranea; cogno&longs;ci pote&longs;t (per Exempla tertia) <lb/>motus Ap&longs;idum qui ex vi illa extranea orietur: & contra. </s> <s>Ut &longs;i <lb/>vis qua corpus revolvitur in Ellip&longs;i &longs;it ut (1/AA), & vis extranea ab­<lb/>lata ut <emph type="italics"/>c<emph.end type="italics"/> A, adeoque vis reliqua ut (A-<emph type="italics"/>c<emph.end type="italics"/> A<emph type="sup"/>4<emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>); erit (in Exemplis ter­<lb/>tiis) <emph type="italics"/>b<emph.end type="italics"/> æqualis 1, <emph type="italics"/>m<emph.end type="italics"/> æqualis 1, <emph type="italics"/>n<emph.end type="italics"/> æqualis 4, adeoque angulus revo­<lb/>lutionis inter Ap&longs;ides æqualis angulo graduum 180 √(1-<emph type="italics"/>c<emph.end type="italics"/>/1-4<emph type="sup"/><emph type="italics"/>c<emph.end type="italics"/><emph.end type="sup"/>). Po­<lb/>natur vim illam extraneam e&longs;&longs;e 357,<emph type="sup"/>45<emph.end type="sup"/> partibus minorem quam vis <lb/>altera qua corpus revolvitur in Ellip&longs;i, id e&longs;t <emph type="italics"/>c<emph.end type="italics"/> e&longs;&longs;e (100/35745), exi&longs;tente A <lb/>vel T æquali 1; & 180 √(1-<emph type="italics"/>c<emph.end type="italics"/>/1-4<emph type="sup"/><emph type="italics"/>c<emph.end type="italics"/><emph.end type="sup"/>) evadet 180 √(35645/35345), &longs;eu 180, 7623, <lb/>id e&longs;t, 180 <emph type="italics"/>gr.<emph.end type="italics"/> 45 <emph type="italics"/>m.<emph.end type="italics"/> 44 <emph type="italics"/>&longs;.<emph.end type="italics"/> Igitur corpus de Ap&longs;ide &longs;umma di&longs;ce­<lb/>dens, motu angulari 180 <emph type="italics"/>gr.<emph.end type="italics"/> 45 <emph type="italics"/>m.<emph.end type="italics"/> 44. <emph type="italics"/>&longs;.<emph.end type="italics"/> perveniet ad Ap&longs;idem <lb/>imam, & hoc motu duplicato ad Ap&longs;idem &longs;ummam redibit: adeo­<lb/>que Ap&longs;is &longs;umma &longs;ingulis revolutionibus progrediendo conficiet <lb/>1 <emph type="italics"/>gr.<emph.end type="italics"/> 31 <emph type="italics"/>m.<emph.end type="italics"/> 28 <emph type="italics"/>&longs;ec.<emph.end type="italics"/></s> </p> <p type="main"> <s>Hactenus de Motu corporum in Orbibus quorum plana per <lb/>centrum Virium tran&longs;eunt. </s> <s>Supere&longs;t ut Motus etiam determine­<lb/>mus in planis excentricis. </s> <s>Nam Scriptores qui Motum gravium <lb/>tractant, con&longs;iderare &longs;olent a&longs;cen&longs;us & de&longs;cen&longs;us ponderum, <lb/>tam obliquos in planis quibu&longs;cunque datis, quam perpendicu­<lb/>lares: & pari jure Motus corporum Viribus quibu&longs;cunque cen- <pb pagenum="132"/> <arrow.to.target n="note108"></arrow.to.target><lb/>tra petentium, & planis excentricis innitentium hic con&longs;iderandus <lb/>venit. </s> <s>Plana autem &longs;upponimus e&longs;&longs;e politi&longs;&longs;ima & ab&longs;olute lubrica <lb/>ne corpora retardent. </s> <s>Quinimo, in his demon&longs;trationibus, vi­<lb/>ce planorum quibus corpora incumbunt quæque tangunt incum­<lb/>bendo, u&longs;urpamus plana his parallela, in quibus centra corpo­<lb/>rum moventur & Orbitas movendo de&longs;cribunt. </s> <s>Et eadem lege <lb/>Motus corporum in &longs;uperficiebus Curvis peractos &longs;ubinde de­<lb/>terminamus. </s> </p> <p type="margin"> <s><margin.target id="note108"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>SECTIO X.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu Corporum in Superficiebus datis, deque Funipendulorum <lb/>Motu reciproco.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLVI. PROBLEMA XXXII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;ita cuju&longs;cunque generis Vi centripeta, datoque tum Virium cen­<lb/>tro tum Plano quocunque in quo corpus revolvitur, & conce&longs;­<lb/>&longs;is Figurarum curvilinearum quadraturis: requiritur Motus cor­<lb/>poris de loco dato, data cum Velocitate, &longs;ecundum rectam in <lb/>Plano illo datam egre&longs;&longs;i.<emph.end type="italics"/></s> </p> <p type="main"> <s>Sit <emph type="italics"/>S<emph.end type="italics"/> centrum Virium, <emph type="italics"/>SC<emph.end type="italics"/> di&longs;tantia minima centri hujus a Plano <lb/>dato, <emph type="italics"/>P<emph.end type="italics"/> corpus de loco <emph type="italics"/>P<emph.end type="italics"/> &longs;ecundum rectam <emph type="italics"/>PZ<emph.end type="italics"/> egrediens, <emph type="italics"/>Q<emph.end type="italics"/><lb/>corpus idem in Trajectoria &longs;ua revolvens, & <emph type="italics"/>PQR<emph.end type="italics"/> Trajectoria <lb/>illa, in Plano dato de&longs;cripta, quam invenire oportet. </s> <s>Jungantur <emph type="italics"/>CQ <lb/>QS,<emph.end type="italics"/> & &longs;i in <emph type="italics"/>QS<emph.end type="italics"/> capiatur <emph type="italics"/>SV<emph.end type="italics"/> proportionalis vi centripetæ qua <lb/>corpus trahitur ver&longs;us centrum <emph type="italics"/>S,<emph.end type="italics"/> & agatur <emph type="italics"/>VT<emph.end type="italics"/> quæ fit parallela <lb/><emph type="italics"/>CQ<emph.end type="italics"/> & occurrat <emph type="italics"/>SC<emph.end type="italics"/> in <emph type="italics"/>T:<emph.end type="italics"/> Vis <emph type="italics"/>SV<emph.end type="italics"/> re&longs;olvetur (per Legum Corol. </s> <s>2.) <lb/>in vires <emph type="italics"/>ST, TV;<emph.end type="italics"/> quarum <emph type="italics"/>ST<emph.end type="italics"/> trahendo corpus &longs;ecundum lineam <lb/>plano perpendicularem, nil mutat motum ejus in hoc plano. </s> <s>Vis <lb/>autem altera <emph type="italics"/>TV,<emph.end type="italics"/> agendo &longs;ecundum po&longs;itionem plani, trahit cor­<lb/>pus directe ver&longs;us punctum <emph type="italics"/>C<emph.end type="italics"/> in plano datum, adeoque facit illud <lb/>in hoc plano perinde moveri ac &longs;i vis <emph type="italics"/>ST<emph.end type="italics"/> tolleretur, & corpus vi <lb/>&longs;ola <emph type="italics"/>TV<emph.end type="italics"/> revolveretur circa centrum <emph type="italics"/>C<emph.end type="italics"/> in &longs;patio libero. </s> <s>Data autem <pb pagenum="133"/>vi centripeta <emph type="italics"/>TV<emph.end type="italics"/> qua corpus <emph type="italics"/>Q<emph.end type="italics"/> in &longs;patio libero circa centrum <lb/> <arrow.to.target n="note109"></arrow.to.target><lb/>datum <emph type="italics"/>C<emph.end type="italics"/> revolvitur, datur per Prop. </s> <s>XLII, tum Trajectoria <emph type="italics"/>PQR<emph.end type="italics"/><lb/>quam corpus de&longs;cribit, tum locus <emph type="italics"/>Q<emph.end type="italics"/> in quo corpus ad datum quod­<lb/>vis tempus ver&longs;abitur, tum denique velocitas corporis in loco illo <lb/><emph type="italics"/>Q<emph.end type="italics"/>; & contra. <emph type="italics"/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note109"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLVII. THEOREMA XV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod Vis centripeta proportionalis &longs;it di&longs;tantiæ corporis a <lb/>centro; corpora omnia in planis quibu&longs;cunque revolventia de­<lb/>&longs;cribent Ellip&longs;es, & revolutiones Temporibus æqualibus peragent; <lb/>quæque moventur in lineis rectis, ultro citroque di&longs;currendo, <lb/>&longs;ingulas eundi & redeundi periodos ii&longs;dem Temporibus ab&longs;ol­<lb/>vent.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam, &longs;tantibus quæ <lb/> <arrow.to.target n="fig88"></arrow.to.target><lb/>in &longs;uperiore Propo&longs;itio­<lb/>ne, vis <emph type="italics"/>SV<emph.end type="italics"/> qua corpus <lb/><emph type="italics"/>Q<emph.end type="italics"/> in plano quovis <emph type="italics"/>PQR<emph.end type="italics"/><lb/>revolvens trahitur ver­<lb/>&longs;us centrum <emph type="italics"/>S<emph.end type="italics"/> e&longs;t ut di­<lb/>ftantia <emph type="italics"/><expan abbr="Sq;">Sque</expan><emph.end type="italics"/> atque adeo <lb/>ob proportionales <emph type="italics"/>SV<emph.end type="italics"/><lb/>& <emph type="italics"/>SQ, TV<emph.end type="italics"/> & <emph type="italics"/>CQ,<emph.end type="italics"/> vis <lb/><emph type="italics"/>TV<emph.end type="italics"/> qua corpus trahi­<lb/>tur ver&longs;us punctum <emph type="italics"/>C<emph.end type="italics"/><lb/>in Orbis plano datum, <lb/>e&longs;t ut di&longs;tantia <emph type="italics"/><expan abbr="Cq.">Cque</expan><emph.end type="italics"/> Vi­<lb/>res igitur, quibus cor­<lb/>pora in plano <emph type="italics"/>PQR<emph.end type="italics"/><lb/>ver&longs;antia trahuntur ver­<lb/>&longs;us punctum <emph type="italics"/>C,<emph.end type="italics"/> &longs;unt pro <lb/>ratione di&longs;tantiarum æquales viribus quibus corpora undiquaque <lb/>trahuntur ver&longs;us centrum <emph type="italics"/>S<emph.end type="italics"/>; & propterea corpora movebuntur ii&longs;­<lb/>dem Temporibus, in ii&longs;dem Figuris, in plano quovis <emph type="italics"/>PQR<emph.end type="italics"/> circa <lb/>punctum <emph type="italics"/>C,<emph.end type="italics"/> atque in &longs;patiis liberis circa centrum <emph type="italics"/>S<emph.end type="italics"/>; adeoque (per <lb/>Corol. </s> <s>2. Prop. </s> <s>X, & Corol. </s> <s>2. Prop. </s> <s>XXXVIII) Temporibus &longs;emper <pb pagenum="134"/> <arrow.to.target n="note110"></arrow.to.target><lb/>æqualibus, vel de&longs;cribent Ellip&longs;es in plano illo circa centrum <emph type="italics"/>C,<emph.end type="italics"/><lb/>vel periodos movendi ultro citroque in lineis rectis per centrum <emph type="italics"/>C<emph.end type="italics"/><lb/>in plano illo ductis, complebunt. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note110"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig88"></figure> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>His affines &longs;unt a&longs;cen&longs;us ac de&longs;cen&longs;us corporum in &longs;uperficiebus <lb/>curvis. </s> <s>Concipe lineas curvas in plano de&longs;cribi, dein circa axes <lb/>quo&longs;vis datos per centrum Virium tran&longs;euntes revolvi, & ea revo­<lb/>lutione &longs;uperficies curvas de&longs;cribere; tum corpora ita moveri ut <lb/>eorum centra in his &longs;uperficiebus perpetuo reperiantur. </s> <s>Si cor­<lb/>pora illa oblique a&longs;cendendo & de&longs;cendendo currant ultro citroque <lb/>peragentur corum motus in planis per axem tran&longs;euntibus, atque <lb/>adeo in lineis curvis quarum revolutione curvæ illæ &longs;uperficies ge­<lb/>nitæ &longs;unt. </s> <s>I&longs;tis igitur in ca&longs;ibus &longs;ufficit motum in his lineis cur­<lb/>vis con&longs;iderare. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLVIII. THEOREMA XVI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si Rota Globo extrin&longs;ecus ad angulos rectos in&longs;i&longs;tat, & more ro­<lb/>tarum revolvendo progrediatur in circulo maximo; longitudo <lb/>Itineris curvilinei, quod punctum quodvis in Rotæ perimetro da­<lb/>tum, ex quo Globum tetigit, confecit, (quodque Cycloidem vel <lb/>Epicycloidem nominare licet) erit ad duplicatum &longs;inum ver&longs;um <lb/>arcus dimidii qui Globum ex eo tempore inter eundum tetigit, <lb/>ut &longs;umma diametrorum Globi & Rotæ ad &longs;emidiametrum Globi.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIX. THEOREMA XVII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si Rota Globo concavo ad rectos angulos intrin&longs;ecus in&longs;i&longs;tat & re­<lb/>volvendo progrediatur in circulo maximo; longitudo Itineris <lb/>curvilinei quod punctum quodvis in Rotæ perimetro datum, ex <lb/>quo Globum tetigit, confecit, erit ad duplicatum &longs;inum ver&longs;um <lb/>arcus dimidii qui Globum toto hoc tempore inter eundum teti­<lb/>git, ut differentia diametrorum Globi & Rotæ ad &longs;emidiame­<lb/>trum Globi.<emph.end type="italics"/></s> </p> <pb pagenum="135"/> <p type="main"> <s>Sit <emph type="italics"/>ABL<emph.end type="italics"/> Globus, <emph type="italics"/>C<emph.end type="italics"/> centrum ejus, <emph type="italics"/>BPV<emph.end type="italics"/> Rota ei in&longs;i&longs;tens, <emph type="italics"/>E<emph.end type="italics"/><lb/> <arrow.to.target n="note111"></arrow.to.target><lb/>centrum Rotæ, <emph type="italics"/>B<emph.end type="italics"/> punctum contactus, & <emph type="italics"/>P<emph.end type="italics"/> punctum datum in pe­<lb/>rimetro Rotæ. </s> <s>Concipe hanc Rotam pergere in circulo maximo <lb/><emph type="italics"/>ABL<emph.end type="italics"/> ab <emph type="italics"/>A<emph.end type="italics"/> per <emph type="italics"/>B<emph.end type="italics"/> ver&longs;us <emph type="italics"/>L,<emph.end type="italics"/> & inter cundum ita revolvi ut ar­<lb/>cus <emph type="italics"/>AB, PB<emph.end type="italics"/> &longs;ibi invicem &longs;emper æquentur, atque punctum illud <lb/><emph type="italics"/>P<emph.end type="italics"/> in perimetro Rotæ datum interea de&longs;cribere Viam curvilineam <lb/><emph type="italics"/>AP.<emph.end type="italics"/> Sit autem <emph type="italics"/>AP<emph.end type="italics"/> Via tota curvilinea de&longs;cripta ex quo Rota <lb/>Globum tetigit in <emph type="italics"/>A,<emph.end type="italics"/> & erit Viæ hujus longitudo <emph type="italics"/>AP<emph.end type="italics"/> ad duplum <lb/> <arrow.to.target n="fig89"></arrow.to.target><lb/>&longs;inum ver&longs;um arcus 1/2 <emph type="italics"/>PB,<emph.end type="italics"/> ut 2 <emph type="italics"/>CE<emph.end type="italics"/> ad <emph type="italics"/>CB.<emph.end type="italics"/> Nam recta <emph type="italics"/>CE<emph.end type="italics"/> (&longs;i <lb/>opus e&longs;t producta) occurrat Rotæ in <emph type="italics"/>V,<emph.end type="italics"/> junganturque <emph type="italics"/>CP, BP, <lb/>EP, VP,<emph.end type="italics"/> & in <emph type="italics"/>CP<emph.end type="italics"/> productam demittatur normalis <emph type="italics"/>VF.<emph.end type="italics"/> Tan­<lb/>gant <emph type="italics"/>PH, VH<emph.end type="italics"/> Circulum in <emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>V<emph.end type="italics"/> concurrentes in <emph type="italics"/>H,<emph.end type="italics"/> &longs;ecetque <lb/><emph type="italics"/>PH<emph.end type="italics"/> ip&longs;am <emph type="italics"/>VF<emph.end type="italics"/> in <emph type="italics"/>G,<emph.end type="italics"/> & ad <emph type="italics"/>VP<emph.end type="italics"/> demittantur normales <emph type="italics"/>GI, HK.<emph.end type="italics"/> <pb pagenum="136"/> <arrow.to.target n="note112"></arrow.to.target><lb/>Centro item <emph type="italics"/>C<emph.end type="italics"/> & intervallo quovis de&longs;cribatur circulus <emph type="italics"/>nom<emph.end type="italics"/> &longs;e­<lb/>cans rectam <emph type="italics"/>CP<emph.end type="italics"/> in <emph type="italics"/>n,<emph.end type="italics"/> Rotæ perimetrum <emph type="italics"/>BP<emph.end type="italics"/> &c. </s> <s>in <emph type="italics"/>o,<emph.end type="italics"/> & Viam curvi­<lb/>lineam <emph type="italics"/>AP<emph.end type="italics"/> in <emph type="italics"/>m;<emph.end type="italics"/> centroque <emph type="italics"/>V<emph.end type="italics"/> & intervallo <emph type="italics"/>Vo<emph.end type="italics"/> de&longs;cribatur circu­<lb/>lus &longs;ecans <emph type="italics"/>VP<emph.end type="italics"/> productam in <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note111"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="margin"> <s><margin.target id="note112"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig89"></figure> <p type="main"> <s>Quoniam Rota eundo &longs;emper revolvitur circa punctum con­<lb/>tactus <emph type="italics"/>B,<emph.end type="italics"/> manife&longs;tum e&longs;t quod recta <emph type="italics"/>BP<emph.end type="italics"/> perpendicularis e&longs;t ad <lb/> <arrow.to.target n="fig90"></arrow.to.target><lb/>lineam illam curvam <emph type="italics"/>AP<emph.end type="italics"/> quam Rotæ punctum <emph type="italics"/>P<emph.end type="italics"/> de&longs;cribit, atque <lb/>adeo quod recta <emph type="italics"/>VP<emph.end type="italics"/> tanget hanc curvam in puncto <emph type="italics"/>P.<emph.end type="italics"/> Circuli <lb/><emph type="italics"/>nom<emph.end type="italics"/> radius &longs;en&longs;im auctus vel diminutus æquetur tandem di&longs;tantiæ <lb/><emph type="italics"/>CP<emph.end type="italics"/>; &, ob &longs;imilitudinem Figuræ evane&longs;centis <emph type="italics"/>Pnomq<emph.end type="italics"/> & Figuræ <lb/><emph type="italics"/>PFGVI,<emph.end type="italics"/> ratio ultima lineolarum evane&longs;centium <emph type="italics"/>Pm, Pn, Po, Pq,<emph.end type="italics"/> <pb pagenum="137"/>id e&longs;t, ratio mutationum momentanearum curvæ <emph type="italics"/>AP,<emph.end type="italics"/> rectæ <lb/> <arrow.to.target n="note113"></arrow.to.target><lb/><emph type="italics"/>CP,<emph.end type="italics"/> arcus circularis <emph type="italics"/>BP,<emph.end type="italics"/> ac rectæ <emph type="italics"/>VP,<emph.end type="italics"/> eadem erit quæ linea­<lb/>rum <emph type="italics"/>PV, PF, PG, PI<emph.end type="italics"/> re&longs;pective. </s> <s>Cum autem <emph type="italics"/>VF<emph.end type="italics"/> ad <emph type="italics"/>CF<emph.end type="italics"/> & <lb/><emph type="italics"/>VH<emph.end type="italics"/> ad <emph type="italics"/>CV<emph.end type="italics"/> perpendiculares &longs;unt, angulique <emph type="italics"/>HVG, VCF<emph.end type="italics"/> prop­<lb/>terea æquales; & angulus <emph type="italics"/>VHG<emph.end type="italics"/> (ob angulos quadrilateri <emph type="italics"/>HVEP<emph.end type="italics"/><lb/>ad <emph type="italics"/>V<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> rectos) angulo <emph type="italics"/>CEP<emph.end type="italics"/> æqualis e&longs;t, &longs;imilia erunt tri­<lb/>angula <emph type="italics"/>VHG, CEP<emph.end type="italics"/>; & inde fiet ut <emph type="italics"/>EP<emph.end type="italics"/> ad <emph type="italics"/>CE<emph.end type="italics"/> ita <emph type="italics"/>HG<emph.end type="italics"/> ad <emph type="italics"/>HV<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>HP<emph.end type="italics"/> & ita <emph type="italics"/>KI<emph.end type="italics"/> ad <emph type="italics"/>KP,<emph.end type="italics"/> & compo&longs;ite vel divi&longs;im ut <emph type="italics"/>CB<emph.end type="italics"/> ad <lb/><emph type="italics"/>CE<emph.end type="italics"/> ita <emph type="italics"/>PI<emph.end type="italics"/> ad <emph type="italics"/>PK,<emph.end type="italics"/> & duplicatis con&longs;equentibus ut <emph type="italics"/>CB<emph.end type="italics"/> ad 2 <emph type="italics"/>CE<emph.end type="italics"/><lb/>ita <emph type="italics"/>PI<emph.end type="italics"/> ad <emph type="italics"/>PV,<emph.end type="italics"/> atque ita adeo <emph type="italics"/>Pq<emph.end type="italics"/> ad <emph type="italics"/>Pm.<emph.end type="italics"/> E&longs;t igitur decremen­<lb/>tum lineæ <emph type="italics"/>VP,<emph.end type="italics"/> id e&longs;t, incrementum lineæ <emph type="italics"/>BV-VP<emph.end type="italics"/> ad incremen­<lb/>tum lineæ curvæ <emph type="italics"/>AP<emph.end type="italics"/> in data ratione <emph type="italics"/>CB<emph.end type="italics"/> ad 2 <emph type="italics"/>CE,<emph.end type="italics"/> & prop­<lb/>terea (per Corol. </s> <s>Lem. </s> <s>IV.) longitudines <emph type="italics"/>BV-VP<emph.end type="italics"/> & <emph type="italics"/>AP,<emph.end type="italics"/> in­<lb/>crementis illis genitæ, &longs;unt in eadem ratione. </s> <s>Sed, exi&longs;tente <emph type="italics"/>BV<emph.end type="italics"/> ra­<lb/>dio, e&longs;t <emph type="italics"/>VP<emph.end type="italics"/> co-&longs;inus anguli <emph type="italics"/>BVP<emph.end type="italics"/> &longs;eu 1/2 <emph type="italics"/>BEP,<emph.end type="italics"/> adeoque <emph type="italics"/>BV-VP<emph.end type="italics"/><lb/>&longs;inus ver&longs;us eju&longs;dem anguli; & propterea in hac Rota, cujus radius <lb/>e&longs;t 1/2 <emph type="italics"/>BV,<emph.end type="italics"/> erit <emph type="italics"/>BV-VP<emph.end type="italics"/> duplus &longs;inus ver&longs;us arcus 1/2 <emph type="italics"/>BP.<emph.end type="italics"/> Ergo <lb/><emph type="italics"/>AP<emph.end type="italics"/> e&longs;t ad duplum &longs;inum ver&longs;um arcus 1/2 <emph type="italics"/>BP<emph.end type="italics"/> ut 2 <emph type="italics"/>CE<emph.end type="italics"/> ad <emph type="italics"/>CB. <lb/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note113"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig90"></figure> <p type="main"> <s>Lineam autem <emph type="italics"/>AP<emph.end type="italics"/> in Propo&longs;itione priore Cycloidem extra <lb/>Globum, alteram in po&longs;teriore Cycloidem intra Globum di&longs;tincti­<lb/>onis gratia nominabimus. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i de&longs;cribatur Cyclois integra <emph type="italics"/>ASL<emph.end type="italics"/> & bi&longs;ecetur <lb/>ea in <emph type="italics"/>S,<emph.end type="italics"/> erit longitudo partis <emph type="italics"/>PS<emph.end type="italics"/> ad longitudinem <emph type="italics"/>VP<emph.end type="italics"/> (quæ du­<lb/>plus e&longs;t &longs;inus anguli <emph type="italics"/>VBP,<emph.end type="italics"/> exi&longs;tente <emph type="italics"/>EB<emph.end type="italics"/> radio) ut 2 <emph type="italics"/>CE<emph.end type="italics"/> ad <emph type="italics"/>CB,<emph.end type="italics"/><lb/>atque adeo in ratione data. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et longitudo &longs;emiperimetri Cycloidis <emph type="italics"/>AS<emph.end type="italics"/> æquabitur <lb/>lineæ rectæ quæ e&longs;t ad Rotæ diametrum <emph type="italics"/>BV,<emph.end type="italics"/> ut 2 <emph type="italics"/>CE<emph.end type="italics"/> ad <emph type="italics"/>CB.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO L. PROBLEMA XXXIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Facere ut Corpus pendulum o&longs;cilletur in Cycloide data.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Intra Globum <emph type="italics"/>QVS,<emph.end type="italics"/> centro <emph type="italics"/>C<emph.end type="italics"/> de&longs;criptum, detur Cyclois <emph type="italics"/>QRS<emph.end type="italics"/><lb/>bi&longs;ecta in <emph type="italics"/>R<emph.end type="italics"/> & punctis &longs;uis extremis <emph type="italics"/>Q<emph.end type="italics"/> & <emph type="italics"/>S<emph.end type="italics"/> &longs;uperficiei Globi hinc <lb/>inde occurrens. </s> <s>Agatur <emph type="italics"/>CR<emph.end type="italics"/> bi&longs;ecans arcum <emph type="italics"/>QS<emph.end type="italics"/> in <emph type="italics"/>O,<emph.end type="italics"/> & produca­<lb/>tur ea ad <emph type="italics"/>A,<emph.end type="italics"/> ut &longs;it <emph type="italics"/>CA<emph.end type="italics"/> ad <emph type="italics"/>CO<emph.end type="italics"/> ut <emph type="italics"/>CO<emph.end type="italics"/> ad <emph type="italics"/>CR.<emph.end type="italics"/> Centro <emph type="italics"/>C<emph.end type="italics"/> in- <pb pagenum="138"/> <arrow.to.target n="note114"></arrow.to.target><lb/>tervallo <emph type="italics"/>CA<emph.end type="italics"/> de&longs;eribatur Globus exterior <emph type="italics"/>ABD,<emph.end type="italics"/> & intra hunc Glo­<lb/>bum a Rota, cujus diameter &longs;it <emph type="italics"/>AO,<emph.end type="italics"/> de&longs;cribantur duæ Semicycloides <lb/><emph type="italics"/>AQ, AS,<emph.end type="italics"/> quæ Globum interiorem tangant in <emph type="italics"/>Q<emph.end type="italics"/> & <emph type="italics"/>S<emph.end type="italics"/> & Globo ex­<lb/>teriori occurrant in <emph type="italics"/>A.<emph.end type="italics"/> A puncto illo <emph type="italics"/>A,<emph.end type="italics"/> Filo <emph type="italics"/>APT<emph.end type="italics"/> longitudinem <lb/><emph type="italics"/>AR<emph.end type="italics"/> æquante, pendeat corpus <emph type="italics"/>T,<emph.end type="italics"/> & ita intra Semicycloides <emph type="italics"/>AQ, <lb/>AS<emph.end type="italics"/> o&longs;cilletur, ut quoties pendulum digreditur a perpendiculo <emph type="italics"/>AR,<emph.end type="italics"/><lb/> <arrow.to.target n="fig91"></arrow.to.target><lb/>Filum parte &longs;ui &longs;uperiore <emph type="italics"/>AP<emph.end type="italics"/> applicetur ad Semicycloidem illam <lb/><emph type="italics"/>APS<emph.end type="italics"/> ver&longs;us quam peragitur motus, & circum eam ceu ob&longs;tacu­<lb/>lum flectatur, parteque reliqua <emph type="italics"/>PT<emph.end type="italics"/> cui Semicyclois nondum obji­<lb/>citur, protendatur in lineam rectam; & pondus <emph type="italics"/>T<emph.end type="italics"/> o&longs;cillabitur in <lb/>Cycloide data <emph type="italics"/>QRS. <expan abbr="q.">que</expan> E. F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note114"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig91"></figure> <p type="main"> <s>Occurrat enim Filum <emph type="italics"/>PT<emph.end type="italics"/> tum Cycloidi <emph type="italics"/>QRS<emph.end type="italics"/> in <emph type="italics"/>T,<emph.end type="italics"/> tum circulo <lb/><emph type="italics"/>QOS<emph.end type="italics"/> in <emph type="italics"/>V,<emph.end type="italics"/> agaturque <emph type="italics"/>CV;<emph.end type="italics"/> & ad Fili partem rectam <emph type="italics"/>PT,<emph.end type="italics"/> e punctis <lb/>extremis <emph type="italics"/>P<emph.end type="italics"/> ac <emph type="italics"/>T,<emph.end type="italics"/> erigantur perpendicula <emph type="italics"/>PB, TW,<emph.end type="italics"/> occurrentia re­<lb/>ctæ <emph type="italics"/>CV<emph.end type="italics"/> in <emph type="italics"/>B<emph.end type="italics"/> & <emph type="italics"/>W.<emph.end type="italics"/> Patet, ex con&longs;tructione & gene&longs;i &longs;imilium Fi­<lb/>gurarum <emph type="italics"/>AS, SR,<emph.end type="italics"/> perpendicula illa <emph type="italics"/>PB, TW<emph.end type="italics"/> ab&longs;cindere de <emph type="italics"/>CV<emph.end type="italics"/> lon­<lb/>gitudines <emph type="italics"/>VB, VW<emph.end type="italics"/> Rotarum diametris <emph type="italics"/>OA, OR<emph.end type="italics"/> æquales. </s> <s>E&longs;t igi­<lb/>tur <emph type="italics"/>TP<emph.end type="italics"/> ad <emph type="italics"/>VP<emph.end type="italics"/> (duplum &longs;inum anguli <emph type="italics"/>VBP<emph.end type="italics"/> exi&longs;tente 1/2 <emph type="italics"/>BV<emph.end type="italics"/> ra- <pb pagenum="139"/>dio) ut <emph type="italics"/>BW<emph.end type="italics"/> ad <emph type="italics"/>BV,<emph.end type="italics"/> &longs;eu <emph type="italics"/>AO+OR<emph.end type="italics"/> ad <emph type="italics"/>AO,<emph.end type="italics"/> id e&longs;t (cum &longs;int <emph type="italics"/>CA<emph.end type="italics"/><lb/> <arrow.to.target n="note115"></arrow.to.target><lb/>ad <emph type="italics"/>CO, CO<emph.end type="italics"/> ad <emph type="italics"/>CR<emph.end type="italics"/> & divi&longs;im <emph type="italics"/>AO<emph.end type="italics"/> ad <emph type="italics"/>OR<emph.end type="italics"/> proportionales,) ut <lb/><emph type="italics"/>CA+CO<emph.end type="italics"/> ad <emph type="italics"/>CA<emph.end type="italics"/> vel, &longs;i bi&longs;ecetur <emph type="italics"/>BV<emph.end type="italics"/> in <emph type="italics"/>E,<emph.end type="italics"/> ut 2 <emph type="italics"/>CE<emph.end type="italics"/> ad <emph type="italics"/>CB.<emph.end type="italics"/><lb/>Proinde, per Corol. </s> <s>1. Prop. </s> <s>XLIX, longitudo partis rectæ Fili <emph type="italics"/>PT<emph.end type="italics"/><lb/>æquatur &longs;emper Cycloidis arcui <emph type="italics"/>PS,<emph.end type="italics"/> & Filum totum <emph type="italics"/>APT<emph.end type="italics"/> æquatur <lb/>&longs;emper Cycloidis arcui dimidio <emph type="italics"/>APS,<emph.end type="italics"/> hoc e&longs;t (per Corol. </s> <s>2. Prop. </s> <s><lb/>XLIX) longitudini <emph type="italics"/>AR.<emph.end type="italics"/> Et propterea vici&longs;&longs;im &longs;i Filum manet &longs;em­<lb/>per æquale longitudini <emph type="italics"/>AR<emph.end type="italics"/> movebitur punctum <emph type="italics"/>T<emph.end type="italics"/> in Cycloide <lb/>data <emph type="italics"/>QRS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note115"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Filum <emph type="italics"/>AR<emph.end type="italics"/> æquatur Semicycloidi <emph type="italics"/>AS,<emph.end type="italics"/> adeoque ad &longs;emi­<lb/>diametrum <emph type="italics"/>AC<emph.end type="italics"/> eandem habet rationem quam &longs;imilis illi Semicy­<lb/>clois <emph type="italics"/>SR<emph.end type="italics"/> habet ad &longs;emidiametrum <emph type="italics"/>CO.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO LI. THEOREMA XVIII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si Vis centripeta tendens undique ad Globi centrum<emph.end type="italics"/> C <emph type="italics"/>&longs;it in locis <lb/>&longs;ingulis ut di&longs;tantia loci cuju&longs;que a centro, & hac &longs;ola Vi a­<lb/>gente corpus<emph.end type="italics"/> T <emph type="italics"/>o&longs;cilletur (modo jam de&longs;cripto) in perimetro Cy­<lb/>cloidis<emph.end type="italics"/> QRS: <emph type="italics"/>dico quod o&longs;cillationum utcunque inæqualium <lb/>æqualia erunt Tempora.<emph.end type="italics"/></s> </p> <p type="main"> <s>Nam in Cycloidis tangentem <emph type="italics"/>TW<emph.end type="italics"/> infinite productam cadat per­<lb/>pendiculum <emph type="italics"/>CX<emph.end type="italics"/> & jungatur <emph type="italics"/>CT.<emph.end type="italics"/> Quoniam vis centripeta qua cor­<lb/>pus <emph type="italics"/>T<emph.end type="italics"/> impellitur ver&longs;us <emph type="italics"/>C<emph.end type="italics"/> e&longs;t ut di&longs;tantia <emph type="italics"/>CT,<emph.end type="italics"/> atque hæc (per Legum <lb/>Corol. </s> <s>2.) re&longs;olvitur in partes <emph type="italics"/>CX, TX,<emph.end type="italics"/> quarum <emph type="italics"/>CX<emph.end type="italics"/> impellen­<lb/>do corpus directe a <emph type="italics"/>P<emph.end type="italics"/> di&longs;tendit filum <emph type="italics"/>PT<emph.end type="italics"/> & per ejus re&longs;i&longs;tentiam <lb/>tota ce&longs;&longs;at, nullum alium edens effectum; pars autem altera <emph type="italics"/>TX,<emph.end type="italics"/><lb/>urgendo corpus tran&longs;ver&longs;im &longs;eu ver&longs;us <emph type="italics"/>X,<emph.end type="italics"/> directe accelerat motum <lb/>ejus in Cycloide; manife&longs;tum e&longs;t quod corporis acceleratio, huic <lb/>vi acceleratrici proportionalis, &longs;it &longs;ingulis momentis ut longitudo <lb/><emph type="italics"/>TX,<emph.end type="italics"/> id e&longs;t, (ob datas <emph type="italics"/>CV, WV<emph.end type="italics"/> ii&longs;que proportionales <emph type="italics"/>TX, TW,<emph.end type="italics"/>) <lb/>ut longitudo <emph type="italics"/>TW,<emph.end type="italics"/> hoc e&longs;t (per Corol. </s> <s>1. Prop. </s> <s>XLIX,) ut longitudo <lb/>arcus Cycloidis <emph type="italics"/>TR.<emph.end type="italics"/> Pendulis igitur duobus <emph type="italics"/>APT, Apt<emph.end type="italics"/> de per­<lb/>pendiculo <emph type="italics"/>AR<emph.end type="italics"/> inæqualiter deductis & &longs;imul dimi&longs;&longs;is, acceleratio­<lb/>nes eorum &longs;emper erunt ut arcus de&longs;cribendi <emph type="italics"/>TR, tR.<emph.end type="italics"/> Sunt au­<lb/>tem partes &longs;ub initio de&longs;criptæ ut accelerationes, hoc e&longs;t, ut totæ <lb/>&longs;ub initio de&longs;cribendæ, & propterea partes quæ manent de&longs;criben- <pb pagenum="140"/> <arrow.to.target n="note116"></arrow.to.target><lb/>dæ & accelerationes &longs;ub&longs;equentes, his partibus proportionales, &longs;unt <lb/>etiam ut totæ; & &longs;ic deinceps. </s> <s>Sunt igitur accelerationes atque <lb/>adeo velocitates genitæ & partes his velocitatibus de&longs;criptæ par­<lb/>te&longs;que de&longs;cribendæ, &longs;emper ut totæ; & propterea partes de&longs;criben­<lb/>dæ datam &longs;ervantes rationem ad invicem &longs;imul evane&longs;cent, id e&longs;t, <lb/>corpora duo o&longs;cillantia &longs;imul pervenient ad perpendiculum <emph type="italics"/>AR.<emph.end type="italics"/><lb/>Cumque vici&longs;&longs;im a&longs;cen&longs;us perpendiculorum de loco in&longs;imo <emph type="italics"/>R,<emph.end type="italics"/> per <lb/>eo&longs;dem arcus Cycloidales motu retrogrado facti, retardentur in <lb/>locis &longs;ingulis a viribus ii&longs;dem a quibus de&longs;cen&longs;us accelerabantur, <lb/>patet velocitates a&longs;cen&longs;uum ac de&longs;cen&longs;uum per eo&longs;dem arcus fa­<lb/>ctorum æquales e&longs;&longs;e, atque adeo temporibus æqualibus fieri; & <lb/>propterea, cum Cycloidis partes duæ <emph type="italics"/>RS<emph.end type="italics"/> & <emph type="italics"/>RQ<emph.end type="italics"/> ad utrumque per­<lb/>pendiculi latus jacentes &longs;int &longs;imiles & æquales, pendula duo o&longs;cil­<lb/>lationes &longs;uas tam totas quam dimidias ii&longs;dem temporibus &longs;emper <lb/>peragent. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note116"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Vis qua corpus <emph type="italics"/>T<emph.end type="italics"/> in loco quovis <emph type="italics"/>T<emph.end type="italics"/> acceleratur vel retar­<lb/>tur in Cycloide, e&longs;t ad totum corporis eju&longs;dem Pondus in loco <lb/>alti&longs;&longs;imo <emph type="italics"/>S<emph.end type="italics"/> vel <emph type="italics"/>Q,<emph.end type="italics"/> ut Cycloidis arcus <emph type="italics"/>TR<emph.end type="italics"/> ad eju&longs;dem arcum <emph type="italics"/>SR<emph.end type="italics"/><lb/>vel <emph type="italics"/>QR.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO LII. PROBLEMA XXXIV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Definire & Velocitates Pendulorum in locis &longs;ingulis, & Tempora <lb/>quibus tum o&longs;cillationes totæ, tum &longs;ingulæ o&longs;cillationum partes <lb/>peraguntur.<emph.end type="italics"/></s> </p> <p type="main"> <s>Centro quovis <emph type="italics"/>G,<emph.end type="italics"/> intervallo <emph type="italics"/>GH<emph.end type="italics"/> Cycloidis arcum <emph type="italics"/>RS<emph.end type="italics"/> æquante, <lb/>de&longs;cribe &longs;emicirculum <emph type="italics"/>HKMG<emph.end type="italics"/> &longs;emidiametro <emph type="italics"/>GK<emph.end type="italics"/> bi&longs;ectum. </s> <s>Et <lb/>&longs;i vis centripeta, di&longs;tantiis locorum a centro proportionalis, tendat <lb/>ad centrum <emph type="italics"/>G,<emph.end type="italics"/> &longs;itque ea in perimetro <emph type="italics"/>HIK<emph.end type="italics"/> æqualis vi centripetæ <lb/>in perimetro Globi <emph type="italics"/>QOS (Vide Fig. </s> <s>Prop.<emph.end type="italics"/> L.) ad ip&longs;ius cen­<lb/>trum tendenti; & eodem tempore quo pendulum <emph type="italics"/>T<emph.end type="italics"/> dimittitur e <lb/>loco &longs;upremo <emph type="italics"/>S,<emph.end type="italics"/> cadat corpus aliquod <emph type="italics"/>L<emph.end type="italics"/> ab <emph type="italics"/>H<emph.end type="italics"/> ad <emph type="italics"/>G:<emph.end type="italics"/> quoniam <lb/>vires quibus corpora urgentur &longs;unt æquales &longs;ub initio & &longs;patiis <lb/>de&longs;cribendis <emph type="italics"/>TR, LG<emph.end type="italics"/> &longs;emper proportionales, atque adeo, &longs;i æ­<lb/>quantur <emph type="italics"/>TR<emph.end type="italics"/> & <emph type="italics"/>LG,<emph.end type="italics"/> æquales in locis <emph type="italics"/>T<emph.end type="italics"/> & <emph type="italics"/>L<emph.end type="italics"/>; patet corpora illa <lb/>de&longs;cribere &longs;patia <emph type="italics"/>ST, HL<emph.end type="italics"/> æqualia &longs;ub initio, adeoque &longs;ubinde per­<lb/>gere æqualiter urgeri, & æqualia &longs;patia de&longs;cribere. </s> <s>Quare, per Prop. </s> <s><lb/>XXXVIII, tempus quo corpus de&longs;cribit arcum <emph type="italics"/>ST<emph.end type="italics"/> e&longs;t ad tempus <pb pagenum="141"/>o&longs;cillationis unius, ut arcus <emph type="italics"/>HI<emph.end type="italics"/> (tempus quo corpus <emph type="italics"/>H<emph.end type="italics"/> perveniet <lb/> <arrow.to.target n="note117"></arrow.to.target><lb/>ad <emph type="italics"/>L<emph.end type="italics"/>) ad &longs;emiperipheriam <emph type="italics"/>HKM<emph.end type="italics"/> (tempus quo corpus <emph type="italics"/>H<emph.end type="italics"/> per­<lb/>veniet ad <emph type="italics"/>M.<emph.end type="italics"/>) Et velocitas corporis penduli in loco <emph type="italics"/>T<emph.end type="italics"/> e&longs;t ad ve­<lb/>loc tatem ipfius in loco infimo <emph type="italics"/>R,<emph.end type="italics"/> (hoc e&longs;t, velocitas corporis <emph type="italics"/>H<emph.end type="italics"/> in <lb/>loco <emph type="italics"/>L<emph.end type="italics"/> ad velocitatem ejus in loco <emph type="italics"/>G,<emph.end type="italics"/> &longs;eu incrementum momenta­<lb/>neum lineæ <emph type="italics"/>HL<emph.end type="italics"/> ad incrementum momentaneum lineæ <emph type="italics"/>HG,<emph.end type="italics"/> arcu­<lb/>bus <emph type="italics"/>HI, HK<emph.end type="italics"/> æquabili fluxu cre&longs;centibus) ut ordinatim applicata <lb/><emph type="italics"/>LI<emph.end type="italics"/> ad radium <emph type="italics"/>GK,<emph.end type="italics"/> &longs;ive ut √<emph type="italics"/><expan abbr="SRq.-TRq.">SRq.-TRque</expan><emph.end type="italics"/> ad <emph type="italics"/>SR.<emph.end type="italics"/> Unde cum, <lb/>in o&longs;cillationibus inæqualibus, de&longs;cribantur æqualibus temporibus <lb/>arcus totis o&longs;cillationum arcubus proportionales; habentur, ex da­<lb/>tis temporibus, & velocitates & arcus de&longs;cripti in o&longs;cillationibus <lb/>univer&longs;is. </s> <s>Quæ erant primo invenienda. </s> </p> <p type="margin"> <s><margin.target id="note117"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s>O&longs;cillentur jam Funipendula <lb/> <arrow.to.target n="fig92"></arrow.to.target><lb/>corpora in Cycloidibus diver&longs;is <lb/>intra Globos diver&longs;os, quorum <lb/>diver&longs;æ &longs;unt etiam Vires ab&longs;olu­<lb/>tæ, de&longs;criptis: &, &longs;i Vis ab&longs;olu­<lb/>ta Globi cuju&longs;vis <emph type="italics"/>QOS<emph.end type="italics"/> dicatur V, <lb/>Vis acceleratrix qua <expan abbr="Pendulũ">Pendulum</expan> urge­<lb/>tur in circumferentia hujus Globi, <lb/>ubi incipit directe ver&longs;us centrum <lb/>ejus moveri, erit ut di&longs;tantia Cor­<lb/>poris penduli a centro illo & Vis ab&longs;oluta Globi conjunctim, hoc <lb/>e&longs;t, ut <emph type="italics"/>CO<emph.end type="italics"/>XV. </s> <s>Itaque lineola <emph type="italics"/>HY,<emph.end type="italics"/> quæ &longs;it ut hæc Vis accelera­<lb/>trix <emph type="italics"/>CO<emph.end type="italics"/>XV, de&longs;cribetur dato tempore; &, &longs;i erigatur normalis <emph type="italics"/>YZ<emph.end type="italics"/><lb/>circumferentiæ occurrens in <emph type="italics"/>Z,<emph.end type="italics"/> arcus na&longs;cens <emph type="italics"/>HZ<emph.end type="italics"/> denotabit datum <lb/>illud tempus. </s> <s>E&longs;t autem arcus hic na&longs;cens <emph type="italics"/>HZ<emph.end type="italics"/> in &longs;ubduplicata ra­<lb/>tione rectanguli <emph type="italics"/>GHY,<emph.end type="italics"/> adeoque ut √<emph type="italics"/>GHXCO<emph.end type="italics"/>XV. </s> <s>Unde Tem­<lb/>pus o&longs;cillationis integræ in Cycloide <emph type="italics"/>QRS<emph.end type="italics"/> (cum &longs;it ut &longs;emiperi­<lb/>pheria <emph type="italics"/>HKM,<emph.end type="italics"/> quæ o&longs;cillationem illam integram denotat, directe, <lb/>utque arcus <emph type="italics"/>HZ,<emph.end type="italics"/> qui datum tempus &longs;imiliter denotat, inver&longs;e) fiet <lb/>ut <emph type="italics"/>GH<emph.end type="italics"/> directe & √<emph type="italics"/>GHXCO<emph.end type="italics"/>XV inver&longs;e, hoc e&longs;t, ob æquales <emph type="italics"/>GH<emph.end type="italics"/><lb/>& <emph type="italics"/>SR,<emph.end type="italics"/> ut √(<emph type="italics"/>SR/CO<emph.end type="italics"/>XV), &longs;ive (per Corol. </s> <s>Prop. </s> <s>L) ut √(<emph type="italics"/>AR/AC<emph.end type="italics"/>XV). <lb/>Itaque O&longs;cillationes in Globis & Cycloidibus omnibus, quibu&longs;­<lb/>cunque cum Viribus ab&longs;olutis factæ, &longs;unt in ratione quæ compo­<lb/>nitur ex &longs;ubduplicata ratione longitudinis Fili directe, & &longs;ubdu­<lb/>plicata ratione di&longs;tantiæ inter punctum &longs;u&longs;pen&longs;ionis & centrum <pb pagenum="142"/> <arrow.to.target n="note118"></arrow.to.target><lb/>Globi inver&longs;e, & &longs;ubduplicata ratione Vis ab&longs;olutæ Globi etiam <lb/>inver&longs;e. <emph type="italics"/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note118"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig92"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc etiam O&longs;cillantium, Cadentium & Revolventium <lb/>corporum tempora po&longs;&longs;unt inter &longs;e conferri. </s> <s>Nam &longs;i Rotæ, qua Cy­<lb/>clois intra globum de&longs;cribitur, diameter con&longs;tituatur æqualis &longs;emi­<lb/>diametro globi, Cyclois evadet Linea recta per centrum globi tran­<lb/>&longs;iens, & O&longs;cillatio jam erit de&longs;cen&longs;us & &longs;ub&longs;equens a&longs;cen&longs;us in hac <lb/>recta. </s> <s>Unde datur tum tempus de&longs;cen&longs;us de loco quovis ad <lb/>centrum, tum tempus huic æquale quo corpus uniformiter cir­<lb/>ca centrum globi ad di&longs;tantiam quamvis revolvendo arcum qua­<lb/>drantalem de&longs;cribit. </s> <s>E&longs;t enim hoc tempus (per Ca&longs;um &longs;ecun­<lb/>dum) ad tempus &longs;emio&longs;cillationis in Cycloide quavis <emph type="italics"/>QRS<emph.end type="italics"/> ut <lb/>1 ad √(<emph type="italics"/>AR/AC<emph.end type="italics"/>). </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Hinc etiam con&longs;ectantur quæ <emph type="italics"/>Wrennus<emph.end type="italics"/> & <emph type="italics"/>Hugenius<emph.end type="italics"/> de <lb/>Cycloide vulgari adinvenerunt. </s> <s>Nam &longs;i Globi diameter augeatur <lb/>in infinitum: mutabitur ejus &longs;uperficies &longs;phærica in planum, Vi&longs;que <lb/>centripeta aget uniformiter &longs;ecundum lineas huic plano perpendi­<lb/>culares, & Cyclois no&longs;tra abibit in Cycloidem vulgi. </s> <s>I&longs;to autem <lb/>in ca&longs;u longitudo arcus Cycloidis, inter planum illud & punctum <lb/>de&longs;cribens, æqualis evadet quadruplicato &longs;inui ver&longs;o dimidii arcus <lb/>Rotæ inter idem planum & punctum de&longs;cribens; ut invenit <emph type="italics"/>Wren­<lb/>nus:<emph.end type="italics"/> Et Pendulum inter duas eju&longs;modi Cycloides in &longs;imili & æ­<lb/>quali Cycloide temporibus æqualibus O&longs;cillabitur, ut demon&longs;travit <lb/><emph type="italics"/>Hugenius.<emph.end type="italics"/> Sed & De&longs;cen&longs;us gravium, tempore O&longs;cillationis unius, <lb/>is erit quem <emph type="italics"/>Hugenius<emph.end type="italics"/> indicavit. </s> </p> <p type="main"> <s>Aptantur autem Propo&longs;itiones a nobis demon&longs;tratæ ad veram <lb/>con&longs;titutionem Terræ, quatenus Rotæ eundo in ejus circulis maxi­<lb/>mis de&longs;cribunt motu Clavorum, perimetris &longs;uis infixorum, Cycloi­<lb/>des extra globum; & Pendula inferius in fodinis & cavernis Terra <lb/>&longs;u&longs;pen&longs;a, in Cycloidibus intra globos O&longs;cillari debent, ut O&longs;cilla­<lb/>tiones omnes evadant I&longs;ochronæ. </s> <s>Nam Gravitas (ut in Libro <lb/>tertio docebitur) decre&longs;cit in progre&longs;&longs;u a &longs;uperficie Terræ, &longs;ur­<lb/>&longs;um quidem in duplicata ratione di&longs;tantiarum a centro ejus, de <lb/>or&longs;um vero in ratione &longs;implici. <pb pagenum="143"/> <arrow.to.target n="note119"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note119"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO LIII. PROBLEMA XXXV.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Conce&longs;&longs;is Figurarum curvilinearum quadraturis, invenire Vires qui­<lb/>bus corpora in datis curvis lineis O&longs;cillationes &longs;emper I&longs;ochro­<lb/>nas peragent.<emph.end type="italics"/></s> </p> <p type="main"> <s>O&longs;cilletur corpus <emph type="italics"/>T<emph.end type="italics"/> in curva quavis linea <emph type="italics"/>STRQ,<emph.end type="italics"/> cujus axis &longs;it <lb/><emph type="italics"/>OR<emph.end type="italics"/> tran&longs;iens per virium centrum <emph type="italics"/>C.<emph.end type="italics"/> Agatur <emph type="italics"/>TX<emph.end type="italics"/> quæ curvam il­<lb/>lam in corporis loco quovis <emph type="italics"/>T<emph.end type="italics"/> contingat, inque hac tangente <emph type="italics"/>TX<emph.end type="italics"/><lb/> <arrow.to.target n="fig93"></arrow.to.target><lb/>capiatur <emph type="italics"/>TY<emph.end type="italics"/> æqualis arcui <emph type="italics"/>TR.<emph.end type="italics"/> Nam longitudo arcus illius ex Fi­<lb/>gurarum quadraturis (per Methodos vulgares) innote&longs;cit. </s> <s>De pun­<lb/>cto <emph type="italics"/>Y<emph.end type="italics"/> educatur recta <emph type="italics"/>YZ<emph.end type="italics"/> tangenti perpendicularis. </s> <s>Agatur <emph type="italics"/>CT<emph.end type="italics"/> per­<lb/>pendiculari illi occurrens in <emph type="italics"/>Z,<emph.end type="italics"/> & erit Vis centripeta proportiona­<lb/>lis rectæ <emph type="italics"/>TZ. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/> <pb pagenum="144"/> <arrow.to.target n="note120"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note120"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig93"></figure> <p type="main"> <s>Nam &longs;i vis, qua corpus trahitur de <emph type="italics"/>T<emph.end type="italics"/> ver&longs;us <emph type="italics"/>C,<emph.end type="italics"/> exponatur per <lb/>rectam <emph type="italics"/>TZ<emph.end type="italics"/> captam ip&longs;i proportionalem, re&longs;olvetur hæc in vires <lb/><emph type="italics"/>TY, YZ<emph.end type="italics"/>; quarum <emph type="italics"/>YZ<emph.end type="italics"/> trahendo corpus &longs;ecundum longitudinem <lb/>Fili <emph type="italics"/>PT,<emph.end type="italics"/> motum ejus nil mutat, vis autem altera <emph type="italics"/>TY<emph.end type="italics"/> motum ejus <lb/>in curva <emph type="italics"/>STRQ<emph.end type="italics"/> directe accelerat vel directe retardat. </s> <s>Proinde <lb/>cum hæc &longs;it ut via de&longs;cribenda <emph type="italics"/>TR,<emph.end type="italics"/> accelerationes corporis vel re­<lb/>tardationes in O&longs;cillationum duarum (majoris & minoris) parti­<lb/>bus proportionalibus de&longs;cribendis, erunt &longs;emper ut partes illæ, & <lb/>propterea facient ut partes illæ &longs;imul de&longs;cribantur. </s> <s>Corpora autem <lb/>quæ partes totis &longs;emper proportionales &longs;imul de&longs;cribunt, &longs;imul de­<lb/>&longs;cribent totas. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i corpus <emph type="italics"/>T<emph.end type="italics"/> Filo rectilineo <emph type="italics"/>AT<emph.end type="italics"/> a centro <emph type="italics"/>A<emph.end type="italics"/> pen­<lb/>dens, de&longs;cribat arcum circularem <emph type="italics"/>STRQ,<emph.end type="italics"/> & interea urgeatur &longs;e­<lb/>cundum lineas parallelas deor&longs;um a vi aliqua, quæ &longs;it ad vim uni­<lb/>formem Gravitatis, ut arcus <emph type="italics"/>TR<emph.end type="italics"/> ad ejus &longs;inum <emph type="italics"/>TN:<emph.end type="italics"/> æqualia e­<lb/>runt O&longs;cillationum &longs;ingularum tempora. </s> <s>Etenim ob parallelas <lb/><emph type="italics"/>TZ, AR,<emph.end type="italics"/> &longs;imilia erunt triangula <emph type="italics"/>ATN, ZTY<emph.end type="italics"/>; & propterea <lb/><emph type="italics"/>TZ<emph.end type="italics"/> erit ad <emph type="italics"/>AT<emph.end type="italics"/> ut <emph type="italics"/>TY<emph.end type="italics"/> ad <emph type="italics"/>TN<emph.end type="italics"/>; hoc e&longs;t, (&longs;i Gravitatis vis unifor­<lb/>mis exponatur per longitudinem datam <emph type="italics"/>AT<emph.end type="italics"/>) vis <emph type="italics"/>TZ,<emph.end type="italics"/> qua O&longs;cil­<lb/>lationes evadent I&longs;ochronæ, erit ad vim Gravitatis <emph type="italics"/>AT,<emph.end type="italics"/> ut arcus <lb/><emph type="italics"/>TR<emph.end type="italics"/> ip&longs;i <emph type="italics"/>TY<emph.end type="italics"/> æqualis ad arcus illius &longs;inum <emph type="italics"/>TN.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Igitur in Horologiis, &longs;i vires a Machina in Pendulum <lb/>ad motum con&longs;ervandum impre&longs;&longs;æ ita cum vi Gravitatis componi <lb/>po&longs;&longs;int, ut vis tota deor&longs;um &longs;emper &longs;it ut linea quæ oritur appli­<lb/>cando rectangulum &longs;ub arcu <emph type="italics"/>TR<emph.end type="italics"/> & radio <emph type="italics"/>AR<emph.end type="italics"/> ad &longs;inum <emph type="italics"/>TN,<emph.end type="italics"/><lb/>O&longs;cillationes omnes erunt I&longs;ochronæ. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO LIV. PROBLEMA XXXVI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Conce&longs;&longs;is Figurarum curvilinearum quadraturis, invenire Tempora <lb/>quibus corpora Vi qualibet centripeta in lineis quibu&longs;cunque cur­<lb/>vis, in plano per centrum Virium tran&longs;eunte de&longs;criptis, de&longs;cen­<lb/>dent & a&longs;cendent.<emph.end type="italics"/></s> </p> <p type="main"> <s>De&longs;cendat corpus de loco quovis <emph type="italics"/>S<emph.end type="italics"/> per lineam quamvis curvam <lb/><emph type="italics"/>STtR,<emph.end type="italics"/> in plano per virium centrum <emph type="italics"/>C<emph.end type="italics"/> tran&longs;eunte datam. </s> <s>Junga­<lb/>tur <emph type="italics"/>CS<emph.end type="italics"/> & dividatur eadem in partes innumeras æquales, &longs;itque <emph type="italics"/>Dd<emph.end type="italics"/> <pb pagenum="145"/>partium illarum aliqua. </s> <s>Centro <emph type="italics"/>C,<emph.end type="italics"/> intervallis <emph type="italics"/>CD, Cd<emph.end type="italics"/> de&longs;criban­</s> </p> <p type="main"> <s> <arrow.to.target n="note121"></arrow.to.target><lb/>tur circuli <emph type="italics"/>DT, dt,<emph.end type="italics"/> lineæ curvæ <emph type="italics"/>STtR<emph.end type="italics"/> occurrentes in <emph type="italics"/>T<emph.end type="italics"/> & <emph type="italics"/>t.<emph.end type="italics"/> Et <lb/>ex data tum lege vis centripetæ, tum <lb/> <arrow.to.target n="fig94"></arrow.to.target><lb/>altitudine <emph type="italics"/>CS<emph.end type="italics"/> de qua corpus cecidit; <lb/>dabitur velocitas corporis in alia qua­<lb/>vis altitudine <emph type="italics"/>CT,<emph.end type="italics"/> per Prop. </s> <s>XXXIX. </s> <s><lb/>Tempus autem, quo corpus de&longs;cribit <lb/>lineolam <emph type="italics"/>Tt,<emph.end type="italics"/> e&longs;t ut lineolæ hujus lon­<lb/>gitudo (id e&longs;t ut &longs;ecans anguli <emph type="italics"/>tTC<emph.end type="italics"/>) <lb/>directe, & velocitas inver&longs;e. </s> <s>Tempori <lb/>huic proportionalis &longs;it ordinatim appli­<lb/>cata <emph type="italics"/>DN<emph.end type="italics"/> ad rectam <emph type="italics"/>CS<emph.end type="italics"/> per punctum <lb/><emph type="italics"/>D<emph.end type="italics"/> perpendicularis, & ob datam <emph type="italics"/>Dd<emph.end type="italics"/><lb/>erit rectangulum <emph type="italics"/>DdXDN,<emph.end type="italics"/> hoc e&longs;t <lb/>area <emph type="italics"/>DNnd,<emph.end type="italics"/> eidem tempori propor­<lb/>tionale. </s> <s>Ergo &longs;i <emph type="italics"/>SNn<emph.end type="italics"/> &longs;it curva illa li­<lb/>nea quam punctum <emph type="italics"/>N<emph.end type="italics"/> perpetuo tangit, <lb/>erit area <emph type="italics"/>SNDS<emph.end type="italics"/> proportionalis tem­<lb/>pori quo corpus de&longs;cendendo de&longs;crip­<lb/>&longs;it lineam <emph type="italics"/>ST<emph.end type="italics"/>; proindeque ex inventa illa area dabitur Tempus. <lb/><emph type="italics"/>q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note121"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig94"></figure> <p type="main"> <s><emph type="center"/>PROPOSITIO LV. THEOREMA XIX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si corpus movetur in &longs;uperficie quacunque curva, cujus axis per <lb/>centrum Virium tran&longs;it, & a corpore in axem demittatur per­<lb/>pendicularis, eique parallela & æqualis ab axis puncto quovis <lb/>dato ducatur: dico quod parallela illa aream tempori proportio­<lb/>nalem de&longs;cribet.<emph.end type="italics"/></s> </p> <p type="main"> <s>Sit <emph type="italics"/>BSKL<emph.end type="italics"/> &longs;uperficies curva, <emph type="italics"/>T<emph.end type="italics"/> corpus in ea revolvens, <emph type="italics"/>STtR<emph.end type="italics"/><lb/>Trajectoria quam corpus in eadem de&longs;cribit, <emph type="italics"/>S<emph.end type="italics"/> initium Trajecto­<lb/>riæ, <emph type="italics"/>OMNK<emph.end type="italics"/> axis &longs;uperficiei curvæ, <emph type="italics"/>TN<emph.end type="italics"/> recta a corpore in axem <lb/>perpendicularis, <emph type="italics"/>OP<emph.end type="italics"/> huic parallela & æqualis a puncto <emph type="italics"/>O<emph.end type="italics"/> quod in <lb/>axe datur educta, <emph type="italics"/>AP<emph.end type="italics"/> ve&longs;tigium Trajectoriæ a puncto <emph type="italics"/>P<emph.end type="italics"/> in lineæ <lb/>volubilis <emph type="italics"/>OP<emph.end type="italics"/> plano <emph type="italics"/>AOP<emph.end type="italics"/> de&longs;criptum, <emph type="italics"/>A<emph.end type="italics"/> ve&longs;tigii initium puncto <emph type="italics"/>S<emph.end type="italics"/><lb/>re&longs;pondens, <emph type="italics"/>TC<emph.end type="italics"/> recta a corpore ad centrum ducta; <emph type="italics"/>TG<emph.end type="italics"/> pars ejus <lb/>vi centripetæ qua corpus urgetur in centrum <emph type="italics"/>C<emph.end type="italics"/> proportionalis; <lb/><emph type="italics"/>TM<emph.end type="italics"/> recta ad &longs;uperficiem cutvam perpendicularis, <emph type="italics"/>TI<emph.end type="italics"/> pars ejus vi <lb/>pre&longs;&longs;ionis, qua corpus urget &longs;uperficiem vici&longs;&longs;imque urgetur ver&longs;us <emph type="italics"/>M<emph.end type="italics"/> <pb pagenum="146"/> <arrow.to.target n="note122"></arrow.to.target><lb/>a &longs;uperficie, proportiona­<lb/> <arrow.to.target n="fig95"></arrow.to.target><lb/>lis; <emph type="italics"/>PHTF<emph.end type="italics"/> recta axi <lb/>parallela per corpus tran­<lb/>&longs;iens, & <emph type="italics"/>GF, IH<emph.end type="italics"/> rectæ <lb/>a punctis <emph type="italics"/>G<emph.end type="italics"/> & <emph type="italics"/>I<emph.end type="italics"/> in pa­<lb/>rallelam illam <emph type="italics"/>PHTF<emph.end type="italics"/><lb/>perpendiculariter demi&longs;­<lb/>&longs;æ. </s> <s>Dico jam quod area <lb/><emph type="italics"/>AOP,<emph.end type="italics"/> radio <emph type="italics"/>OP<emph.end type="italics"/> ab ini­<lb/>tio motus de&longs;cripta, &longs;it <lb/>tempori proportionalis. </s> <s><lb/>Nam vis <emph type="italics"/>TG<emph.end type="italics"/> (per Le­<lb/>gum Corol. </s> <s>2.) re&longs;olvitur <lb/>in vires <emph type="italics"/>TF, FG<emph.end type="italics"/>; & vis <lb/><emph type="italics"/>TI<emph.end type="italics"/> in vires <emph type="italics"/>TH, HI:<emph.end type="italics"/><lb/>Vires autem <emph type="italics"/>TF, TH<emph.end type="italics"/><lb/>agendo &longs;ecundum lineam <lb/><emph type="italics"/>PF<emph.end type="italics"/> plano <emph type="italics"/>AOP<emph.end type="italics"/> per­<lb/>pendicularem mutant &longs;o­<lb/>lummodo motum cor­<lb/>poris quatenus huic plano perpendicularem. </s> <s>Ideoque motus ejus <lb/>quatenus &longs;ecundum po&longs;itionem plani factus, hoc e&longs;t, motus pun­<lb/>cti <emph type="italics"/>P<emph.end type="italics"/> quo Trajectoriæ ve&longs;tigium <emph type="italics"/>AP<emph.end type="italics"/> in hoc plano de&longs;cri­<lb/>bitur, idem e&longs;t ac &longs;i vires <emph type="italics"/>TF, TH<emph.end type="italics"/> tollerentur, & corpus &longs;olis vi­<lb/>ribus <emph type="italics"/>FG, HI<emph.end type="italics"/> agitaretur; hoc e&longs;t, idem ac &longs;i corpus in plano <lb/><emph type="italics"/>AOP,<emph.end type="italics"/> vi centripeta ad centrum <emph type="italics"/>O<emph.end type="italics"/> tendente & &longs;ummam virium <lb/><emph type="italics"/>FG<emph.end type="italics"/> & <emph type="italics"/>HI<emph.end type="italics"/> æquante, de&longs;criberet curvam <emph type="italics"/>AP.<emph.end type="italics"/> Sed vi tali de&longs;cribi­<lb/>tur area <emph type="italics"/>AOP<emph.end type="italics"/> (per Prop. </s> <s>1.) tempori proportionalis. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note122"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <figure id="fig95"></figure> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> Eodem argumento &longs;i corpus a viribus agitatum ad centra <lb/>duo vel plura in eadem quavis recta <emph type="italics"/>CO<emph.end type="italics"/> data tendentibus, de&longs;cri­<lb/>beret in &longs;patio libero lineam quamcunque curvam <emph type="italics"/>ST<emph.end type="italics"/>; foret area <lb/><emph type="italics"/>AOP<emph.end type="italics"/> tempori &longs;emper proportionalis. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO LVI. PROBLEMA XXXVII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Conce&longs;&longs;is Figurarum curvilinearum quadraturis, dati&longs;que tum lege <lb/>Vis centripetæ ad centrum datum tendentis, tum &longs;uperficie cur­<lb/>va cujus axis per centrum illud træn&longs;it; invenieuda est Traje­<lb/>ctoria quam corpus in eadem &longs;uperficie de&longs;cribet, de loco dato, data <lb/>cum Velocitate, ver&longs;us plagam in &longs;uperficie illa datam egre&longs;&longs;um.<emph.end type="italics"/></s> </p> <pb pagenum="147"/> <p type="main"> <s>Stantibus quæ in &longs;uperiore Propo&longs;itione con&longs;tructa &longs;unt, exeat <lb/> <arrow.to.target n="note123"></arrow.to.target><lb/>corpus de loco <emph type="italics"/>S<emph.end type="italics"/> in Trajectoriam inveniendam <emph type="italics"/>STtR<emph.end type="italics"/>; &, ex da­<lb/>ta ejus velocitate in altitudine <emph type="italics"/>SC,<emph.end type="italics"/> dabitur ejus velocitas in alia <lb/>quavis altitudine <emph type="italics"/>TC.<emph.end type="italics"/> Ea cum velocitate, dato tempore quam <lb/>minimo, de&longs;cribat corpus Trajectoriæ &longs;uæ particulam <emph type="italics"/>Tt,<emph.end type="italics"/> &longs;itque <lb/><emph type="italics"/>Pp<emph.end type="italics"/> ve&longs;tigium ejus in plano <emph type="italics"/>AOP<emph.end type="italics"/> de&longs;criptum. </s> <s>Jungatur <emph type="italics"/>Op,<emph.end type="italics"/> & <lb/>Circelli centro <emph type="italics"/>T<emph.end type="italics"/> intervallo <emph type="italics"/>Tt<emph.end type="italics"/> in &longs;uperficie curva de&longs;cripti &longs;it <emph type="italics"/>PpQ<emph.end type="italics"/><lb/>ve&longs;tigium Ellipticum in eodem plano <emph type="italics"/>OAPp<emph.end type="italics"/> de&longs;criptum. </s> <s>Et ob <lb/>datum magnitudine & po&longs;itione Circellum, dabitur Ellip&longs;is illa <lb/><emph type="italics"/><expan abbr="Ppq.">Ppque</expan><emph.end type="italics"/> Cumque area <emph type="italics"/>POp<emph.end type="italics"/> &longs;it tempori proportionalis, atque ad­<lb/>eo ex dato tempore detur, dabitur <emph type="italics"/>Op<emph.end type="italics"/> po&longs;itione, & inde dabitur <lb/>communis ejus & Ellip&longs;eos inter&longs;ectio <emph type="italics"/>p,<emph.end type="italics"/> una cum angulo <emph type="italics"/>OPp,<emph.end type="italics"/><lb/>in quo Trajectoriæ ve&longs;tigium <emph type="italics"/>APp<emph.end type="italics"/> &longs;ecat lineam <emph type="italics"/>OP.<emph.end type="italics"/> Inde au­<lb/>tem invenietur Trajectoriæ ve&longs;tigium illud <emph type="italics"/>APp,<emph.end type="italics"/> eadem methodo <lb/>qua curva linea <emph type="italics"/>VIKk,<emph.end type="italics"/> in Propo&longs;itione XLI, ex &longs;imilibus datis <lb/>inventa fuit. </s> <s>Tum ex &longs;ingulis ve&longs;tigii punctis <emph type="italics"/>P<emph.end type="italics"/> erigendo ad pla­<lb/>num <emph type="italics"/>AOP<emph.end type="italics"/> perpendicula <emph type="italics"/>PT<emph.end type="italics"/> &longs;uperficiei curvæ occurrentia in <emph type="italics"/>T,<emph.end type="italics"/><lb/>dabuntur &longs;ingula Trajectoriæ puncta <emph type="italics"/>T. q.E.I.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note123"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <p type="main"> <s><emph type="center"/>SECTIO XI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu Corporum Viribus centripetis &longs;e mutuo petentium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Hactenus expo&longs;ui Motus corporum attractorum ad centrum Im­<lb/>mobile, quale tamen vix extat in rerum natura. </s> <s>Attractiones enim <lb/>fieri &longs;olent ad corpora; & corporum trahentium & attractorum <lb/>actiones &longs;emper mutuæ &longs;unt & æquales, per Legem tertiam: ad­<lb/>eo ut neque attrahens po&longs;&longs;it quie&longs;cere neque attractum, &longs;i duo &longs;int <lb/>corpora, &longs;ed ambo (per Legum Corollarium quartum) qua&longs;i at­<lb/>tractione mutua, circum gravitatis centrum commune revolvantur: <lb/>& &longs;i plura &longs;int corpora (quæ vel ab unico attrahantur vel omnia <lb/>&longs;e mutuo attrahant) hæc ita inter &longs;e moveri debeant, ut gravitatis <lb/>centrum commune vel quie&longs;cat vel uniformiter moveatur in direc­<lb/>tum. </s> <s>Qua de cau&longs;a jam pergo Motum exponere corporum &longs;e mu­<lb/>tuo trahentium, con&longs;iderando Vires centripetas tanquam Attractio­<lb/>nes, quamvis forta&longs;&longs;e, &longs;i phy&longs;ice loquamur, verius dicantur Im­<lb/>pul&longs;us. </s> <s>In Mathematicis enim jam ver&longs;amur, & propterea mi&longs;&longs;is <lb/>di&longs;putationibus Phy&longs;icis, familiari utimur &longs;ermone, quo po&longs;&longs;imus <lb/>a Lectoribus Mathematicis facilius intelligi. <pb pagenum="148"/> <arrow.to.target n="note124"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note124"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO LVII. THEOREMA XX.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corpora duo &longs;e invicem trahentia de&longs;cribunt, & circum commune <lb/>centrum gravitatis, & circum &longs;e mutuo, Figuras &longs;imiles.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s>Sunt enim di&longs;tantiæ a communi gravitatis centro reciproce pro­<lb/>portionales corporibus, atque adeo in data ratione ad invicem, & <lb/>componendo, in data ratione ad di&longs;tantiam totam inter corpora. </s> <s><lb/>Feruntur autem hæ di&longs;tantiæ circum terminos &longs;uos communi motu <lb/>angulari, propterea quod in directum &longs;emper jacentes non mutant <lb/>inclinationem ad &longs;e mutuo. </s> <s>Lineæ autem rectæ, quæ &longs;unt in data <lb/>ratione ad invicem, & æquali motu angulari circum terminos &longs;uos <lb/>feruntur, Figuras circum eo&longs;dem terminos (in planis quæ una cum <lb/>his terminis vel quie&longs;cunt vel motu quovis non angulari moven­<lb/>tur) de&longs;cribunt omnino &longs;imiles. </s> <s>Proinde &longs;imiles &longs;unt Figuræ quæ <lb/>his di&longs;tantiis circumactis de&longs;cribuntur. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO LVIII. THEOREMA XXI.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Si corpora duo Viribus quibu&longs;vis &longs;e mutuo trahunt, & interea re­<lb/>volvuntur circa gravitatis centrum commune: dico quod Fi­<lb/>guris, quas corpora &longs;ic mota de&longs;cribunt circum &longs;e mutuo, potest <lb/>Figura &longs;imilis & æqualis, circum corpus alterutrum immotum, <lb/>Viribus ii&longs;dem de&longs;eribi.<emph.end type="italics"/></s> </p> <p type="main"> <s>Revolvantur corpora <emph type="italics"/>S, P<emph.end type="italics"/> circa commune gravitatis centrum <lb/><emph type="italics"/>C,<emph.end type="italics"/> pergendo de <emph type="italics"/>S<emph.end type="italics"/> ad <emph type="italics"/>T<emph.end type="italics"/> deque <emph type="italics"/>P<emph.end type="italics"/> ad <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> A dato puncto <emph type="italics"/>s<emph.end type="italics"/> ip&longs;is <lb/> <arrow.to.target n="fig96"></arrow.to.target><lb/><emph type="italics"/>SP, TQ<emph.end type="italics"/> æquales & parallelæ ducantur &longs;emper <emph type="italics"/>sp, sq<emph.end type="italics"/>; & Curva <lb/><emph type="italics"/>pqv<emph.end type="italics"/> quam punctum <emph type="italics"/>p,<emph.end type="italics"/> revolvendo circum punctum immotum <emph type="italics"/>s,<emph.end type="italics"/> <pb pagenum="149"/>de&longs;cribit, erit &longs;imilis & æqualis Curvis quas corpora <emph type="italics"/>S, P<emph.end type="italics"/> de&longs;cri­<lb/> <arrow.to.target n="note125"></arrow.to.target><lb/>bunt circum &longs;e mutuo: proindeque (per Theor. </s> <s>XX) &longs;imilis Curvis <lb/><emph type="italics"/>ST<emph.end type="italics"/> & <emph type="italics"/>PQV,<emph.end type="italics"/> quas eadem corpora de&longs;cribunt circum commune <lb/>gravitatis centrum <emph type="italics"/>C:<emph.end type="italics"/> id adeo quia proportiones linearum <emph type="italics"/>SC, CP<emph.end type="italics"/><lb/>& <emph type="italics"/>SP<emph.end type="italics"/> vel. <emph type="italics"/>sp<emph.end type="italics"/> ad invicem dantur. </s> </p> <p type="margin"> <s><margin.target id="note125"></margin.target>LIBER <lb/>PRIMUS.</s> </p> <figure id="fig96"></figure> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Commune illud gravitatis centrum <emph type="italics"/>C,<emph.end type="italics"/> per Legum Co­<lb/>rollarium quartum, vel quie&longs;cit vel movetur uniformiter in direc­<lb/>tum. </s> <s>Ponamus primo quod id quie&longs;cit, inque <emph type="italics"/>s<emph.end type="italics"/> & <emph type="italics"/>p<emph.end type="italics"/> locentur cor­<lb/>pora duo, immobile in <emph type="italics"/>s,<emph.end type="italics"/> mobile in <emph type="italics"/>p,<emph.end type="italics"/> corporibus <emph type="italics"/>S<emph.end type="italics"/> & <emph type="italics"/>P<emph.end type="italics"/> &longs;imilia <lb/>& æqualia. </s> <s>Dein tangant rectæ <emph type="italics"/>PR<emph.end type="italics"/> & <emph type="italics"/>pr<emph.end type="italics"/> Curvas <emph type="italics"/>PQ<emph.end type="italics"/> & <emph type="italics"/>pq<emph.end type="italics"/> in <lb/><emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>p,<emph.end type="italics"/> & producantur <emph type="italics"/>CQ<emph.end type="italics"/> & <emph type="italics"/>sq<emph.end type="italics"/> ad <emph type="italics"/>R<emph.end type="italics"/> & <emph type="italics"/>r.<emph.end type="italics"/> Et, ob &longs;imilitudi­<lb/>nem Figurarum <emph type="italics"/>CPRQ, sprq,<emph.end type="italics"/> erit <emph type="italics"/>RQ<emph.end type="italics"/> ad <emph type="italics"/>rq<emph.end type="italics"/> ut <emph type="italics"/>CP<emph.end type="italics"/> ad <emph type="italics"/>sp,<emph.end type="italics"/> ad­<lb/>eoque in data ratione. </s> <s>Proinde &longs;i vis qua corpus <emph type="italics"/>P<emph.end type="italics"/> ver&longs;us cor­<lb/>pus <emph type="italics"/>S,<emph.end type="italics"/> atque adeo ver&longs;us centrum intermedium <emph type="italics"/>C<emph.end type="italics"/> attrahitur, e&longs;&longs;et <lb/>ad vim qua corpus <emph type="italics"/>p<emph.end type="italics"/> ver&longs;us centrum <emph type="italics"/>s<emph.end type="italics"/> attrahitur in eadem illa ra­<lb/>tione data; hæ vires æqualibus temporibus attraherent &longs;emper cor­<lb/>pora de tangentibus <emph type="italics"/>PR, pr<emph.end type="italics"/> ad arcus <emph type="italics"/>PQ, pq,<emph.end type="italics"/> per intervalla ip&longs;is <lb/>proportionalia <emph type="italics"/>RQ, <expan abbr="rq;">rque</expan><emph.end type="italics"/> adeoque vis po&longs;terior efficeret ut corpus <lb/><emph type="italics"/>p<emph.end type="italics"/> gyraretur in Curva <emph type="italics"/>pqv,<emph.end type="italics"/> quæ &longs;imilis e&longs;&longs;et Curvæ <emph type="italics"/>PQV,<emph.end type="italics"/> in qua <lb/>vis prior efficit ut corpus <emph type="italics"/>P<emph.end type="italics"/> gyretur, & revolutiones ii&longs;dem tem­<lb/>poribus complerentur. </s> <s>At quoniam vires illæ non &longs;unt ad invi­<lb/>cem in ratione <emph type="italics"/>CP<emph.end type="italics"/> ad <emph type="italics"/>sp,<emph.end type="italics"/> &longs;ed (ob &longs;imilitudinem & æqualitatem <lb/>corporum <emph type="italics"/>S<emph.end type="italics"/> & <emph type="italics"/>s, P<emph.end type="italics"/> & <emph type="italics"/>p,<emph.end type="italics"/> æqualitatem di&longs;tantiarum <emph type="italics"/>SP, sp<emph.end type="italics"/>) <lb/>&longs;ibi mutuo æquales; corpora æqualibus temporibus æqualiter tra­<lb/>hentur de tangentibus: & propterea, ut corpus po&longs;terius <emph type="italics"/>p<emph.end type="italics"/> trahatur <lb/>per intervallum majus <emph type="italics"/>rq,<emph.end type="italics"/> requiritur tempus majus, idque in &longs;ub­<lb/>duplicata ratione intervallorum; propterea quod (per Lemma de­<lb/>cimum) &longs;patia, ip&longs;o motus initio de&longs;cripta, &longs;unt in duplicata ratione <lb/>temporum. </s> <s>Ponatur igitur velocitas corporis <emph type="italics"/>p<emph.end type="italics"/> e&longs;&longs;e ad velocita­<lb/>tem corporis <emph type="italics"/>P<emph.end type="italics"/> in &longs;ubduplicata ratione di&longs;tantiæ <emph type="italics"/>sp<emph.end type="italics"/> ad di&longs;tantiam <lb/><emph type="italics"/>CP,<emph.end type="italics"/> eo ut temporibus quæ &longs;int in eadem &longs;ubduplicata ratione de­<lb/>&longs;cribantur arcus <emph type="italics"/>pq, PQ,<emph.end type="italics"/> qui &longs;unt in ratione integra: Et corpora <lb/><emph type="italics"/>P, p<emph.end type="italics"/> viribus æqualibus &longs;emper attracta de&longs;cribent circum centra <lb/>quie&longs;centia <emph type="italics"/>C<emph.end type="italics"/> & <emph type="italics"/>s<emph.end type="italics"/> Figuras &longs;imiles <emph type="italics"/>PQV, pqv,<emph.end type="italics"/> quarum po&longs;terior <lb/><emph type="italics"/>pqv<emph.end type="italics"/> &longs;imilis e&longs;t & æqualis Figuræ quam corpus <emph type="italics"/>P<emph.end type="italics"/> circum corpus <lb/>mobile <emph type="italics"/>S<emph.end type="italics"/> de&longs;cribit. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Ponamus jam quod commune gravitatis centrum, una <lb/>cum &longs;patio in quo corpora moventur inter &longs;e, progreditur unifor­<lb/>miter in directum; &, per Legum Corollarium &longs;extum, motus <lb/>omnes in hoc &longs;patio peragentur ut prius, adeoque corpora de&longs;cri- <pb pagenum="150"/> <arrow.to.target n="note126"></arrow.to.target><lb/>bent circum &longs;e mutuo Figuras ea&longs;dem ac prius, & propterea Figuræ <lb/><emph type="italics"/>pqv<emph.end type="italics"/> &longs;imiles & æquales. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note126"></margin.target>DE MOTU <lb/>CORPORUM</s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc corpora duo Viribus di&longs;tantiæ &longs;uæ proportionali­<lb/>bus &longs;e mutuo trahentia, de&longs;cribunt (per Prop. </s> <s>X,) & circum com­<lb/>mune gravitatis centrum, & circum &longs;e mutuo, Ellip&longs;es concentri­<lb/>cas: & vice ver&longs;a, &longs;i tales Figuræ de&longs;cribuntur, &longs;unt Vires di&longs;tan­<lb/>tiæ proportionales. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et corpora duo Viribus quadrato di&longs;tantiæ &longs;uæ recipro­<lb/>ce proportionalibus de&longs;cribunt (per Prop. </s> <s>XI, XII, XIII) & circum <lb/>commune gravitatis centrum, & circum &longs;e mutuo, Sectiones conicas <lb/>umbilicum habentes in centro circum quod Figuræ de&longs;cribuntur. </s> <s>Et <lb/>vice ver&longs;a, &longs;i tales Figuræ de&longs;cribuntur, Vires centripetæ &longs;unt qua­<lb/>drato di&longs;tantiæ reciproce proportionales. </s> </p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Corpora duo quævis cirum gravitatis centrum com­<lb/>mune gyrantia, radiis & ad centrum illud & ad &longs;e mutuo ductis, <lb/>de&longs;cribunt areas temporibus proportionales. </s> </p> <p type="main"> <s><emph type="center"/>PROPOSITIO LIX. THEOREMA XXII.<emph.end type="center"/></s> </p> <p type="main"> <s><emph type="italics"/>Corporum duorum<emph.end type="italics"/> S <emph type="italics"/>&<emph.end type="italics"/> P <emph type="italics&